At a glance. One lesson of about 60 minutes. It reviews terminating and recurring decimals from Year 7 (Unit 5, lesson 7) and powers and prime factors from Year 7 (Unit 1). Students long-divide fractions in the figure, state the rule for which denominators end, then place numbers in the nested sets. Lesson 2 turns repeating decimals back into fractions.
Calculator: no calculator. Nothing here needs a calculator. The figure does the long division, and every question can be answered from the prime factors or by short division.
What students will learn
Know
- A rational number can be written p/q with p and q integers and q not 0; its decimal ends or repeats.
- A fraction in simplest form ends exactly when its denominator has only the prime factors 2 and 5.
- An irrational number has a decimal that neither ends nor repeats; the rational and irrational numbers make the real numbers.
Understand
- Going on forever is not the test for irrational: a repeating block is the test for rational.
- Dividing by b has fewer than b possible remainders, so a decimal must end or repeat.
- The sets nest: natural, integer, rational, real.
Do
- Say whether a fraction's decimal ends or repeats from the prime factors of its denominator.
- Find the length of a repeating block and the number of places of an ending decimal.
- Classify a number as natural, integer, rational, irrational or real, and justify it.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: definitions, and ending against repeating decimals (questions 1 to 6).
- Criterion B. Investigating patterns: the figure. Ask students to tabulate denominators 2 to 16 as ending or repeating and state the rule they find, then test it on 25 and 24.
- Criterion C. Communicating: justifying a classification in words, such as why 0.2666… is rational (questions 6 and 9).
- Criterion D. Applying mathematics in real-life contexts: measures that end or repeat when shared, such as 7/16 of a cup of oil, and why a ruler marked in fractions cannot show √2 exactly (the hook and question 2).
The one thing to do with the figure
Ask for a prediction before the first slide. “Will 1/6 end, or go on forever? What on the bottom tells you?” Take 2 × 3 from the room, then let the figure show the repeating 6.
Then set the numerator to 6 and the denominator to 8 and ask why the figure says 3/4. Finally set 4 and 2: a whole number is a rational number with nothing after the point.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. Include 1/7 and 9/11. |
| 10 min | Key idea: the remainder argument, then the prime factor rule. |
| 10 min | Nested sets and the square root exception; the worked examples and the faded 13/50. |
| 5 min | Question 1, the check. If many choose A, B or D, go back to the definition. |
| 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 | 0.25; 32; no, 91 = 7 × 13. |
|---|---|
| Worked example | 7/40 = 0.175 (ends); 11/30 = 0.3666… (repeats). |
| Faded example | 13/50 = 0.26. |
| 1. Recognising a rational number | C. A and B are irrational numbers; D is described as having no repeating block. |
| 2. Fraction to a terminating decimal | 0.4375. 7/16 = 4375/10000. |
| 3. Which fraction ends | C, 9/32. 12, 30 and 15 all contain a 3. |
| 4. Length of a repeating block | 3. The block is 135. |
| 5. Decimal places from prime factors | 4. 11/80 = 1375/10000 = 0.1375. |
| 6. Going on forever is not irrational | B. A and D treat "never ends" as irrational; C says it ends, which it does not. |
| 7. Counting irrational numbers | 3. √7, π and √12. |
| 8. Counting unit fractions that end | 7. d = 2, 4, 5, 8, 10, 16, 20. |
| 9. A number with no repeating block | A. B repeats, C = 3/2, D is an integer. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Treated any decimal that goes on as irrational, or named a surd as rational. | Ask: is there a block that repeats? Show 1/3 and √2 side by side. |
| 2.2857 in question 2 | Divided the bottom by the top. | Ask which is bigger, 7 or 16, and whether the fraction is more or less than 1. |
| A, B or D in question 3 | Looked at the numerator or ignored the 3, or did not test the denominator's factors. | Write the prime factors of each denominator before choosing. |
| 37 or 6 in question 4 | Counted the denominator or two copies of the block. | Colour the first block only, then count. |
| 3 or 5 or 1 in question 5 | Counted the wrong thing among the prime factors. | Make the bottom a power of 10: how many tens does 80 need to become 10 000? |
| A or D in question 6 | Said that never ending means irrational. | Divide 4 by 15 digit by digit and watch the remainder come back. |
| 4, 5 or 2 in question 7 | Counted a root sign, or a repeating decimal, as irrational; or missed 12 not being a perfect square. | List the perfect squares to 20, then test each root. |
| 12, 6 or 5 in question 8 | Counted every even denominator, or stopped at 10. | Prime-factorize each d and tick only those with 2 and 5 alone. |
Questions worth asking
- Why must a fraction's decimal end or repeat? There are only b − 1 non-zero remainders, so one comes back. Listen for “a remainder repeats”.
- Is 0.999… a rational number? Yes; it is 1 (next lesson). Listen for the block 9.
- Is √2 × √2 irrational? It is 2, which is rational. Irrational numbers can multiply to a rational one.
- Can you list all the irrational numbers? No; there is no list, and between any two numbers there are both kinds.
Differentiation
A way in for anyone stuck
- Give a printed table of the denominators 2 to 16 with the prime factors written in, and highlight the 2s and 5s.
- Start with unit fractions only (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10), then other numerators.
- Let students check an ending decimal by multiplying back: 0.4375 × 16 = 7.
Stretch
- Question 8, then: for which d from 2 to 20 is the repeating block of 1/d longest? (d = 19, with a block of 18 digits; d = 17 has 16.)
- Find two irrational numbers whose sum is rational. (√2 and −√2 give 0.)
Students learning in English as an additional language
“Rational” here comes from “ratio”, a number that is a ratio of two integers. “Irrational” means not a ratio. It does not mean unreasonable. Say “repeats” and “ends” with a gesture.
| Word | Plain meaning | Example |
|---|---|---|
| rational number | a number that can be written as a fraction of two integers | 3/8, −5 and 0.333… |
| irrational number | a number that cannot be written as a fraction; its decimal never ends or repeats | π and √2 |
| real numbers | all the rational and irrational numbers together | every point on the number line |
| terminating decimal | a decimal that ends | 0.375 |
| repeating decimal | a decimal in which a block of digits repeats forever | 0.142857142857… |
| integer | a whole number, positive, negative or zero | −7, 0 and 12 |
| natural number | a counting number; the IB Diploma includes 0, and some courses start at 1 | 0, 1, 2, 3 |
Sentence starters for explaining aloud or in writing:
- This number is rational because …
- The decimal repeats because the denominator has a factor …
- It goes on forever, but it is not irrational because …
Two things not to say
“Irrational means it goes on forever.” Repeating decimals go on forever and are rational.
“All square roots are irrational.” √49 = 7, and √(9/4) = 3/2.
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.
- TerminatingWrite 3/5 as a decimal.
0.6. - RationalWrite 0.5 as a fraction in simplest form.
1/2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA plank is 3/16 of a metre long. Write 3/16 as a decimal.
0.1875. - Earlier work (Year 7, Unit 1, lesson 4)Write 24 as a product of prime factors.
2 × 2 × 2 × 3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWithout dividing, which of 7/12, 7/16 and 7/25 end?
7/16 and 7/25: 16 = 2 × 2 × 2 × 2 and 25 = 5 × 5. 12 has a 3. - Counter-exampleIs the sum of two irrational numbers always irrational?
No: √2 + (−√2) = 0, which is rational.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under it, “Show me the steps” gives the same states in words.
No device? Long division on paper gives every state; the denominator table is a good board task.