At a glance. One lesson of about 60 minutes. The core is fraction to decimal and the rule for which decimals stop. Students who are ready then turn recurring decimals back into fractions, which is going further.
Calculator: scientific calculator. A scientific calculator shows what a display does with a recurring decimal: it cuts and rounds. The lesson uses it once, after an estimate, to see why the display is not the fraction.
What students will learn
Know
- Every fraction is a division, and its decimal terminates or recurs.
- A fraction in simplest form terminates exactly when its bottom has only the prime factors 2 and 5.
- A repeating block of k digits is that block over k nines.
Understand
- A repeated remainder forces repeating digits.
- A recurring decimal is an exact number, and a calculator display is a rounded copy.
- Subtracting a shifted copy cancels the infinite tail.
Do
- Convert a fraction to a decimal by division and say whether it stops.
- Predict termination from the prime factors of the bottom.
- Convert a recurring decimal to a fraction.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: terminating and recurring decimals, and their fractions (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to run through bottoms 2 to 16 and find which make decimals stop, then to state a rule using prime factors.
- Criterion C. Communicating: explaining in words why 0.333... is exactly 1/3 (question 9), using the words recurring, remainder and exact.
- Criterion D. Applying mathematics in real-life contexts: reading a calculator display, and the time in question 4, where the rounded 0.33 hours is 19.8 minutes and not 20 (the hook, the calculator note and question 4).
The one thing to do with the figure
Ask for a prediction before the first slide. “1/4, 1/5, 1/6, 1/8: which go on for ever?” Take votes, then run the four. Ask what the remainders do in each.
Then fix the top number at 1 and sweep the bottom from 2 to 16, and ask students to list the bottoms that stop. Link the list to the prime factors line. In the second figure, ask for the fraction before the figure shows it. The first figure computes the fractions in questions 2, 6 and 8, so ask for those answers before it is opened, or cover the sliders.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, prediction and the first figure. Include one bottom with a long block, such as 7 or 13. |
| 10 min | Key idea and the calculator note; students check 1 ÷ 7 and find the block. |
| 10 min | Second figure and the worked examples; students finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, go back to the factors line. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 15%; 2/5; 2 × 2 × 2 × 5. |
|---|---|
| Worked examples | 7/40 = 0.175; 0.4333... = 13/30. |
| Faded example | 0.7272... = 8/11. |
| 1. Which fraction terminates | C, 7/20. A, B and D have bottoms with a prime factor of 3 or 7. |
| 2. Fraction to a terminating decimal | 0.3125. 16 = 2 × 2 × 2 × 2. |
| 3. Which decimal is exactly one third | C, 0.3̇. A and B stop too soon, and D is rounded up; only the repeating 3 is exactly 1/3. |
| 4. A recurring decimal in context | 20. 60 ÷ 3 = 20. The rounded 0.33 hours is only 19.8 minutes. |
| 5. Recurring decimal to fraction | 33. 15/99 = 5/33. |
| 6. A digit far along a repeating decimal | 3. Digits 2, 3, 0, 7, 6, 9 repeat. |
| 7. Which numerators terminate | 3. 3/12, 6/12, 9/12. |
| 8. Mixed recurring decimal to fraction | 5. 5 ÷ 6 = 0.8333... |
| 9. A recurring decimal is exact | B. A and C treat a recurring decimal as inexact; D thinks a fraction and a decimal must differ. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Judged by the size of the bottom, or did not factor it. | Write each bottom as a product of primes, and underline any prime that is not 2 or 5. |
| 3.2 in question 2 | Divided the bottom by the top. | Ask whether 5/16 is more or less than 1, and which division gives that. |
| 0.31 in question 2 | Stopped before the remainder reached 0. | Continue the division to the remainder: ask whether it is 0. |
| A, B or D in question 3 | Wrote a stopping decimal for 1/3: 0.3 and 0.33 stop short, and 0.34 is rounded up. | Ask for 3 × 0.33 and 3 × 0.3. What does 3 × 0.333... have to be? |
| 33, 19.8 or 18 in question 4 | Used a shortened or rounded decimal for 1/3, or read the digits as minutes. | Ask: is 0.33 exactly 1/3? Then take a third of 60 minutes. |
| 99 or 3 in question 5 | Left it unsimplified, or gave the common factor. | Ask: is it in simplest form? What do you divide by? |
| 2 or 9 or 0 in question 6 | Used the wrong position in the block. | Write the block on a strip and count 20 along, wrapping round. |
| 5 or 2 in question 7 | Counted even tops, or missed one. | Simplify every fraction first, then factor each bottom. |
| 8 or 83 in question 8 | Read the digit, or used 83/100. | Ask whether 0.83 stops, and whether 0.8333... does. |
| A, C or D in question 9 | Thought a recurring decimal is not exact, or a fraction and a decimal differ. | Ask for 3 × 1/3 and 3 × 0.333... and compare. |
Questions worth asking
- Why must a recurring decimal come back to a remainder it has seen? There are only d possible remainders (0 to d − 1), so after d steps one must repeat. Listen for “only so many remainders”.
- Why does 3/12 stop but 2/12 not? 3/12 = 1/4 and 2/12 = 1/6. Simplify first.
- Is 0.999... less than 1? No: it equals 1. 9x = 9 from x = 0.999...
- Which has the longer block, 1/7 or 1/13? Both repeat every 6 digits. The longest block for a bottom d has at most d − 1 digits.
Differentiation
A way in for anyone stuck
- Give a long-division scaffold with the remainders in a column, so students can see when a remainder returns.
- Start with unit fractions: 1/2 to 1/10.
- Colour the repeating block so students can see where it starts.
Stretch
- Find the number of digits in the block of 1/7, 1/13 and 1/17. What do you notice? (6, 6, 16.)
- Write 0.12̇3̇ = 0.1232323... as a fraction. (61/495.)
Students learning in English as an additional language
“Terminate” means to end, and “recur” means to come back again. Say “stops” and “repeats” with them. Write each word with its everyday meaning on the word wall.
| Word | Plain meaning | Example |
|---|---|---|
| terminating decimal | a decimal with a last digit: it stops | 0.125 is terminating |
| recurring decimal | a decimal with a block of digits that repeats for ever | 0.2727... is recurring |
| recurring dot | a dot over the first and last digit of the repeating block | 0.2̇7̇ means 0.2727... |
| repeating block | the group of digits that repeats | in 0.142857142857... the block is 142857 |
| remainder | what is left over in a division | 10 ÷ 4 has a remainder of 2 |
| prime factor | a prime number that divides exactly | the prime factors of 12 are 2 and 3 |
Sentence starters for explaining aloud or in writing:
- The remainder came back as …, so the digits …
- The bottom number is … = …, so the decimal … because …
- Two digits repeat, so the bottom is … and the top is …
Two things not to say
“It is just 0.33.” Rounding changes the number. Say “0.33 to two decimal places” when you round.
“A decimal that goes on for ever is irrational.” Recurring decimals are rational: they are fractions. Irrational numbers come later.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Stop or repeat?Does 3/8 stop? Does 5/6?
3/8 = 0.375 stops. 5/6 = 0.8333... repeats. - FractionWrite 0.6666... as a fraction.
2/3. 9x = 6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA calculator shows 2 ÷ 9 = 0.2222222222. A student says the answer is exactly 0.2222222222. What is the exact value, and what is the difference between the display and the exact value?
2/9. The display is a rounded copy: the exact decimal never ends. The difference is about 0.0000000000222..., which is 2/9 × 0.0000000001. - Earlier work (Lesson 6 of this unit)Work out 1/4 ÷ 1/8 and write the answer as a fraction and a decimal.
2, which is 2.0 and stops.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningShow that 0.4999... = 0.5. Use x = 0.4999...
10x = 4.999..., 10x − x = 4.5, 9x = 4.5, x = 0.5. - Working backA fraction in simplest form has a bottom number of 28. Does it stop? What bottoms could make n/28 stop after simplifying?
28 = 2 × 2 × 7, so it repeats. Only if 7 divides n, giving a bottom of 4: 7/28 = 1/4, 14/28 = 1/2, 21/28 = 3/4.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
A scientific calculator is useful once in this lesson, for the calculator note. No device? Do long division on paper and mark each remainder.