Standards for Measurement ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Scientific Notation Measurement and Uncertainty Significant Figures Significant Figures in Calculations Metric System Problem-Solving Measuring Mass and Volume Measurement of Temperature Density Qualitative Observation – a less.
Download ReportTranscript Standards for Measurement ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Scientific Notation Measurement and Uncertainty Significant Figures Significant Figures in Calculations Metric System Problem-Solving Measuring Mass and Volume Measurement of Temperature Density Qualitative Observation – a less.
Slide 1
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 2
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 3
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 4
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 5
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 6
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 7
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 8
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 9
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 10
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 11
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 12
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 13
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 14
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 15
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 16
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 17
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 18
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 19
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 20
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 21
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 22
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 23
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 24
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 25
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 2
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 3
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 4
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 5
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 6
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 7
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 8
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 9
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 10
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 11
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 12
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 13
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 14
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 15
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 16
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 17
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 18
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 19
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 20
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 21
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 22
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 23
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 24
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832
Slide 25
Standards for Measurement
◦
◦
◦
◦
◦
◦
◦
◦
◦
Scientific Notation
Measurement and Uncertainty
Significant Figures
Significant Figures in Calculations
Metric System
Problem-Solving
Measuring Mass and Volume
Measurement of Temperature
Density
Qualitative Observation – a less specific
observation describing the quality of
something.
Quantitative Observations – a measurement
of a specific quantity of something.
Question: What are some qualitative and
quantitative observations of our classroom?
Science often must use very large and very
small numbers.
◦ The average distance from the Sun to the Earth is
150,000,000 kilometers.
◦ The diameter of an atom is about 0.0000000001m.
Writing these out is not always easy.
Rules for converting a large or small number into
scientific notation:
1. Move the decimal point in the original number
until it is located after the first non-zero digit.
2. Multiply this by the number ten raised to an
exponent (called the “power”) equal to the
number of decimal places that was required in
step one.
3. The sign on the exponent is positive if you
moved the decimal point to the left and
negative if you moved it to the right.
Write each of the numbers into scientific
notation.
◦ Ex) 150,000,000 km
◦ Ex) 0.0000000001 m
To convert a number from scientific notation
back to a decimal number:
◦ Move the decimal point back to the right for a
positive exponent – will need to add zero’s at the
end of the number.
◦ Move the decimal point back to the left for a
negative exponent – will need to add zero’s before
the number.
Write the following scientific notation
numbers as a decimal equivalent.
◦ Ex) 4.5 x 105
◦ Ex) 6.3 x 10-6
On your calculator – use either
the EE or EXP key for entering
scientific notation.
Ex) 4.5 x 105 is entered as:
4
5
EE
5
Each colored box represents a the key on
the calculator to push.
Multiply or dividing two (or more) scientific
notation numbers.
Ex) (4.5 x 105) * (3.6 x 107) =
Ex) (8.1 x 1012) (4.9 x 10-5) =
Numbers can be either Exact or Measured.
An Exact number is one that is either a
counted number or a number that is part of
an established relationship.
◦ Ex) 21 chairs in a room
◦ Ex) 12 inches in a foot
◦ Ex) 100 centimeters in a meter
A Measured number is one that is obtained
using any measuring instrument like a ruler,
graduated cylinder, or thermometer.
Describe as either an Exact or Measured
number.
◦
◦
◦
◦
◦
◦
Ex)
Ex)
Ex)
Ex)
Ex)
Ex)
12 apples in a bag
25.0mL of water in a 25-mL graduated cylinder
16 ounces in a pound
The mass of a U.S. quarter is 5.67 grams
1000 grams in one kilogram
The indoor temperature is 74.8oF
All measured numbers are made up of certain
and one uncertain digit.
◦ Certain digits – all will agree. That is: we all agree
that is at least 45 but not 46 degrees Fahrenheit.
◦ Uncertain digit – not all will agree! It is each
person’s best “guess” between two markings.
◦ Can only have ONE uncertain digit!
Each marking is
0.2mL.
We would (hopefully)
agree that it is
between 6.6 and
6.8mL.
Must provide a best
“guess” to nearest
0.05mL.
What if it seems to
be “exactly” on a
mark???
Many of the measuring
devices used today are
electronic like a scale
or a digital
thermometer.
The last digit is always
the uncertain digit and
should always be
included even when it
is a zero.
All of the certain digits plus the one uncertain
digit are called significant digits.
◦ Thus, our volume from earlier – 6.65mL would have
three significant figures.
◦ Our temperature, 36.60oC, would have four
significant figures.
◦ And our scale reading, 45.22g, would also have
four significant figures.
Note: Exact numbers have an infinite
significance because they are exact.
Rules for counting significant figures are:
1. All non-zero digits are significant.
2. Zeros are significant when:
a. Between non-zero digits like in 205.
b. At the end of a number WITH a decimal point like
in 0.250 or 12.00 or 850.
3.
Zeros are not significant when:
a. Before the first non-zero digit like in 0.0025.
b. After non-zero digits without a decimal point at
the end like 95,000 or 45,000,000
How many significant digits does each
measured number have?
◦
◦
◦
◦
◦
◦
◦
0.000305
125.0
1.0
0.023012
0.0001
100
45.050
A calculation involving measured numbers
will almost always need to be rounded to a
certain place.
Rules:
1. When the first digit to be rounded after
those that will be retained is a 4 or less,
then drop all of those extra digits.
2. When the first digit to be rounded is 5+, the
drop all of those digits and adjust the last
digit retained up by one.
3.
If it is EXACTLY 5, then round the final
retained digit to an even number.
Ex) 45.62 rounded to two digits is ______
Ex) 89.145 rounded to three digits is ______
Ex) 9.15 rounded to two digits is _____
Ex) 84.5 rounded to two digits is _____
Rules for Multiplication and Division
◦ The answer must be rounded to the same number
of significant figures as the number with the least
amount of significant figures.
◦ Reason? The result of a calculation cannot be more
precise than the least precise measurement.
◦ Ex) 45.6 x 0.023 = 10.488 (calculator answer)
◦ Ex) 8.02 45.23 = 0.1773159… (calculator answer)
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 89.00 x 3.25 = 289.25
◦ 6.516 482 = 0.01351867…
◦ 1.25 x 80. = 100
◦ 35.7 x 89.52 0.028 = 114,138
Rules for Addition and Subtraction
◦ The answer cannot contain any more decimal places
as the value with the fewest decimal places.
◦ Can get more or less number of significant figures
than either of the two numbers you are adding or
subtracting.
◦ Example: Let’s say you had a briefcase containing
roughly $1.1 million dollars. You open the
briefcase and pull out a $5 bill. How much is in the
briefcase?
Round each to the correct number of
significant figures. Answers are “calculator”
answers.
◦ 45.2 + 3.5 = 48.7
◦ 73.21 – 71.58 = 1.63
◦ 151 + 3.28 = 154.28
◦ 173.18 – 167.98 = 5.2
For a mixed calculation, you must follow each
of two rules individually and follow order of
operations.
Ex) (45.23 – 3.15) 4825 = 0.87212435…
◦ You must do the subtraction first!
Ex) (11.35 + 2.55) x 0.12 = 1.668
◦ You must do the addition first!
Ex) (23.33 – 22.93) x 4.58 = 1.832