Estimating to check

Classroom resources

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Students will be able to

• I can estimate a calculation by rounding each number to its first digit.
• I can compare an estimate with the exact answer and say whether it is reasonable.
• I can use an estimate to find a slip, such as a decimal point in the wrong place.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need rounding (previous lesson) and multiplying by multiples of 10 (Year 6).

The one idea: round to the leading digit and keep the size, calculate the easy numbers, and compare with the exact answer. The commonest mistakes are losing the size (Lina rounds 19.6 to 2), losing a zero in the calculation, and trusting a calculator answer because it looks plausible.

Estimation is only worth teaching if it is used. In class, make students estimate before pressing = for the whole lesson.

Running the lesson

1. Predict: students who chose 200 lost a zero; 90 added the numbers.
2. Run: ask whether the exact answer is more or less than the estimate, and why.
3. Investigate: have students write their estimate before the calculator answer is shown.
4. Modify: ask when rounding both numbers up gives an estimate that is too big.
5. Make: collect pairs of numbers; check which have an exact answer above and below 2000.

The hard part (on purpose)

Keeping the size when rounding: 19.6 is about 20, not 2. Ask 'how many zeros should the estimate have?' before computing.

When students get stuck

Answers to the checks

1. estimate 38 × 52: Which is the best estimate for $38 \times 52$?
   2000. 38 is about 40 and 52 is about 50, and 40 × 50 = 2000. 90 is 38 + 52.
2. estimate and exact: Set a = 49 and b = 21. What is the estimate? Is it more or less than the exact answer?
   
   49 ≈ 50 and 21 ≈ 20, so the estimate is 50 × 20 = 1000. The exact answer is 1029, a little more.
3. Round, then calculate: Estimate $6.8 \times 31$ by rounding each number to its first digit.
   210
   6.8 ≈ 7 and 31 ≈ 30, so 7 × 30 = 210.
4. Find the mistake: Lina estimates 19.6 × 4.1. Click the line where her working goes wrong.
   line 2
   Line 2: 19.6 is about 20, not 2: rounding keeps the size. 20 × 4 = 80, so 19.6 × 4.1 is about 80.
5. Is it reasonable?: You type $3.2 \times 48$ and the calculator shows 15.36. Is that reasonable?
   No: 3 × 50 = 150, so the decimal point is in the wrong place
   3.2 × 48 ≈ 3 × 50 = 150. The real answer is 153.6: the calculator shows 15.36 only if the 3.2 was typed as 0.32.
6. Add them: Estimate 387 + 512 by rounding to the first digit. Set a near 387 and b near 512: what is the estimate?
   
   387 ≈ 400 and 512 ≈ 500, so the estimate is 900. The exact answer is 899.
7. Keep the size: Estimate $4.9 \times 81$.
   400
   4.9 ≈ 5 and 81 ≈ 80, so 5 × 80 = 400.
8. Divide: Estimate $612 \div 29$.
   20
   612 ≈ 600 and 29 ≈ 30, so 600 ÷ 30 = 20.
9. Bigger or smaller?: Is an estimate always smaller than the exact answer?
   No: 38 × 52 ≈ 2000, but the exact answer is 1976
   Rounding goes up for some numbers and down for others, so an estimate can be above or below. What matters is that it is close.
10. Two numbers: Estimate $58 \times 19$.
    1200
    58 ≈ 60 and 19 ≈ 20, so 60 × 20 = 1200.
11. A decimal times: Estimate $7.9 \times 4.2$.
    32
    7.9 ≈ 8 and 4.2 ≈ 4, so 8 × 4 = 32.
12. Check the calculator: The calculator shows $504 \div 6.3 = 8$. Is that reasonable?
    No: 500 ÷ 6 is about 80, so 8 has lost a zero
    504 ÷ 6.3 = 80. Estimating 500 ÷ 6 ≈ 80 shows 8 has lost a zero.
13. Review: rounding: Review: round 6.45 to 1 decimal place.
    6.5
    The next digit is 5, so 6.5.
14. close but under: Set a and b so that the estimate is exactly 2000 but the exact answer is less than 2000.
    
    For example 38 × 52 = 1976: the estimate 40 × 50 = 2000 is a little bigger.

Solutions

One possible solution for each task. Other correct programs are fine.

other operations, other sizes › Add them

Open task: any program that does what the step asks.

other operations, other sizes › Keep the size

Open task: any program that does what the step asks.

other operations, other sizes › Divide

Open task: any program that does what the step asks.

other operations, other sizes › Bigger or smaller?

Open task: any program that does what the step asks.

close but under

Open task: any program that does what the step asks.

Stretch and support

Stretch: Round one number up and the other down so that the estimate is closer to the exact answer. When is that a good idea?

Support: Round to the nearest ten or hundred, and write the rounded numbers before multiplying.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Estimating to check".

The idea: Estimation to check reasonableness: I can Students use estimation strategies (front-end, rounding, compatible numbers) to check whether a calculated solution is reasonable.

What understanding looks like:
- Produces a reasonable estimate before or after computing.
- Identifies whether a given answer is unreasonable based on estimation.
- Explains the estimation strategy used.

Mistakes students often make here:
- A student rounds 19.6 to 2 (Lina's slip).
- A student computes 5 × 80 = 40.
- A student says any calculator answer must be reasonable.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Context-appropriate accuracy), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Estimating to check".

Exit ticket

Estimate 4.8 × 31. The calculator shows 148.8. Is it reasonable?

5 × 30 = 150, so yes.

What each step is for

• estimate 38 × 52: Committing to an estimate shows whether they round to the first digit and keep the size (2000), lose a zero (200) or gain one (20 000).
• estimate and exact: Moving a and b shows the estimate (rounded numbers) staying close to the exact answer, and both bars the same size.
• estimates catch slips: Keeping the size of each rounded number (19.6 is 20, not 2) is the whole skill; an estimate of the wrong size cannot catch a decimal slip.
• other operations, other sizes: Estimating works for adding, dividing and with decimals; an estimate can be above or below the exact answer.
• estimating: Estimates for multiplying and for a decimal product, a calculator check with a division, and a rounding review.
• close but under: Finding numbers whose estimate is exactly 2000 but whose exact product is less shows an estimate is a close value, not a bound.

Where it leads

Next: Ratios, where estimating also helps check sharing and scaling.

Where it leads: estimating in modelling problems (money, measurement), and significant figures in Year 8 and 9.