011i - Identifying Terminating and Recurring Decimals

Lesson 11 — Identifying Terminating and Recurring Decimals

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

Learning Intentions

• To use digital tools to convert fractions to decimals and classify the result as terminating or recurring.
• To understand why the denominator of a fraction, written in simplest form, determines whether its decimal terminates or recurs.

Success Criteria

I can:

1. Use a calculator to convert a fraction to a decimal.
2. Classify a decimal as terminating or recurring from its calculator display, without being fooled by rounding.
3. State the prime factorisation rule that predicts whether a fraction terminates.
4. Predict, without dividing, whether a given fraction will terminate or recur.

Warmup

(5 minutes — calculator round, pairs)

Convert each fraction to a decimal on your calculator and write down exactly what appears on the screen:

Sort your five results into two groups. What rule are you using to sort them?

Discussion: Most students will separate “decimals that stop” from “decimals that keep going”. Flag question 3 and 5 for a closer look — some calculators show (a rounded, cut-off display), which is different from , which is exactly correct and simply ends. This distinction is the hook for Activity 1.

Activities

Activity 1 — Explicit Instruction: what is the Calculator Really Showing Us? (10 min)

I do / We do / You do.

I do: Perform long division for on the board, tracking the remainder at each step.

The remainder hits 0, so the division stops exactly: . This is a terminating decimal.

Now perform :

The remainder returns to 1, a remainder we have already seen — so the digit repeats forever. This is a recurring decimal, written .

Key idea: A calculator screen only has room for 8–10 digits. When it shows , it has truncated or rounded an infinitely repeating decimal — it has not “finished”. Students must learn to recognise the tell-tale repeated digit or block, not just trust that the screen stopped.

We do: As a class, perform long division for and , tracking remainders each time, and identify the point where a remainder repeats.

You do: Use long division to classify , , as terminating or recurring. Confirm each with a calculator.

Activity 2 — Guided Practice: the Prime Factorisation Rule (10 min)

I do: Write each denominator (fraction already in simplest form) as a product of primes.

Rule: A fraction in simplest form terminates if and only if the prime factorisation of its denominator contains only 2s and/or 5s. Any other prime factor (3, 7, 11, 13, …) forces the decimal to recur.

We do: Complete this table together, factorising each denominator before deciding.

Fraction	Denominator factorised	Terminates or recurs?
		Terminates
		Recurs
		Terminates
		Recurs
		Terminates

Note and : one stray factor of 3 is enough to force recurrence, even though 2s are also present.

You do: Without dividing, predict whether each fraction terminates or recurs, giving the prime factorisation as evidence: , , , , . Verify three of your predictions with a calculator.

Activity 3 — Inquiry: the Trap of Unsimplified Fractions (14 min)

Pairs, then whole-class share.

Give students and and ask them to predict, using the denominator rule, whether each terminates.

A student factorises and predicts recurs. The calculator shows . What went wrong with the prediction?

Let pairs investigate, then formalise: the rule only works on the simplified denominator. , and alone terminates.

Extend with a harder case: .

Socratic scaffolding for pairs who are unsure how to proceed:

Prompt	Purpose
What are you trying to find out?	Whether terminates or recurs — but the rule needs the simplified fraction first.
What do you know about the rule?	It only applies once the fraction cannot be simplified further.
Can you find a common factor of 21 and 35?	Both are divisible by .
What is the simplified fraction?	.
Now factorise the new denominator	is prime, and it is a .
So what do you conclude?	It terminates. Check with a calculator: . ✓
Looking back — why did the unsimplified form look misleading?	contains a , which would (wrongly) suggest recurring. Simplifying removed the “hidden” common factor of 7 that was disguising the true denominator.

Finish with a whole-class list-building task: as a group, find five fractions between and that terminate, and five that recur, using only the rule (calculator to verify only).

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Use long division to show whether terminates or recurs.
2. Without dividing, predict whether terminates or recurs. Justify using prime factorisation.
3. A calculator shows . Explain why this display does not prove the decimal terminates.
4. Simplify first, then predict whether it terminates or recurs.

Answers: 1. Remainders , terminates at ; 2. , contains a factor of 3, so it recurs; 3. The screen only holds a limited number of digits — the final is a rounded digit, not proof the pattern stopped; long division of never reaches a remainder of ; 4. , and , so it terminates ().

Common Misconceptions

Misconception	How to pre-empt it
”The calculator screen stopped, so the decimal terminates.”	Explicitly compare (exact) with displayed as (truncated). Model long division to reveal the difference.
Applying the prime factorisation rule to the unsimplified denominator.	Insist on the instruction “simplify first, factorise second” every time, as in Activity 3.
”A prime denominator always means recurring.”	Contrast (terminates) with (recurs) — both prime, different outcomes, because only 2 and 5 are special.
Believing recurring decimals are “random” digits that never repeat exactly.	Show the remainder cycle in long division returning to a value already seen — this is what forces the exact repetition.
Thinking a decimal with many digits shown must be recurring.	Show (short, terminates) versus -style traps; also show a recurring decimal with a very short block, e.g. .

Enrichment — Competition-Style Problems

E1 (AMC Junior style). How many of the fractions are terminating decimals?

Answer

Terminating denominators (only 2s and 5s): — that is 5 fractions ().

E2 (Kangaroo style). A fraction , where is a whole number between 1 and 20, terminates. What is the greatest possible value of ?

Answer

Check denominators from 20 downward for “only 2s and 5s”: works. , giving .

E3 (Challenge). Find the smallest whole number such that is a terminating decimal and is not a factor of 7.

Answer

We need ‘s prime factors to be only 2s and 5s, and should not divide 7 evenly (ruling out trivial being irrelevant since 7 doesn’t divide into powers of 2/5 anyway). Trying small values: gives — terminates. is the smallest.

E4 (Investigation). Explain why always terminates, for any whole numbers , by considering what happens when you multiply the numerator and denominator to make the denominator a power of 10.

Answer

Multiply top and bottom so the denominator becomes where : e.g. ; multiply by (since we need two more factors of 5) to get . Any fraction over a power of 10 has a finite decimal expansion — it must terminate.

Homework

1. Use long division to classify each as terminating or recurring: (a) (b) (c) .
2. Without dividing, predict terminating or recurring, showing the prime factorisation of each denominator: (a) (b) (c) (d) .
3. Simplify first, then predict: (a) (b) (c) .
4. A student says: ” must terminate because 13 is a small number.” Explain the error in this reasoning.
5. Reasoning. Explain, in your own words, why multiplying the numerator and denominator of a fraction by the same number never changes whether the decimal terminates or recurs, even though it can change how the factorisation looks mid-working (before simplifying).
6. Challenge. List every value of from 1 to 30 for which terminates. How many are there?

Answers: Q1 — (a) terminates, (b) recurs, (c) recurs, . Q2 — (a) , terminates (b) is prime, not 2 or 5, recurs (c) , terminates (d) , recurs. Q3 — (a) , terminates (b) , recurs (c) , terminates. Q4 — size is irrelevant; only the prime factors matter, and is prime but is not or , so it recurs. Q5 — the value of the fraction is unchanged, and it is the value’s simplest-form denominator that determines the decimal type, not the appearance of an intermediate, unsimplified form. Q6 — — 9 values.