MATHEMATICS Approximations - Rounding • The aim of this powerpoint is to teach you techniques for approximating numbers. EITHER • Take notes as you.
Download ReportTranscript MATHEMATICS Approximations - Rounding • The aim of this powerpoint is to teach you techniques for approximating numbers. EITHER • Take notes as you.
Slide 1
MATHEMATICS
Approximations - Rounding
Slide 2
• The aim of this powerpoint is to teach you techniques for
approximating numbers.
EITHER
• Take notes as you go along, include some examples and
write down any questions and your answers (which you
can mark as you go along)
OR
• At the end of the powerpoint, printout the notes called
Calc2a
Slide 3
Approximations
• To approximate means to give a rough estimate, not an
exact value.
• The symbol ‘≈’ means approximately equal to.
• It is useful to be able to approximate (or ‘round’ values up
or down) as one method for checking that answers to
calculations are sensible as well as make values easier to
work with.
Slide 4
Rounding – Method 1
• Look at the degree of rounding required.
• Imagine the two rounded values that lie either side of the
actual value quoted on a number line.
• Which of the two values is the actual value closest to?
• If the actual value is exactly half-way between them, then
round UP.
Slide 5
Example 1
• Round 76 to the nearest ten.
• The two ‘tens’ values either side of 76 are 70 and 80.
• Which of the two values is the actual value closest to?
75
70
• ANS: 76 is closest to 80.
76
80
Slide 6
Example 2
• Round 1949 to the nearest hundred.
• The two ‘hundred’ values either side are 1900 and 2000.
• Which of the two values is the actual value closest to?
1950
1900
1949
• ANS: 1949 is closest to 1900.
2000
Slide 7
Example 3
• Round 21500 to the nearest thousand.
• The two ‘thousand’ values either side are 21000 and
22000.
• Which of the two values is the actual value closest to?
21500
21000
21500
22000
• ANS: 21500 is exactly halfway so round UP to 22000.
Slide 8
Rounding – Method 2
• Find the digits up to and including the ‘rounding’ column.
• If the number on the right of these digits is 5 or more, add
1 on to these digits.
• If the number on the right of these digits is 4 or less, leave
the digits as they are.
• All the other digits on the right of these now become zeros
(or ignored if they are zeros at the end after the decimal
point).
Slide 9
Example 1
• Round 120.63 to the nearest whole number. (i.e. unit)
• The digits up to and including the units column are 120.
• The number on the right is 6 (which is 5 or more) so add 1
on to the ‘120’ to get ‘121’
• The digits on the right now become zeros (and can be
ignored as they are after the decimal point).
• ANS: 120.63 rounds to 121 (to the nearest whole number)
Slide 10
Example 2
• Round 1749.7 to the nearest hundred.
• The digits up to and including the hundreds column are
17.
• The number on the right is 4 (which is LESS than 5) so 17
stays as it is…
• The digits on the right now become zeros (and any zeros
after the decimal point can be ignored).
• ANS: 1749.7 rounds to 1700 (to the nearest hundred)
Slide 11
Example 3
• Round 5.9718 to the nearest tenth
• The digits up to and including the tenths column are 5(.)9.
• The number on the right is 7 (which is 5 or more) so add 1
on to the ‘5(.)9’ to get ‘6(.)0’
• The digits on the right now become zeros.
• ANS: 5.9718 rounds to 6.0 (to the nearest tenth)
• You could quote the answer as 6, but it is good practice to include the
tenths value (even though it is 0) as you were asked to round the
value to the nearest tenth.
Slide 12
Practice Questions
Work out the answers to each of these questions before
moving on to the next slides to check them.
Q1. Round each of these values to the nearest 10.
a) 753 ≈
b) 1378 ≈
c) 5702 ≈
Q2. Round each of these values to the nearest 100.
a) 1838 ≈
b) 23750 ≈
c) 12792 ≈
Q3. Round each of these values to the nearest 1000.
a) 48753 ≈
b) 32178 ≈
c) 129609 ≈
Slide 13
Practice Question 1
Round each of these values to the nearest 10.
a) 753 ≈
Between 750 – 760
753 is closest to 750
b) 1378 ≈
Is it 1370 or 1380?
1378 is closest to 1380
c) 5702 ≈
‘570’ are the digits up to and including the tens
2 is on the right but is less then 5
570 stays as it is but the 2 becomes a zero
5702 is closest to 5700
NB. Rounding to 10 means the answer must end in ‘0’
Slide 14
Practice Question 2
Round each of these values to the nearest 100.
a) 1838 ≈ Between 1800–1900
1838 is closest to 1800
b) 23750 ≈ Is it 23700 or 23800?
23750 rounds up to 23800
c) 12792 ≈ ‘127’ are the digits up to and including the hundreds
9 on the right is more than 5 so…
…add 1 on to 127
…then the ’92’ become zeros
12792 is closest to 12800
NB. Rounding to 100 means the answer must end in ‘00’
Slide 15
Practice Question 3
Round each of these values to the nearest 1000.
a) 48753 ≈ Between 48000–49000 48753 is closest to 49000
b) 32178 ≈ Is it 32000 or 33000?
32178 rounds to 32000
c) 129609 ≈ ‘129’ are the digits up to and including the thousands
6 on the right is more than 5 so…
…add 1 on to 129 to get 130
…then the ’609’ become zeros
129609 is closest to 130000
NB. Rounding to 1000 means the answer must end in ‘000’
Slide 16
More Practice Questions
Work out the answers to each of these questions before
moving on to the next slides to check them.
Q4. Round each of these values to the nearest unit.
a) 8.63 ≈
b) 12.15 ≈
c) 0.499 ≈
Q5. Round each of these values to the nearest tenth.
a) 0.8501 ≈
b) 10.449 ≈
c) 3.083 ≈
Slide 17
Practice Question 4
Round each of these values to the nearest unit.
a) 8.63 ≈
Between 8 – 9
8.63 is closest to 9
b) 12.15 ≈
Is it 12 or 13?
12.15 is closest to 12
c) 0.499 ≈
‘0’ is the digit up to and including the units
4 is on the right but is less then 5
0 stays as it is but the ‘499’ become zeros (and as they all
come after the decimal point they can be ignored)
0.499 is closest to 0.000 0
Slide 18
Practice Question 5
Round each of these values to the nearest tenth.
a) 0.8501 ≈ Between 0.8 – 0.9
0.8501 is closest to 0.9
b) 10.449 ≈ Is it 10.4 or 10.5?
10.449 is closest to 10.4
c) 3.083 ≈
‘3.0’ are the digits up to and including the tenths
8 on the right is more then 5 so…
…add 1 on to the (3.)0 to get (3.)1
…then the ‘83’ become zeros and can be ignored
3.083 is closest to 3.1
Slide 19
What next? (Page 1)
• If you haven’t made any notes or copied any examples,
questions and answers out during this presentation, print
out the notes called Calc2a. Read through them and
make sure you answer any questions.
• Work through pages 5 to 9 of the MyMaths lesson called
Estimating Amounts found at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingAmounts&taskID=1219
• Work through the MyMaths lesson (and then the online
homework) called Rounding to 10, 100 found at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=accuracy/rounding10&taskID=1003
http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=accuracy/rounding10OH&taskID=1003
Continued on the next page…
Slide 20
What next? (Page 2)
• Please work through pages 1 to 4 (making notes and copying
examples as well as questions and answers) of the MyMaths lesson
(and then the online homework) called Estimating Amounts at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingAmounts&taskID=1219
http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=estimating/estimatingAmountsOH&taskID=1219
• To recognise when you might need to round an answer up or down
according to the situation, please work through (making notes and
copying examples as well as questions and answers) the MyMaths
lesson (and then the online homework) called Solving Problems by
Rounding found at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingGold&taskID=1373
http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=estimating/estimatingGoldOH&taskID=1373
• Save and complete the worksheets called Round10-100.xlsx and
Round-S1.xlsx
• Now move on to the Calc2b-dp-sf powerpoint
MATHEMATICS
Approximations - Rounding
Slide 2
• The aim of this powerpoint is to teach you techniques for
approximating numbers.
EITHER
• Take notes as you go along, include some examples and
write down any questions and your answers (which you
can mark as you go along)
OR
• At the end of the powerpoint, printout the notes called
Calc2a
Slide 3
Approximations
• To approximate means to give a rough estimate, not an
exact value.
• The symbol ‘≈’ means approximately equal to.
• It is useful to be able to approximate (or ‘round’ values up
or down) as one method for checking that answers to
calculations are sensible as well as make values easier to
work with.
Slide 4
Rounding – Method 1
• Look at the degree of rounding required.
• Imagine the two rounded values that lie either side of the
actual value quoted on a number line.
• Which of the two values is the actual value closest to?
• If the actual value is exactly half-way between them, then
round UP.
Slide 5
Example 1
• Round 76 to the nearest ten.
• The two ‘tens’ values either side of 76 are 70 and 80.
• Which of the two values is the actual value closest to?
75
70
• ANS: 76 is closest to 80.
76
80
Slide 6
Example 2
• Round 1949 to the nearest hundred.
• The two ‘hundred’ values either side are 1900 and 2000.
• Which of the two values is the actual value closest to?
1950
1900
1949
• ANS: 1949 is closest to 1900.
2000
Slide 7
Example 3
• Round 21500 to the nearest thousand.
• The two ‘thousand’ values either side are 21000 and
22000.
• Which of the two values is the actual value closest to?
21500
21000
21500
22000
• ANS: 21500 is exactly halfway so round UP to 22000.
Slide 8
Rounding – Method 2
• Find the digits up to and including the ‘rounding’ column.
• If the number on the right of these digits is 5 or more, add
1 on to these digits.
• If the number on the right of these digits is 4 or less, leave
the digits as they are.
• All the other digits on the right of these now become zeros
(or ignored if they are zeros at the end after the decimal
point).
Slide 9
Example 1
• Round 120.63 to the nearest whole number. (i.e. unit)
• The digits up to and including the units column are 120.
• The number on the right is 6 (which is 5 or more) so add 1
on to the ‘120’ to get ‘121’
• The digits on the right now become zeros (and can be
ignored as they are after the decimal point).
• ANS: 120.63 rounds to 121 (to the nearest whole number)
Slide 10
Example 2
• Round 1749.7 to the nearest hundred.
• The digits up to and including the hundreds column are
17.
• The number on the right is 4 (which is LESS than 5) so 17
stays as it is…
• The digits on the right now become zeros (and any zeros
after the decimal point can be ignored).
• ANS: 1749.7 rounds to 1700 (to the nearest hundred)
Slide 11
Example 3
• Round 5.9718 to the nearest tenth
• The digits up to and including the tenths column are 5(.)9.
• The number on the right is 7 (which is 5 or more) so add 1
on to the ‘5(.)9’ to get ‘6(.)0’
• The digits on the right now become zeros.
• ANS: 5.9718 rounds to 6.0 (to the nearest tenth)
• You could quote the answer as 6, but it is good practice to include the
tenths value (even though it is 0) as you were asked to round the
value to the nearest tenth.
Slide 12
Practice Questions
Work out the answers to each of these questions before
moving on to the next slides to check them.
Q1. Round each of these values to the nearest 10.
a) 753 ≈
b) 1378 ≈
c) 5702 ≈
Q2. Round each of these values to the nearest 100.
a) 1838 ≈
b) 23750 ≈
c) 12792 ≈
Q3. Round each of these values to the nearest 1000.
a) 48753 ≈
b) 32178 ≈
c) 129609 ≈
Slide 13
Practice Question 1
Round each of these values to the nearest 10.
a) 753 ≈
Between 750 – 760
753 is closest to 750
b) 1378 ≈
Is it 1370 or 1380?
1378 is closest to 1380
c) 5702 ≈
‘570’ are the digits up to and including the tens
2 is on the right but is less then 5
570 stays as it is but the 2 becomes a zero
5702 is closest to 5700
NB. Rounding to 10 means the answer must end in ‘0’
Slide 14
Practice Question 2
Round each of these values to the nearest 100.
a) 1838 ≈ Between 1800–1900
1838 is closest to 1800
b) 23750 ≈ Is it 23700 or 23800?
23750 rounds up to 23800
c) 12792 ≈ ‘127’ are the digits up to and including the hundreds
9 on the right is more than 5 so…
…add 1 on to 127
…then the ’92’ become zeros
12792 is closest to 12800
NB. Rounding to 100 means the answer must end in ‘00’
Slide 15
Practice Question 3
Round each of these values to the nearest 1000.
a) 48753 ≈ Between 48000–49000 48753 is closest to 49000
b) 32178 ≈ Is it 32000 or 33000?
32178 rounds to 32000
c) 129609 ≈ ‘129’ are the digits up to and including the thousands
6 on the right is more than 5 so…
…add 1 on to 129 to get 130
…then the ’609’ become zeros
129609 is closest to 130000
NB. Rounding to 1000 means the answer must end in ‘000’
Slide 16
More Practice Questions
Work out the answers to each of these questions before
moving on to the next slides to check them.
Q4. Round each of these values to the nearest unit.
a) 8.63 ≈
b) 12.15 ≈
c) 0.499 ≈
Q5. Round each of these values to the nearest tenth.
a) 0.8501 ≈
b) 10.449 ≈
c) 3.083 ≈
Slide 17
Practice Question 4
Round each of these values to the nearest unit.
a) 8.63 ≈
Between 8 – 9
8.63 is closest to 9
b) 12.15 ≈
Is it 12 or 13?
12.15 is closest to 12
c) 0.499 ≈
‘0’ is the digit up to and including the units
4 is on the right but is less then 5
0 stays as it is but the ‘499’ become zeros (and as they all
come after the decimal point they can be ignored)
0.499 is closest to 0.000 0
Slide 18
Practice Question 5
Round each of these values to the nearest tenth.
a) 0.8501 ≈ Between 0.8 – 0.9
0.8501 is closest to 0.9
b) 10.449 ≈ Is it 10.4 or 10.5?
10.449 is closest to 10.4
c) 3.083 ≈
‘3.0’ are the digits up to and including the tenths
8 on the right is more then 5 so…
…add 1 on to the (3.)0 to get (3.)1
…then the ‘83’ become zeros and can be ignored
3.083 is closest to 3.1
Slide 19
What next? (Page 1)
• If you haven’t made any notes or copied any examples,
questions and answers out during this presentation, print
out the notes called Calc2a. Read through them and
make sure you answer any questions.
• Work through pages 5 to 9 of the MyMaths lesson called
Estimating Amounts found at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingAmounts&taskID=1219
• Work through the MyMaths lesson (and then the online
homework) called Rounding to 10, 100 found at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=accuracy/rounding10&taskID=1003
http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=accuracy/rounding10OH&taskID=1003
Continued on the next page…
Slide 20
What next? (Page 2)
• Please work through pages 1 to 4 (making notes and copying
examples as well as questions and answers) of the MyMaths lesson
(and then the online homework) called Estimating Amounts at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingAmounts&taskID=1219
http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=estimating/estimatingAmountsOH&taskID=1219
• To recognise when you might need to round an answer up or down
according to the situation, please work through (making notes and
copying examples as well as questions and answers) the MyMaths
lesson (and then the online homework) called Solving Problems by
Rounding found at:
http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=estimating/estimatingGold&taskID=1373
http://www.mymaths.co.uk/tasks/library/loadTask.asp?title=estimating/estimatingGoldOH&taskID=1373
• Save and complete the worksheets called Round10-100.xlsx and
Round-S1.xlsx
• Now move on to the Calc2b-dp-sf powerpoint