At a glance. One lesson of about 60 minutes. It assumes the number sets from Year 9 and the Standard course's classification of numbers. Students write recurring decimals as fractions, test the sets for closure, and complete a proof by contradiction.
Calculator: no calculator. No calculator: the lesson is about exact form and about reasons, and a display would hide the repeating block.
What students will learn
Know
- A recurring decimal is a rational number; its fraction is found by lining up the tails.
- ℚ is closed under +, −, × and ÷ (by non-zero); the irrational numbers are not.
- A proof by contradiction assumes the opposite and reaches an impossibility.
Understand
- Rational and irrational are decided by a reason, not by how a decimal looks.
- "Always" needs a proof and "sometimes" needs one example of each outcome.
- The same idea (cancel the endless part) works for every block length.
Do
- Convert terminating, recurring and mixed recurring decimals to fractions in lowest terms.
- Test a statement about the number sets with a counter-example or an argument.
- Follow and complete a proof that a square root is irrational.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: converting recurring decimals and recognizing rational results (questions 1 to 9).
- Criterion B. Investigating patterns: the figures and the what-if. Students find when a sum or product of irrational numbers is rational, and state a general rule with its reason.
- Criterion C. Communicating: writing a proof in logical steps, with each line justified (worked example 2 and question 7).
- Criterion D. Applying mathematics in real-life contexts: reading the repeating digits of a quotient from a calculator display to decide a far digit (question 4), and turning a spreadsheet's recurring decimals back into exact shares of a school (question 6).
The one thing to do with the figure
Ask for a prediction before every press. For the first figure, ask “how many places must I shift so that the tails match?” before choosing 0.4545… or 0.123123…. For the 0.1666… button, ask why one shift is not enough.
In the second figure, give students the pair √3 and −√3 and ask for the sum before they set it. Then ask them to find two more irrational pairs that give a rational sum, and one that gives a rational product.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: 1 ÷ 7 and the two predictions. Collect them on the board. |
| 12 min | The first figure with predictions; the key idea; worked example 1. |
| 10 min | The second figure, then the warning card; worked examples 2 and 3. |
| 5 min | Question 1, the check. If many choose A, return to 0.4545… in the first figure: two digits repeat, so the shift is 100. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 0.35; irrational, between 6 and 7; 3/8. |
|---|---|
| Hook | 0.6363… = 63/99 = 7/11, so the denominator is 11; √3 + (−√3) = 0, which is rational. |
| Worked examples | 13/30; √5 irrational by contradiction; 3√2 irrational. |
| Faded example | 20/33. |
| 1. Lining up the tails | B. A moves one digit; C moves three; D is the 99 in 99x, not the multiplier. |
| 2. Recurring block to a fraction | 33. 21/99 = 7/33. |
| 3. Subtracting shifted decimals | 65. 72 − 7, the tails cancelling. |
| 4. A digit far along a recurring decimal | 3. 100 ÷ 5 = 20 exactly, so the last digit of the block. |
| 5. A mixed recurring decimal | 90. x = 53/90. |
| 6. Adding recurring decimals | 71. 44/99 + 27/99 = 71/99. |
| 7. Completing a contradiction proof | 7. q² = 7k². |
| 8. Which results are rational | 4. 6, 5, 0 and 0 are rational. |
| 9. A rational minus an irrational | C. A and D can be rational (for example √2); B is always 1. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Shifted by the wrong number of places, or gave the 99 of 99x. | Count the digits of the block, then ask what number of places puts the tails on top of each other. |
| 99 or 100 in question 2 | Did not simplify, or put the block over 100. | Ask: is 21/99 in lowest terms? Does 0.2121… stop at 0.21? |
| 72 or 7 in question 3 | Gave one of the two shifted numbers instead of the difference. | Write both numbers, line up the decimal points, and subtract. |
| 4 or 6 in question 4 | Started the count at the first digit of a block, or one digit early. | Write the block and the position of every multiple of 5. |
| 99 or 9 in question 5 | Treated the whole of 58 as the block, or shifted by 1 only. | Ask which digits repeat. Does the 5 repeat? |
| 7 or 31 in question 6 | Added the tops and the bottoms, of 4/9 and 3/11 or of 4/9 and 27/99. | Convert both to ninety-ninths first. |
| 49 or 1 in question 7 | Forgot to divide by the 7 on the left, or dropped the 7. | Substitute p = 7k into 7q² = p² line by line. |
| 3 or 2 in question 8 | Did not simplify √18 or √3 × √12. | Simplify every number before classifying. |
| A, B or D in question 9 | Found one example where the result is rational, or did not notice s ÷ s = 1. | Try s = √2 in each option. |
Questions worth asking
- Why does subtracting x remove the endless part? Listen for “both numbers have exactly the same digits after the point, so they cancel”.
- Why is 0.9999… equal to 1? Let x = 0.999…; 10x − x = 9, so x = 1. The two numbers are the same point on the number line.
- What is the difference between “sometimes” and “always”? One counter-example kills “always”; “sometimes” needs one example of each outcome.
- Where does the proof use “lowest terms”? To get the contradiction: both p and q divisible by the prime would not be in lowest terms.
Differentiation
A way in for anyone stuck
- Give a table with columns x, 10x, 100x and have students fill in the digits, with the repeating block coloured.
- Start with single-digit repeats before two-digit blocks and mixed decimals.
- For the proof, give the five sentences on cards and have students put them in order before writing their own.
Stretch
- Prove that √2 + √3 is irrational. (Hint: if it equals r, square both sides.)
- Without dividing, explain how you know 1/13 repeats and cannot have a block longer than 12 digits.
Students learning in English as an additional language
“Recurring” means “happening again and again”. “Closed” here means “you cannot get out of the set by using the operation”. “Contradiction” means “two statements that cannot both be true”. Say “lowest terms” as “the fraction cannot be simplified”.
| Word | Plain meaning | Example |
|---|---|---|
| recurring decimal | a decimal whose digits repeat in a block for ever | 0.8333… |
| closed | an operation on members of a set always gives a member of the same set | whole numbers are closed under + |
| contradiction | two statements that cannot both be true | p is even and p is odd |
| proof by contradiction | assume the opposite of what you want to show and reach a contradiction | assume √5 = p/q |
| lowest terms | a fraction with no common factor in its top and bottom | 7/12 |
| counter-example | one case that shows a statement is not always true | 2, for “every prime number is odd” |
Sentence starters for explaining aloud or in writing:
- The repeating block is …, so I multiply by …
- This is not closed, because …
- Suppose it were rational. Then … which is impossible because …
Two things not to say
“Irrational numbers are the ones with long decimals.” Terminating and recurring decimals are rational however long they are; say “a decimal that never repeats”.
“It works for these cases, so it is proved.” Examples can disprove a statement but never prove it for all numbers.
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.
- FractionsWrite 0.5555… as a fraction.
5/9. - ClassifyingIs 0.101001000100001… rational?
No: the digits never repeat in a block.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextOn a calculator 5 ÷ 13 = 0.384615384615…, and the block 384615 repeats. What is the 50th digit after the point?
The block has 6 digits and 50 = 8 × 6 + 2, so it is the second digit of the block: 8. - Earlier work (Year 9)Simplify √75.
5√3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningProve that 1 + √2 is irrational, given that √2 is.
Suppose 1 + √2 = r, rational. Then √2 = r − 1 is rational, a contradiction. - ExtendingWrite 0.04545… as a fraction.
10x = 0.4545… = 5/11, so x = 5/110 = 1/22.
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 or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed; every answer is exact.