GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher
Estimation and approximation
No calculator, and someone asks, "Roughly how much was all that?" Learn to answer in seconds, and to tell whether you're too high or too low.
Worked example · the estimating method
Problem
A circular patio has radius 5.9 m. Each paving slab covers 0.48 m². Estimate how many slabs are needed to cover the patio. (Area of a circle = π × radius².)
Predict, then check
Commit to one before you look.
A hall has 182 adults and 17 children, and you want an estimate of the total number of people. Which approach should give the closer estimate?
Maths · Estimating
Too big, too small, or can't tell?
No exact answers allowed. Sort each estimate using only the direction every value was rounded.
Still to sort
Overestimate (0)
The estimate is greater than the exact answer.
Where the line is: Every single rounding pushes the answer up.
Underestimate (0)
The estimate is less than the exact answer.
Where the line is: Every single rounding pushes the answer down.
Rounding directions don't settle it (0)
Some roundings push the answer up and some push it down.
Where the line is: This doesn't mean it's impossible to know. It means the directions alone can't tell you.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Round every value to one significant figure first, then calculate. You can even tell whether you came out too big or too small.
What you need to know
- Rounding: look at the digit just right of your rounding column. Under 5, round down; 5 or more, round up.
- Truncating chops digits off without looking at what follows, replacing them with zeros where needed.
Have a goRound 2,481 to the nearest 100. Then truncate 2,481 so only the thousands digit is kept. Do you get the same answer?
No: rounding gives 2,500 and truncating gives 2,000.
Rounding looks at the 8 just right of the hundreds column and goes up; truncating never looks at it and just cuts.
- Significant figures: counting starts at the first non-zero digit. Zeros before it don't count, but a zero after it does.
Have a goSam says 0.0306 has four significant figures "because there are four digits after the point". Sam sounds very sure. How many does it really have?
Three: the 3, the 0 and the 6.
Counting starts at the first non-zero digit (the 3), so the zero before it doesn't count, but the zero after it does.
- One non-zero digit does not mean one significant figure: 4,983 is 5,000 to 2 s.f., and 5,004 is 5,000 to 3 s.f.
- To estimate, round every value to one significant figure first, then calculate. It's quick enough to do without a calculator.
- Never round the answer again: 8.25 × 4 × 52 ≈ 8 × 4 × 50 = 1,600, and 1,600 stays.
Have a goEstimate 3.9 × 71. Once you have the result, should you round it again?
The estimate is 4 × 70 = 280, and it stays as 280.
The rounding was done to the values before calculating, so the result is left as it comes out.
- The less rounding you do, the closer your estimate usually is to the true answer.
- Two handy tools: use 3 for π, and for a decimal divisor multiply top and bottom by the same power of 10.
- An overestimate is bigger than the exact answer and an underestimate is smaller. Check how every value was rounded to tell which.
- Careful: rounding up a value you subtract, or divide by, pushes the answer down, so the 'all up' rule can't be applied blindly.
The big picture
An estimate isn't a guess. You round every value to one significant figure first, then calculate with those, and leave the answer alone. Along the way you'll see how rounding differs from truncating, how to count significant figures, and how to tell whether an estimate is too big or too small without knowing the exact answer.
Key points
Worked example
Problem
Estimate the total cost of 19 notebooks at £2.85 each and 28 pens at £1.15 each.
⚠ Watch out
Working out the exact answer and then rounding it. That isn't estimating. Round each value to 1 significant figure first, calculate with those, and leave the result as it comes out.
Memory hook
Round the ingredients, not the cake: tidy up the numbers going in, and leave whatever comes out alone.
Check yourself
Estimate (31.4 × 5.2) ÷ 0.62 without a calculator. Then check: (30 × 5) ÷ 0.6 = 150 ÷ 0.6 = 1,500 ÷ 6 = 250, and 250 stays.
Flashcards
(14)When rounding, which digit decides whether you round up or down?
Round 5,433 to the nearest 1,000 and then to the nearest 100.
What is truncation, and what is 3,678 truncated to one significant figure?
Which digit is the first significant figure?
Write 0.02568 to 2 significant figures.
4,983 and 5,004 both round to 5,000. To how many significant figures each?
How do you make a quick estimate of a calculation?
Is working out the exact answer and then rounding it a valid estimate?
What's the link between how much you round and how close an estimate is?
What value do you use for π in an estimate?
How can you simplify a division by a decimal when estimating?
What are an overestimate and an underestimate?
How can you tell over or under without the exact answer?
When does 'all rounded up means overestimate' stop working?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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Keep me postedMore Edexcel GCSE Maths topics
- Angle properties and parallel lines
- Area of any triangle (½ab sin C)
- Calculating with roots and indices
- Circle definitions and properties
- Circumference, area of circle and 3D solids
- Conditional probability
- Geometrical problems on coordinate axes
- Gradients and areas under curves
- Gradients and intercepts of linear functions
- Iterative methods
- Limits of accuracy and bounds
- Measuring lines, angles and bearings
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