At a glance. One lesson of about 60 minutes. It follows Lesson 1 (rational numbers and repeating decimals) and uses simplifying fractions (Year 7, Unit 5) and solving a one-step equation (Year 8, Unit 4). Students use the figure to see why the tails cancel, then handle blocks of one and two digits and decimals where only the last digits repeat.
Calculator: no calculator. Nothing here needs a calculator. A calculator may be used afterwards to check a fraction by dividing, but it cannot find the fraction, and it hides why the tails cancel.
What students will learn
Know
- A block of n repeating digits goes over n nines: 0.4545… = 45/99.
- To write a repeating decimal as a fraction: multiply by a power of 10, subtract so the tails cancel, solve, simplify.
- 0.999… = 1.
Understand
- A repeating decimal is exact: the multiply-and-subtract step removes the endless tail.
- The multiplier must shift the tail by exactly one block so that the digits line up.
- Every repeating decimal is rational, and every fraction is a repeating or an ending decimal.
Do
- Write a repeating decimal with a one- or two-digit block as a fraction in simplest form.
- Handle a decimal where only some digits repeat, using two multipliers.
- Explain why the subtraction removes the tail.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the method for pure and mixed repeating decimals (questions 1 to 7).
- Criterion B. Investigating patterns: the figure. Ask students to find the fraction for every one-digit block, then every two-digit block that is a multiple of 11, and state the rule for the denominator.
- Criterion C. Communicating: explaining why 100x and not 10x, and writing the subtraction line by line (questions 6 and 7).
- Criterion D. Applying mathematics in real-life contexts: repeating decimals from distances and machine rates (question 2 and the extra questions).
The one thing to do with the figure
Ask for a prediction first. “0.7777… has the block 7. What fraction is it?” Take 7/10 and 7/9 from the room, and let the figure's subtraction decide.
Then set the digits to 1 and the block to 9 and ask what the figure says. Leave the argument about whether 0.999… is really 1 to the students, using the working on the screen as the evidence.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, predictions first. Include the one-digit block 9. |
| 10 min | Key idea and the first worked example, student by student on mini-whiteboards. |
| 10 min | The mixed repeating example (0.8333…) and the faded 0.7222… |
| 5 min | Question 1, the check. If many choose B or D, return to the denominator rule. |
| 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 | 0.075 (ends); 2/3; 7. |
|---|---|
| Worked examples | 7/11; 5/6. |
| Faded example | 13/18. |
| 1. A repeating decimal as a fraction | C, 2/11. 18/99 simplifies by dividing by 9. |
| 2. Adding repeating decimals | 3. 6/9 = 2/3. |
| 3. Denominator of a two-digit block | 33. 12/99 = 4/33. |
| 4. Part of the decimal repeats | 15. 42/90 = 7/15. |
| 5. A whole number and a repeating part | 11. 9/9 + 2/9 = 11/9. |
| 6. Why multiply by 100 | A. B, C and D give reasons that do not match the length of the block. |
| 7. A zero before the repeat | 90. x = 1/90. |
| 8. Fraction to a repeating block | 6. 8/33 = 24/99 = 0.2424… |
| 9. Only the last digit repeats | B, 4/45. 0.0888… = 0.888… ÷ 10 = (8/9) ÷ 10 = 8/90. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Put the block on the bottom, over 100, or over 9 for a two-digit block. | Ask how many digits repeat, then how many nines go underneath. |
| 9 or 6 in question 2 | Gave the denominator before simplifying, or the numerator. | Write 4/9 and 2/9, add the tops to get 6/9, then simplify and read the bottom. |
| 99, 100 or 9 in question 3 | Did not simplify, or used the wrong number of nines. | Count the digits in the block; then find the greatest common factor. |
| 90 or 99 in question 4 | Did not simplify, or took 46 as the block. | Ask which digits repeat; underline them. |
| 12 or 3 in question 5 | Added the whole number's digits to the top of the ninths. | Split the number into 1 and 0.222… and add fractions. |
| B, C or D in question 6 | Picked a reason that does not match the block length. | Test 10x and 7x on 0.2727… and look at the tails. |
| 9, 99 or 100 in question 7 | Treated the 0 before the repeat as part of the block, or ignored the repeat. | Multiply by 10 first to make 0.111… and name it. |
| 24 or 4 in question 8 | Gave the block, not the digit sum, or one digit. | Underline the question word: sum of the digits. |
Questions worth asking
- Why do the digits after the point cancel? Because the shift is exactly one block. Listen for “the tails are the same”.
- Is 0.999… less than 1, by a tiny amount? No; there is no number between them, and the working shows x = 1. Listen for the subtraction.
- What would be the denominator for a three-digit block? 999. Listen for “three nines”.
- Does 1/7 have a block of 6 digits? Yes; 142857/999999 = 1/7 (stretch).
Differentiation
A way in for anyone stuck
- Provide a template: x = …; …x = …; subtract; solve; simplify. Fill it in together for 0.4444…
- Use one-digit blocks (over 9) before two-digit blocks (over 99).
- Let students check each answer by dividing top by bottom on paper for two or three digits.
Stretch
- Question 7, then: write 0.142857142857… as a fraction by hand and simplify. (142857/999999 = 1/7.)
- Why does a block of n digits go over n nines? Write the answer with 10n.
- A block of three digits: write 0.123123123… as a fraction. (1000x − x = 123, so x = 123/999 = 41/333.)
Students learning in English as an additional language
“Repeating” and “recurring” mean the same here. “Block” is the group of digits that repeat. “Tail” is the endless part after the point. Say “the tails match” each time the figure subtracts.
| Word | Plain meaning | Example |
|---|---|---|
| repeating decimal | a decimal in which a block of digits repeats forever | 0.4545… |
| block | the group of digits that repeats | 45 in 0.4545… |
| tail | the digits that go on after the point | the 4545… in 0.4545… |
| simplest form | a fraction where the top and the bottom share no factor except 1 | 5/11 |
| denominator | the bottom number of a fraction | 11 in 5/11 |
| subtract | take one number away from another | 100x − x |
Sentence starters for explaining aloud or in writing:
- I multiplied by … because the block has … digits.
- The tails cancel because …
- The fraction in simplest form is … because …
Two things not to say
“Put the digits over 9s.” That works for a pure block only, and without the reason it fails on 0.2777…
“0.999… is very close to 1.” It is equal to 1, and the working shows it.
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.
- One digitWrite 0.7777… as a fraction.
7/9. - Two digitsWrite 0.151515… as a fraction in simplest form.
15/99 = 5/33.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA pump moves 0.5454… of a litre each second. Write it as a fraction.
54/99 = 6/11. - Earlier work (Lesson 1)Does 9/16 end or repeat? Write it as a decimal if it ends.
It ends, since 16 = 2 × 2 × 2 × 2: 9/16 = 0.5625.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningLet x = 0.999… Show that x = 1.
10x = 9.999…, so 10x − x = 9, 9x = 9 and x = 1. - Mixed numberWrite 2.444… as an improper fraction.
2 + 4/9 = 18/9 + 4/9 = 22/9.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write the two rows on the board with the digits after the point lined up, and subtract.