061i - Problem Solving with Rounding and Estimation

Lesson 61 — Problem Solving and Consolidation: Rounding and Estimation

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

Learning Intentions

• To apply rounding and estimation in multi-step practical problems.
• To use estimation habitually as a checking tool.

Success Criteria

I can:

1. Round to a stated accuracy within a larger problem.
2. Estimate before calculating and compare afterwards.
3. Decide whether a context requires rounding up, down, or to the nearest.
4. Identify and explain calculation errors using estimation.

Warmup

(6 minutes — three kinds of rounding, pairs)

The answer to each calculation is . Give the sensible final answer for each context.

1. The number of buses needed to carry the students.
2. The number of complete shelves that can be filled.
3. The average number of goals per game.
4. The length of a shelf in metres, to one decimal place.

Answers: 1. — round up; a partial bus is still a whole bus needed; 2. — round down; the eighth shelf is incomplete; 3. — leave it; an average need not be a whole number; 4. m — round to the nearest.

The principle: the context, not the digits, decides which way to round. This is the single most examined idea in this topic.

Activities

Activity 1 — Contextual Rounding Problems (14 min)

Pairs. Every answer needs the exact calculation, the rounding decision, and a justification.

Problem 1. A lift carries at most people. How many trips are needed to carry people?

Problem 2. Tickets cost 12.50$80$?

Problem 3. A cake recipe needs kg of flour. How many complete cakes can be made from a kg bag?

Problem 4. A car travels km using L of fuel. Find the fuel consumption in L per km, to one decimal place.

Problem 5. Fencing comes in m panels. A boundary is m long. How many panels must be bought, and how much is left over?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	Litres used per km — a rate.
Estimate first.	About L per km, which is L per km.
Set up the calculation.	.
Carry it out.
Round as instructed.	L/ km.
Compare with the estimate.	is close to ✓
Looking back	Which rounding rule applied here? Nearest — the value is a measurement, not a count of objects.

Answers: 1. trips (up); 2. tickets (down); 3. cakes (down); 4. L/ km (nearest); 5. panels (up); m, so m spare.

Activity 2 — Estimation as Error Detection (12 min)

Pairs. Realistic receipts and calculations, each containing one error.

Task. Each of these shopping calculations contains exactly one error. Use estimation to find which line is wrong, then correct it.

Receipt A

Item	Working	Total
kg apples at 4.20$/kg		12.60$
tins at 1.85$		9.25$
loaves at 3.90$		78.00$
Total		99.85$

Receipt B

Item	Working	Total
notebooks at 2.35$		9.40$
pens at 1.15$		1.38$
calculator		24.95$
Total		35.73$

Socratic scaffolding:

Prompt	Purpose
Estimate each line before checking any arithmetic.	Line by line: , , .
Which line disagrees with its estimate?	The loaves: should be about , not .
What kind of error is it?	A misplaced decimal point.
Correct it and re-total.	7.80= 12.60 + 9.25 + 7.80 = $29.65$.
Looking back	The totals were also wrong — one bad line corrupts everything after it. Estimate the total too: 30$ ✓

Answers: A — the loaves line should be 7.80$29.65$13.809.40 + 13.80 + 24.95 = $48.15$.

Activity 3 — Inquiry: the Tolerance Problem (12 min)

Pairs. What rounding hides.

A carpenter measures a shelf as m long, to one decimal place.

1. What is the shortest the shelf could actually be? The longest?
2. The gap it must fit is measured as m, also to one decimal place. Is the shelf guaranteed to fit?
3. Two shelves, each measured as m, are placed end to end. Between what lengths could the total lie?

Socratic scaffolding:

Prompt	Purpose
Understand: what does ” to one decimal place” mean?	The true value rounds to — it lies in a range.
Find the range.	From up to (but not including) .
So can we know the exact length?	No — only that it is within of .
Now Q2. Worst case for fitting?	Shelf at its longest (), gap at its shortest ().
Does it fit then?	No — the shelf could be up to about m too long.
So is the fit guaranteed?	No. Equal rounded measurements do not mean equal true values.
Q3: combine the ranges.	Shortest: . Longest: just under .
Looking back	Rounding errors accumulate when measurements are combined — each piece contributes its own uncertainty.

Answers: 1. m and just under m; 2. No — the shelf could be up to nearly m longer than the gap; 3. Between m and just under m.

Closing discussion. This is why carpenters measure to the millimetre and machinists to fractions of a millimetre: the accuracy of a measurement must match the tolerance the job demands. Connect back to Lesson 59’s “appropriate accuracy” — too little accuracy is as much a problem as false precision.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A ferry holds passengers. How many trips for people?
2. How many 3.75$50$?
3. A calculation gives . Round it to (a) 1 d.p. (b) 2 d.p.
4. A number rounds to to one decimal place. State the smallest and largest values it could be.
5. Reasoning. A student says “both pieces measured m, so together they are exactly m.” Explain the flaw.

Answers: 1. trips; 2. tickets; 3. (a) (b) ; 4. and just under ; 5. Each measurement is only accurate to m, so the true total lies anywhere between and just under m — the errors can accumulate.

Common Misconceptions

Misconception	How to pre-empt it
Always rounding to the nearest, regardless of context.	The warmup sorts the three cases explicitly. Ask “what does a partial unit mean here?”
Rounding up for tickets/purchases when money runs out.	You cannot buy tickets with 80$ — the seventh is unaffordable. Round down.
Rounding intermediate steps, compounding error.	Keep full accuracy until the final answer; round once, at the end.
Believing a rounded measurement is exact.	The tolerance inquiry addresses this directly.
Assuming two equal rounded values are truly equal.	Q2 of the inquiry — the shelf may not fit.
Ignoring the estimate when it clashes with the calculation.	A clash means something is wrong. Investigate, never ignore.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Buses hold passengers. What is the smallest number of buses needed for people?

Answer

, so buses.

E2 (AMC Junior style). A whole number, when rounded to the nearest ten, gives . What are the smallest and largest possible values of the number?

Answer

Smallest ; largest .

E3 (Challenge). Two numbers each round to (nearest whole number). What are the smallest and largest possible values of their sum?

Answer

Each lies in . Sum ranges from up to (but not including) . So the sum could be anything from to just under — it need not round to .

E4 (Challenge). A rectangle’s sides are measured as cm and cm, each to the nearest centimetre. Find the smallest and largest possible values of its area.

Answer

The “true” area could differ from the naive by more than either way.

E5 (Challenge). A shop rounds every bill to the nearest cents. A customer’s true bill is 17.63$. What do they pay? What is the most a single rounding can cost or save a customer?

Answer

17.63$17.6523821, 2, 4, 6, 7, 922.52$ cents).

Homework

1. Decide the sensible answer for each: (a) boxes needed to pack items, per box (b) 45 \div $6.80 = 6.6250 \div 35 = 7.14$ average points per game.
2. A minibus seats . How many minibuses for a group of ?
3. How many m lengths of timber can be cut from a m plank? How much is wasted?
4. A car uses L over km. Find its consumption in L/ km, to 1 d.p. Estimate first.
5. Each line of this receipt contains the shop’s working. Find the single error by estimation, correct it, and re-total: pears at 0.85 = $5.102$3.20= $64.00$1.60$70.70$.
6. A number rounds to (1 d.p.). State its smallest and largest possible values.
7. Two pieces of rope each measure m to one decimal place. Between what lengths does their combined length lie?
8. Reasoning. Explain why intermediate results should not be rounded during a multi-step calculation.
9. Reasoning. A café bill of 27.525$ cents. What is paid, and why do shops use this rule?
10. Challenge. A square’s side is measured as cm to the nearest centimetre. Find the smallest and largest possible values of its perimeter and area.

Answers: Q1 — (a) boxes (up) (b) tickets (down) (c) (leave). Q2 — . Q3 — lengths; m wasted. Q4 — estimate ; exact L/ km. Q5 — rice should be 6.405.10 + 6.40 + 1.60 = $13.1012.3512.457.107.30$27.5012[5.5, 6.5)[22, 26)[30.25, 42.25)\ \text{cm}^2$.