060i - Estimation Strategies

Lesson 60 — Estimation Strategies to Check Reasonableness

Strand: Number | Descriptor: AC9M7N05 | Duration: 45 minutes

Learning Intentions

• To use estimation to judge whether a calculated answer is reasonable.
• To select an appropriate estimation strategy for a given calculation.

Success Criteria

I can:

1. Round numbers sensibly before estimating.
2. Estimate the result of a calculation before performing it.
3. Judge whether an answer is reasonable and explain why.
4. Identify errors such as a misplaced decimal point using estimation.

Warmup

(6 minutes — spot the unreasonable answer, pairs)

Each answer is wrong. Without calculating exactly, say why each is obviously unreasonable.

2. 19.95 \times 4 = $7.98$
5. of

Answers: 1. About , so is far too small; 2. About 20 \times 4 = $80600 \div 6 = 100129188080$.

The principle to name: estimation does not give the answer — it tells you roughly where the answer should be, so you can spot when something has gone badly wrong.

Activities

Activity 1 — Explicit Instruction: Estimation Strategies (14 min)

Strategy 1 — Round to leading digits. Round each number to one significant figure, then calculate mentally.

Strategy 2 — Round to compatible numbers. Choose nearby numbers that divide neatly.

Strategy 3 — Front-end estimation. Use only the leading digits, then adjust.

Strategy 4 — Benchmark fractions and percentages. Use familiar reference points.

Strategy 5 — Bracket the answer. Find values you know it must lie between.

Which strategy when?

Calculation type	Best strategy
Multiplication of large numbers	Leading digits
Division	Compatible numbers
Adding several numbers	Front-end
Fractions or percentages of a quantity	Benchmarks
Checking a suspicious answer	Bracketing

We do — estimate each, then calculate exactly and compare:

4. of
5. of

Important habit to build: estimate before calculating, not after. An estimate made afterwards tends to be bent towards whatever answer you already have.

Activity 2 — Checking for Reasonableness (10 min)

Pairs. For each, estimate first, then decide whether the given answer is reasonable.

4. 34.95 \times 6 = $20.97$
5. of
6. of

Socratic scaffolding for Q2:

Prompt	Purpose
Estimate first. What are the numbers close to?	About and about .
So what should the answer be near?	About .
What is the given answer?	— roughly ten times too small.
What kind of error does that suggest?	A misplaced decimal point.
What is the correct answer?	.
Looking back	Estimation catches decimal-point errors instantly, which is its most valuable use.

Answers: 1. Reasonable — estimate , and is close ✓; 2. Unreasonable — should be about ; the answer is ; 3. Reasonable — estimate ✓; 4. Unreasonable — about 35 \times 6 = $210$209.70\tfrac13240 = 809090\tfrac23450 = 300$ ✓.

Activity 3 — Inquiry: Fermi Estimation (10 min)

Pairs. Estimating quantities nobody has measured.

Estimate each. You will not be able to look anything up — reason from what you know, state your assumptions, and show your steps.

1. How many heartbeats does a Year 7 student have in a year?
2. How many sheets of A4 paper would cover the classroom floor?
3. How many minutes have you spent at school since Year 7 began?

Socratic scaffolding for Q1:

Prompt	Purpose
Understand: what do you need to know?	Heartbeats per minute, and minutes per year.
Estimate a resting heart rate.	About – beats per minute. Use .
Minutes in an hour, hours in a day?	and , so minutes per day.
Round for easy arithmetic.	About minutes per day.
Days in a year?	, round to for estimation.
Multiply.	.
State it sensibly.	About million beats a year.
Looking back	The true figure is around million — our estimate is the right order of magnitude ✓

Answers (approximate, methods vary): 1. About million; 2. An A4 sheet is roughly , so a room needs about sheets; 3. Roughly hours × days × weeks = hours ≈ minutes per year.

Discussion. These are called Fermi problems. The aim is not a precise answer but the right order of magnitude — knowing whether something is in the thousands or the millions. Scientists and engineers use this constantly to sanity-check results.

Checks for Understanding

(5 minutes — exit ticket)

1. Estimate .
2. Estimate .
3. Estimate of .
4. Reasoning. A student calculates . Use estimation to show this is wrong, and give the correct answer.
5. Reasoning. Why should you estimate before calculating rather than after?

Answers: 1. About (exact ); 2. About (exact ); 3. About of (exact ); 4. The numbers are near and , so the answer should be near . The given answer is ten times too small — the correct value is ; 5. An estimate made afterwards is easily influenced by the answer you already have, so it no longer provides an independent check.

Common Misconceptions

Misconception	How to pre-empt it
Treating an estimate as the exact answer.	Always use , never , for estimates.
Estimating after calculating, so the check is not independent.	Build the habit: estimate first, every time.
Rounding so heavily that the estimate is useless.	Rounding to is too crude. Keep one significant figure.
Believing estimation is only for people who cannot calculate.	Professionals use it constantly to catch errors. Emphasise its purpose.
Ignoring an estimate that disagrees with a calculation.	Disagreement means one of them is wrong — investigate rather than pick.
Thinking a “close” estimate proves the answer correct.	Estimation catches big errors, not small ones. It rules out, rather than confirms.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which is closest to : , , , or ?

Answer

. The exact value is , so is closest.

E2 (AMC Junior style). Estimate to the nearest whole number, then calculate exactly.

Answer

E3 (Challenge). A shop sells items at 18.95$8906.50$. Use estimation to show this is wrong, and find the correct total.

Answer

Estimate: 20 = $100047 \times 18.95 = $890.65$ — a misplaced decimal point.

E4 (Challenge). Without a calculator, decide which is larger: or .

Answer

and . So the first is larger. (Exact: and ✓)

E5 (Fermi challenge). Estimate how many words are in a typical novel. State your assumptions.

Answer

About words per page × about pages ≈ words. Typical novels range from to words, so the estimate is the right order of magnitude.

Homework

1. Estimate each, then calculate exactly and compare: (a) (b) (c) (d) of .
2. For each, estimate first and state whether the given answer is reasonable: (a) (b) (c) (d) 24.95 \times 8 = $19.96$
3. Estimate: (a) of (b) of (c) of .
4. Which is closest to : , , or ?
5. A student calculates . Use estimation to show this is wrong and give the correct answer.
6. Estimate, showing your reasoning: (a) how many seconds you have been alive (b) how many pages are in all the books in your bedroom.
7. Reasoning. Explain why estimation is especially good at catching decimal point errors.
8. Reasoning. Explain why a good estimate does not prove that an answer is correct.
9. Challenge. A rectangular field measures m by m. Estimate its area, then calculate exactly. Turf costs 8$ per square metre — estimate the total cost before calculating it.

Answers: Q1 — (a) , exact (b) , exact (c) , exact (d) , exact . Q2 — (a) reasonable, (b) unreasonable, ; correct (c) reasonable, (d) unreasonable, 200$199.60\approx 4542\approx 300302.6\approx 2001806000 \div 30 = 200206.42004243.66\approx 200 \times 90 = 18,000\ \text{m}^217,578\ \text{m}^2\approx $144,000$140,624$.