013i - Problem Solving with Terminating and Recurring Decimals

Lesson 13 — Problem Solving and Consolidation: Terminating and Recurring Decimals

Strand: Number | Descriptor: AC9M8N03 | Duration: 45 minutes

Learning Intentions

• To apply knowledge of terminating and recurring decimals to solve applied and reasoning problems.
• To consolidate fraction-to-decimal conversion, the prime factorisation rule, and bar notation from Lessons 10–12.

Success Criteria

I can:

1. Classify and order a mixed set of fractions and decimals, including recurring decimals.
2. Use digital tools appropriately to verify hand-worked predictions.
3. Solve a multi-step applied problem involving terminating or recurring decimals, explaining my reasoning.
4. Justify whether a real-world measurement is best expressed as a fraction or a rounded decimal.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. A fraction with a prime denominator produces a recurring decimal.
2. Simplifying a fraction can change whether its decimal terminates.
3. A recurring decimal can be written exactly using bar notation.
4. Rounding a recurring decimal to 2 decimal places gives an exact value.

Answers: 1. Sometimes — true for , false for . 2. Never — the value stays the same, so the outcome (based on the fully simplified denominator) is fixed; only an unsimplified denominator can look misleading. 3. Always — that is precisely what bar notation is for. 4. Never — rounding always introduces a small error for a genuinely recurring decimal, since the digits never actually stop.

Activities

Activity 1 — Guided Consolidation: Ordering a Mixed Set (12 min)

Explicit reminder, then practice.

Remind students of the toolkit built across Lessons 10–12: convert fractions to decimals (by division or the prime-factorisation check), use bar notation for exact recurring values, and compare decimals digit by digit from the left.

I do: Order , , , from smallest to largest.

Comparing digit by digit: , so the order is .

You do: Order each set from smallest to largest, showing all conversions:

1. , , ,
2. , , ,
3. , , ,

Activity 2 — Applied Problem-solving (19 min)

Pairs. Every answer must include working and a one-sentence justification.

Problem 1. A recipe calls for splitting kg of flour equally among bakers. Each baker’s calculator shows kg. Is this the exact amount? If a baker uses this rounded figure every day for a year (365 days), estimate the total shortfall across all three bakers combined, to the nearest gram.

Problem 2. A stopwatch records three athletes’ average lap times as min, min and min. Convert all three to decimals and rank the athletes from fastest to slowest lap.

Problem 3. An electrician cuts a m cable into equal pieces. Should the length of each piece be recorded as a fraction, a rounded decimal, or a recurring decimal on the job sheet? Justify your choice, considering that a second electrician will use the recorded length to cut further cable.

Problem 4. A pizza shop divides profits of 1009$ staff. The manager wants to pay each staff member a whole number of cents. Explain why this is impossible to do exactly, and propose a fair way to handle the leftover amount.

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	Whether 100 \div 9$ can be split into equal whole-cent amounts, and what to do if not.
What is as a decimal?	, a recurring decimal.
Devise a plan	Since money only allows two decimal places (cents), round 11.11$ per person and check the total.
Carry out the plan	11.11 = $99.99$.
What has happened to the missing amount?	100.00 - $99.99 = $0.01$ is left over, because the true value per person recurs forever and cannot be represented exactly in cents.
Looking back — is this a flaw in the maths or in the money system?	The maths is exact ( recurring forever); it is the currency system, which only allows two decimal places, that cannot represent the exact share. A real solution: give one staff member an extra cent, chosen fairly (e.g. by roster).

Answers: 1 — Not exact; the true value is , so using undercounts by about kg per split, which is negligible even over a year — shortfall is far less than 1 gram total (illustrating that truncation error can be practically insignificant even though it is not zero). 2 — , , as given; fastest to slowest: . 3 — A fraction ( m) or exact recurring decimal ( m) should be used for further calculation, since a rounded decimal compounds error if the second electrician cuts six more pieces from it; a rounded decimal is fine only for a single, final display. 4 — Impossible because recurs forever and cents only allow 2 decimal places; a fair fix is to pay 8 staff 11.11$11.12$, or rotate who receives the extra cent.

Checks for Understanding

(8 minutes — exit ticket, collected)

1. Order from smallest to largest: , , , .
2. A m rope is cut into equal pieces. Write the exact length of each piece using bar notation, and explain why a rounded decimal would not be suitable for further precise cutting.
3. Convert and to a common form and state which is larger.
4. Reasoning. A student says ” and are basically the same number, just rounded differently.” Explain what is wrong with this statement.

Answers: 1. , so ordering all: — note exactly, so these two are equal; final order: . 2. Each piece is m exactly; a rounded decimal (e.g. m) would leave a small but compounding error if used to mark out multiple pieces. 3. , which is larger than ? Check carefully: , so is larger. 4. is exactly six tenths, a terminating decimal; , a completely different value close to — they are not versions of the same number, they differ by more than .

Common Misconceptions

Misconception	How to pre-empt it
Comparing recurring decimals only by their first 2–3 digits without checking whether they are equal or genuinely different.	Model writing at least 4–5 decimal places, or converting to a common fraction form, before concluding an order.
Treating a rounded decimal as safe to reuse in further exact calculations.	Use Problem 3 (the cable) explicitly: rounding early and reusing the rounded value compounds error across repeated steps.
Believing recurring decimals cannot be compared exactly to terminating decimals.	Show that converting both to decimals (or both to fractions) to enough places always resolves the comparison.
Assuming any “neat-looking” decimal like must be more accurate than a recurring one like .	Emphasise that recurring decimals are exact values, not less precise than terminating ones — precision is about the underlying value, not the appearance.
In money contexts, assuming an exact equal split is always possible.	Use Problem 4 to show recurring decimals can make an exact equal split impossible within a currency’s smallest unit.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Which of the following is largest: , , , ?

Answer

; ; ; . Largest is .

E2 (Kangaroo style). Three siblings share 50$16.666666667$50.00$ using whole-cent amounts?

Answer

50 \div 3 = $16.\overline{6}$16.67$50.01$16.66$ instead, an adjustment of 1 cent.

E3 (Challenge). A number line marks and . A point is placed at and another at . Which point is further from , and by how much (as an exact fraction)?

Answer

is slightly more than . Difference: . The point at is further from by exactly .

E4 (Investigation). A student claims that , when computed by first converting both to rounded decimals (), gives a less accurate answer than adding the fractions directly. Test this claim and explain the size of the error.

Answer

Exact: . Rounded decimal approach: — in this case the rounding errors happen to cancel out, giving the same result. This is not guaranteed in general; students should test a case where the errors do not cancel, e.g. , compared to the exact — here the rounded sum is slightly too large.

Homework

1. Order from smallest to largest: , , , .
2. A m plank is cut into equal shelves. Write the exact length of one shelf using bar notation and as a fraction.
3. Convert and to decimals to at least 4 places and state which is larger.
4. A charity raises 2508$ volunteers. Determine whether the split is exact in whole cents, and if not, propose a fair resolution.
5. Reasoning. Explain why converting a recurring decimal to its bar notation form before comparing it to another number is generally safer than comparing rounded calculator displays.
6. Challenge. Find the decimal expansion of and classify it as terminating or recurring. If a builder rounds this to 2 decimal places () and then multiplies by to check their work, will they get back exactly ? Explain what this shows about reusing rounded recurring decimals in further calculations.

Answers: Q1 — Converting all to decimals: , , , . Order: . Q2 — m m. Q3 — , which is larger than . Q4 — exactly — this one is exact, a useful contrast case (terminating, since ). Q6 — , a recurring decimal; , not exactly , showing that rounding a recurring decimal before reusing it in a calculation introduces a small but real error — the exact fraction should be used instead.