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Place value and estimating.pptx
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Place value and estimating
Number
Lesson 2 of 7
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Last Lesson and Before
Number
Lesson 2 of 7
Answer each one, then check.
1
Round 3.846 to 1 decimal place.
2
Work out 4500 ÷ 100.
3
Work out 0.3 × 0.2.
4
Work out 7 ÷ 0.1.
5
Last lesson: list the meals from 2 starters (A, B) and 2 mains (X, Y).
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Last Lesson and Before - Answers
Number
Lesson 2 of 7
1
Round 3.846 to 1 decimal place.
3.8
2
Work out 4500 ÷ 100.
45
3
Work out 0.3 × 0.2.
0.06
4
Work out 7 ÷ 0.1.
70 - dividing by 0.1 is the same as multiplying by 10.
5
Last lesson: list the meals from 2 starters (A, B) and 2 mains (X, Y).
AX, AY, BX, BY
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Learning Objectives
Number
Lesson 2 of 7
1
Use a known calculation to work out related calculations.
2
Round numbers to decimal places and significant figures.
3
Estimate the answer to a calculation by rounding to 1 significant figure.
4
Estimate square roots of numbers that are not square numbers.
5
Decide whether an estimate is too big or too small.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Place Value
Number
Lesson 2 of 7
The value of a digit depends on its position.
Multiplying by 10
Every digit moves one place to the left: 3.7 × 10 = 37.
Dividing by 10
Every digit moves one place to the right: 3.7 ÷ 10 = 0.37.
Multiplying by 0.1
The same as dividing by 10: 45 × 0.1 = 4.5.
Dividing by 0.1
The same as multiplying by 10: 45 ÷ 0.1 = 450.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
One Fact, Many Answers
Number
Lesson 2 of 7
Start from 38 × 142 = 5396 and use place value.
Calculation
Answer
Why
38 × 142
5396
The fact we are given
3.8 × 142
539.6
One number is 10 times smaller, so the answer is 10 times smaller
3.8 × 1.42
5.396
10 times smaller and 100 times smaller: 1000 times smaller
0.38 × 14.2
5.396
100 times smaller and 10 times smaller: 1000 times smaller
5396 ÷ 142
38
Division is the inverse of multiplication
5396 ÷ 14.2
380
Dividing by a number 10 times smaller gives an answer 10 times bigger
53.96 ÷ 3.8
14.2
5396 ÷ 38 = 142, and both numbers are 100 times smaller
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Using a Known Fact
Number
Lesson 2 of 7
Given that 23 × 47 = 1081, work out (a) 2.3 × 0.47 and (b) 108.1 ÷ 4.7.
1
(a) Compare with the fact: 2.3 is 23 ÷ 10 and 0.47 is 47 ÷ 100
÷ 10 and ÷ 100 make ÷ 1000
2
So divide the answer by 1000
1081 ÷ 1000 = 1.081
3
(b) From the fact, 1081 ÷ 47 = 23
division undoes multiplication
4
108.1 is 1081 ÷ 10 and 4.7 is 47 ÷ 10
both divided by 10
5
Dividing both numbers by 10 does not change the answer
108.1 ÷ 4.7 = 23
ANSWER
(a) 1.081 (b) 23
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
PART ONE
Rounding
Decimal places and significant figures.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Significant Figures
Number
Lesson 2 of 7
The first significant figure is the first digit that is not zero.
Counting
4 is the first significant figure of 45 678 and of 0.004 12.
Zeros in the middle count
7.0496 has 5 significant figures; the 0 counts.
Zeros at the start do not
0.003 082 has 4 significant figures: 3, 0, 8, 2.
Keep the place value
45 678 to 2 significant figures is 46 000, not 46.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Rounding to Significant Figures
Number
Lesson 2 of 7
Number
1 s.f.
2 s.f.
3 s.f.
45 678
50 000
46 000
45 700
0.003 082
0.003
0.0031
0.003 08
7.0496
7
7.0
7.05
299 512
300 000
300 000
300 000
0.098 76
0.1
0.099
0.0988
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
PART TWO
Estimating
A rough answer that tells you if the real one is sensible.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
How to Estimate
Number
Lesson 2 of 7
To estimate, round every number to 1 significant figure, then work out the calculation.
Round first
Round each number to 1 s.f. before doing anything else.
Never round to 0
0.048 rounds to 0.05, not 0.
Dividing by a decimal
150 ÷ 0.5 = 300: dividing by 0.5 is the same as doubling.
Show it
Write the rounded calculation down: the method marks are for the rounded numbers, not just the answer. Use ≈ for "is approximately equal to".
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Estimating a Calculation
Number
Lesson 2 of 7
Work out an estimate for (48.7 × 3.14)/0.52
1
Round each number to 1 significant figure
48.7 ≈ 50, 3.14 ≈ 3, 0.52 ≈ 0.5
2
Write the rounded calculation
(50 × 3)/0.5
3
Work out the top
50 × 3 = 150
4
Divide by 0.5 (the same as doubling)
150 ÷ 0.5 = 300
5
Compare with the calculator
The exact answer is 294.07…, close to 300
ANSWER
≈ 300
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Too Big or Too Small?
Number
Lesson 2 of 7
Look at which way each number was rounded.
Multiplying
Rounding the numbers UP gives an overestimate; rounding them DOWN gives an underestimate.
Dividing by a number rounded DOWN
The answer gets bigger: an overestimate.
Dividing by a number rounded UP
The answer gets smaller: an underestimate.
Mixed rounding
If some numbers went up and some went down, you cannot always tell without working it out.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Estimating a Square Root
Number
Lesson 2 of 7
40 is not a square number, so √40 lies between two whole numbers.
√36 = 6
√49 = 7
√36 = 6
The square number just below 40
√40 ≈ 6.3
40 is 4 of the 13 steps from 36 to 49, so about 0.3 of the way
√42.25 = 6.5
Halfway between 6 and 7 - 40 is below this
√49 = 7
The square number just above 40
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Estimating a Square Root
Number
Lesson 2 of 7
Estimate √70 to 1 decimal place without a calculator.
1
Find the square numbers either side of 70
64 = 8² and 81 = 9²
2
So √70 is between 8 and 9
8 < √70 < 9
3
How far is 70 from 64, out of the gap from 64 to 81?
6 out of 17
4
6 out of 17 is a bit more than a third
about 0.35
5
Check by squaring
8.4² = 70.56, which is close to 70
ANSWER
√70 ≈ 8.4 (the exact value is 8.366…)
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Estimating a Crowd
Number
Lesson 2 of 7
Nobody counts a crowd one person at a time.
Count a sample
Count the people in one block of seats: about 400.
Count the blocks
The stadium has about 150 blocks.
Multiply
400 × 150 = 60 000.
Report it sensibly
"About 60 000" - rounded to 1 significant figure, because the inputs were only estimates.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Number
Lesson 2 of 7
CASE STUDY
Enrico Fermi and "Fermi Problems"
The Italian-American physicist Enrico Fermi, who won the 1938 Nobel Prize in Physics, was famous for making quick, surprisingly accurate estimates from almost no information. He liked to ask his students questions such as "How many piano tuners are there in Chicago?" and expected them to reason it out: the population of the city, how many homes have a piano, how often a piano is tuned, and how many pianos one tuner can tune in a year. Each number is rounded, but the errors tend to cancel out, and the answer is usually the right size. Questions like this are now called Fermi problems.
1938
Fermi wins the Nobel Prize in Physics
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Key Terms
Number
Lesson 2 of 7
Place value
The value of a digit, which depends on its position in the number.
Decimal place (d.p.)
A digit after the decimal point.
Significant figure (s.f.)
Any digit of a number starting from the first non-zero digit.
Estimate
An approximate answer found by rounding the numbers first.
Approximately equal to
The symbol ≈, used for an estimate.
Overestimate
An estimate bigger than the exact answer.
Underestimate
An estimate smaller than the exact answer.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Number
Lesson 2 of 7
YOUR TASK
Estimation Station
12 minutes
Estimate each calculation, then use a calculator to find the exact answer and say whether your estimate was over or under. (a) 6.8 × 41.2 (b) 897 ÷ 2.9 (c) (19.6 × 0.48)/0.21 (d) √(98.6 + 23.1) (e) 3.14 × 7.9²
1
Estimate first, writing the rounded calculation.
2
Then use a calculator.
3
Say whether the estimate was over or under, and why.
WHAT A GOOD ANSWER SHOWS
(a) 7 × 40 = 280 (exact 280.16). (b) 900 ÷ 3 = 300 (exact 309.3). (c) (20 × 0.5)/0.2 = 50 (exact 44.8). (d) √120 ≈ 11 (exact 11.03). (e) 3 × 8² = 192 (exact 195.97).
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Can I...?
Number
Lesson 2 of 7
Use a known fact to work out related calculations.
Multiply and divide by 0.1 and 0.01.
Round to a given number of decimal places.
Round to a given number of significant figures.
Estimate a calculation by rounding to 1 s.f.
Divide by a decimal such as 0.5 or 0.2.
Estimate a square root.
Say whether an estimate is over or under.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Summary & Exam Focus
A known fact plus place value gives many related answers.
Significant figures start at the first non-zero digit; keep the place value when rounding.
Estimate by rounding each number to 1 significant figure.
Estimate square roots using the square numbers either side.
Over or under: look at which way each number was rounded.
EXAM FOCUS
Work out an estimate for (3.78 × 48.2)/0.52 (3 marks)
Write the rounded numbers down before you calculate. An answer with no rounded calculation shown can score zero, even if it is right.
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