010i - Converting Fractions to Decimals

Lesson 10 — Converting Fractions to Decimals

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

Learning Intentions

• To convert fractions to decimals using the division algorithm.
• To recognise and correctly notate terminating and recurring decimals.

Success Criteria

I can:

1. Convert a fraction to a decimal by dividing the numerator by the denominator.
2. Identify whether a decimal terminates or recurs.
3. Use correct recurring decimal notation (dot or bar notation).
4. Predict whether a fraction will terminate, based on the prime factors of its denominator.

Warmup

(5 minutes — mini whiteboards, rapid recall)

1. State the decimal equivalent of .
2. State the decimal equivalent of .
3. State the decimal equivalent of .
4. Try on your calculator. What do you notice about the digits after the decimal point?

Teacher note: Question 4 is today’s hook — most students will have memorised Q1–3 but will not have a ready method for Q4. Do not resolve it yet; Activity 2 formalises the idea of a recurring decimal.

Activities

Activity 1 — Explicit Instruction: Converting Using Division (terminating decimals) (10 min)

I do: Convert to a decimal using short division, treating the fraction as .

Model the division process explicitly on the board: remainder ; bring down a zero to make ; remainder ; bring down a zero to make ; remainder ; bring down a zero to make ; remainder — the division terminates because the remainder becomes exactly .

A terminating decimal is one where the division eventually leaves a remainder of , so the decimal stops.

We do: Together convert and using the same long-division method, tracking each remainder.

You do: Convert to decimals, showing the division working: , , .

(Answers: ; ; .)

Activity 2 — Explicit Instruction: Recurring Decimals and Notation (10 min)

I do: Convert using long division, and notice the remainder repeats without ever reaching .

Trace the remainders explicitly: remainder ; bring down a zero to make again — the remainder repeats forever, so the digit repeats forever too.

A recurring decimal is one where the division never leaves a remainder of — instead, the remainders eventually repeat in a cycle, causing a block of digits to repeat forever.

Introduce notation: (a dot or bar is placed over the repeating digit or block). For a longer repeating block, e.g. , we write (bar over the entire repeating block) or, using dot notation, (a dot over the first and last digit of the repeating block).

We do: Together convert and , tracing the repeating remainder and writing the answer in bar notation.

You do: Convert to decimals using bar notation: , , .

(Answers: ; ; .)

Activity 3 — Inquiry Task: Predicting Terminating Vs Recurring (14 min)

Pairs.

Convert each of the following fractions to a decimal and record whether it terminates or recurs: , , , , , , , , , , , .

Then, for each denominator, write its prime factorisation (e.g. ). Look for a pattern connecting the prime factors of the denominator to whether the decimal terminates.

Socratic scaffolding for the investigation:

Prompt	Purpose
Understand the problem	We are looking for a rule, based on the denominator alone, that predicts termination without doing the division.
Devise a plan	Sort your results into two lists — “terminates” and “recurs” — then write the prime factorisation of every denominator in each list.
Carry out the plan	Terminating denominators: — prime factors are all s and/or s only. Recurring denominators: — each includes a prime factor other than or (e.g. or ).
Look for the pattern	A fraction in simplest form terminates exactly when its denominator’s only prime factors are and/or .
Looking back — test the rule	Predict, without dividing: will terminate? — only s and s, so yes. Check: ✓. Will terminate? — includes a , so no, it should recur. Check: ✓.

Why does this rule work? Decimals are built on place values of tenths, hundredths, thousandths — powers of . A fraction can be rewritten with a denominator that is a power of only if its original denominator’s prime factors are entirely s and/or s; otherwise, no power of is ever exactly divisible by the denominator, and the division falls into a repeating cycle instead.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Convert to a decimal, showing your division working.
2. Convert to a decimal, using correct recurring decimal notation.
3. Without dividing, predict whether terminates or recurs. Justify using prime factors.
4. Without dividing, predict whether terminates or recurs. Justify using prime factors.

Answers: 1. . 2. . 3. Terminates — , only the prime factor . 4. Recurs — , includes the prime factor .

Common Misconceptions

Misconception	How to pre-empt it
Rounding a recurring decimal and treating the rounded value as exact, e.g. writing .	Insist on bar/dot notation for any exact recurring answer; reserve rounded decimals for when the question explicitly asks for an approximation.
Assuming every fraction terminates, based only on experience with “friendly” fractions like halves and quarters.	Use the Activity 3 investigation to build the general prime-factor rule, tested on unfamiliar denominators.
Placing the recurring bar/dot over the wrong digit(s), especially when the repeating block does not start immediately after the decimal point (e.g. , not ).	Always trace the long division remainders explicitly and mark exactly where the cycle of remainders begins repeating.
Believing the denominator alone (without checking it is in simplest form) determines termination, e.g. judging by the factors of without simplifying first.	Always simplify the fraction fully before applying the prime-factor rule — , which terminates, even though would suggest otherwise.
Thinking the division process “goes wrong” or is a mistake when the remainder doesn’t reach zero.	Reframe explicitly: a repeating remainder is not an error — it is the signature of a recurring decimal, a completely valid and exact way to represent the fraction.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Without dividing, determine whether terminates.

Answer

— only prime factors and , so it terminates. (In fact .)

E2 (Kangaroo style). Find the fraction (in simplest form) equal to .

Answer

Let . Then , so , giving , so .

E3 (Challenge). Find the length of the repeating block (the number of digits that repeat) in the decimal expansion of .

Answer

— the repeating block has digits. (A nice fact: the block length for , for prime , always divides ; here it divides exactly.)

E4 (Investigation). Convert , , and to decimals. What pattern do you notice in the repeating blocks?

Answer

, , — each repeating block is a multiple of (i.e. ), increasing by each time the numerator increases by .

Homework

1. Convert to decimals, showing your division working: (a) (b) (c) .
2. Convert to decimals using correct recurring decimal notation: (a) (b) (c) .
3. Without dividing, state whether each fraction terminates or recurs, justifying with prime factors: (a) (b) (c) (d) .
4. Simplify fully, then decide whether it terminates, justifying using the simplified denominator.
5. Reasoning. Explain why will always terminate, for any positive integer , but will never terminate.
6. Challenge. Find the smallest positive integer such that is a terminating decimal (with and the fraction not already reducible to remove the factor of trivially — i.e. find the smallest that cancels the in ).

Answers: 1(a) (b) (c) . 2(a) (b) (c) . 3(a) Terminates — . (b) Recurs — , includes a . (c) Terminates — . (d) Recurs — . 4. ; terminates, since , only the prime factor . 5. has only the prime factor , so always terminates; always includes the prime factor , so always recurs. 6. ; to cancel the , the numerator must be a multiple of (the largest factor of built only from s) — the smallest such is , giving , which terminates.