• • • • •

1.1 Fractions: Defining Terms

Fraction: part of a whole - example Numerator: number on top Denominator: number on bottom 3 5 Proper Fraction: numerator is less than the denominator Improper Fraction: numerator is equal to or greater than the denominator

1.1 Problem Solving with Fractions •  Mixed Number Consists of a whole number and a proper fraction – example 1 2 3

1.2 Changing the Form of a Fraction • Converting a mixed number to an improper fraction: 3 3 8  3  8 8  3  27 8 • Converting an improper fraction to a mixed number: 35 3 9 9 35 Divide 9 into 35: 35 3 8 9  9 27 8

1.2 Changing the Form of a Fraction • • Multiplying or dividing the numerator (top) and the denominator (bottom) of a fraction by the same number does not change the value of a fraction.

Writing a fraction in lowest terms: 1. Factor the top and bottom completely 2. Divide the top and bottom by the greatest common factor

1.2 Changing the Form of a Fraction • A number can be divided evenly by: 2 – if the last digit is 0, 2, 4, 6, 8 3 – if the sum of the digits is divisible by 3 4 – if the last two digits form a number that is divisible by 4 5 – if the last digit is 0 or 5 6 – if the number is divisible by 2 and 3

1.2 Changing the Form of a Fraction • A number can be divided evenly by: 7 – double the last digit and subtract it from a number formed by the other digits. This number must be zero or divisible by 7 8 – if the last three digits form a number that is divisible by 8 9 – if the sum of the digits is divisible by 9 10 – if the last digit is 0

1.2 Changing the Form of a Fraction • A prime number can only be divided evenly by itself and the number 1 • Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, etc.

• Factor trees can be used to factor a number to its prime factorization

1.2 Changing the Form of a Fraction – Factor Trees 18 3 6 2 3

1.3 Adding and Subtracting Fractions • Adding fractions with the same denominator:

a c



a



c b



b b

• Subtracting fractions with the same denominator:

a c



a



c b



b b

• • 1.3 Adding and Subtracting Fractions To add or subtract fractions with different denominators - get a common denominator.

Using the least common denominator: 1. Factor both denominators completely 2. Multiply the largest number of repeats of each prime factor together to get the LCD 3. Multiply the top and bottom of each fraction by the number that produces the LCD in the denominator

1.3 Adding and Subtracting Fractions – no common factors in denominator • Adding fractions with different denominators:

a c



a



d



b



c b



d b



d

• Subtracting fractions with different denominators:

a c



a



d



b



c b



d b



d

1.3 Adding and Subtracting Fractions • Try these: 1 9  5 9  ?

5 7  2 21  ?

5  9 1 4  ?

1.4 Multiplying and Dividing Fractions • Multiplying fractions:

a c



b



d a b

 

c d

• Dividing fractions:

a c



a b



d b



d c



a



d b



c

1.4 Multiplying and Dividing Fractions • Complex Fractions: The numerator, denominator or both are some sort of fraction (proper, improper, or mixed) • Example: 3 1 3 5  8 5  3 10  8 5  10 3  16 3  5 1 3 10

1.4 Multiplying and Dividing Fractions • Try these: 12 16 (simplify) 7 9  3 14  ?

9 10  3 5  ?

2.1 Reading, Writing, and Rounding Decimals • Place value: the position of the number in relation to the decimal place 12345 .

67890 • What power of 10 does the 4 represent?

• What does the 8 represent?

• What about the 1?

2.1 Reading, Writing, and Rounding Decimals • Translating a decimal to words: 12 .

32 • In words: Twelve and thirty-two hundredths • Translate the following: 37 .

281

2.1 Reading, Writing, and Rounding Decimals • Rounding a decimal: – Look at the digit to the right of the place to which you are rounding – If the digit is less than 5 all the digits to the right of the place you are rounding become zero – If the digit is 5 or greater, the place you are rounding to is increased by 1 and all the digits to the right of the place you are rounding become zero – Drop zeros to the right of the decimal place

2.1 Reading, Writing, and Rounding Decimals • Round 5.455 to the nearest tenth: • 5 is next to the tenths place so increase 4 by 1 to get 5 .

500 • Drop the zeros: 5 .

5

2.2 Adding and Subtracting Decimals • Write each number so that the decimal points are in a vertical line • Add the numbers as if there were no decimal points.

• Place the decimal point in the answer in line with the other decimal points 0 .

125  1 .

25  1 .

375

2.2 Adding and Subtracting Decimals • Examples: .

0312  10 .

1 

.

6  .

56  11 .

2912 10 .

125  1 .

925  8 .

2

2.3 Multiplying and Dividing Decimals • To multiply decimals: – Multiply the numbers as if there were no decimal points.

– Count the number of decimal places in each number and add them together – Put that many decimal places in the answer 0 .

125  1 .

2 125  12 250 1250 1500 Answer: 0.15

2.3 Multiplying and Dividing Decimals • To divide decimals: – Write the numbers in long division format – Move the decimal in the divisor to the right until you have a whole number – Move the decimal in the dividend to the right the same number of places – Divide as if the decimal points were not there – Place the decimal in the answer just above the decimal in the dividend

2.3 Multiplying and Dividing Decimals • Example: 3 0 .

8  8 3 .

75 30 .

00 24 .

6 .

0 5 .

6 .

40 .

40

2.4 Converting Fractions and Decimals • Converting decimals to fractions: 0 .

125  125 1000  25 200  5 40  1 8 • Converting fractions to decimals: 3 8  8

.

375 3

.

000 2

.

4

.

60

.

56

.

040

.

040

2.5 Converting Decimals and Percents

• Write a decimal as a percent by moving the decimal point 2 places to the right and attaching a percent sign: • Example: 0 .

382  38 .

2 %

2.5 Converting Decimals and Percents

• Write a percent as a decimal by moving the decimal point 2 places to the left and removing the percent sign: • Example: 3 .

41  341 %

2.6 Converting Fractions and Percents • Write a fraction as a percent by converting the fraction to a decimal and then converting the decimal to a percent: • Example: 3 8  8

.

375 3

.

000 2

.

4 0 .

375  37 .

5 %

.

60

.

56

.

040

.

040

2.6 Converting Fractions and Percents • Write a percent as a fraction by first changing the percent to a decimal then changing the decimal to the fraction and reduce: • Example: 45 %  0 .

45  45 100  9 20  5  5  9 20

• Supplement: Chapter 1 1.1 Scientific Notation 1.

Writing a number in scientific notation: Move the decimal point to the right of the first non zero digit.

2.

3.

4.

Count the places you moved the decimal point.

The number of places that you counted in step 2 is the exponent (without the sign) If your original number (without the sign) was smaller than 1, the exponent is negative. If it was bigger than 1, the exponent is positive

Supplement: Chapter 1 1.1 Scientific Notation • Converting to scientific notation (examples): 6200000  6 .

2  10 ?

.

00012  1 .

2  10 ?

• Converting back – just undo the process: 6 .

203  10 23  620 , 300 , 000 , 000 , 000 , 000 , 000 , 000 1 .

86  10 5  186 , 000

Supplement: Chapter 1 1.1 Scientific Notation • Multiplication with scientific notation:  4  10 5   5  10  8    4  5    10 5  10  8   20  10  3  2  10 1  10  3  2  10  2 • Division with scientific notation:  4  5   10 12 10 4    4 5  10 12 10 4  .

8  10 12  4  .

8  10 8  8  10 7

Supplement: Chapter 1 1.2 Uncertainty in Measurements • Accuracy: correctness of a measurement Example: The statue of liberty is 12.135 inches tall – the measurement is very precise but inaccurate • Precision: degree of correctness Examples: 3.2 cm is more precise than 3 cm but less precise than 3.24 cm

Supplement: Chapter 1 1.2 Uncertainty in Measurements • Absolute error Measurement Absolute error 23 mg 23.2 mg 2.035 mg 0.5 mg 0.05 mg 0.0005 mg

Supplement: Chapter 1 1.2 Uncertainty in Measurements • Lower limit = measurement – absolute error • Upper limit = measurement + absolute error • Relative error: Relative Error 

Absolute Error Measuremen t

 100 %

Supplement: Chapter 1 1.3 Estimation • “  ” means “approximately equal to” • Interval estimate: look at the first digit to get the interval Example: 347 + 231 + 583 Low Estimate: 300 + 200 + 500 = 1000 High Estimate: 400 + 300 + 600 = 1300 The actual sum is between 1000 and 1300

Supplement: Chapter 1 1.3 Estimation • Rounding was covered in section 2.1 of the text and can be used to find an estimate • Example – find an estimate by rounding to the tens place: 347 + 231 + 583 347 + 231 + 583  350 + 230 + 580 = 1160 An estimation is 1160