At a glance. One lesson of about 60 minutes. Students list factors and multiples, define primes by counting factors, explain why 1 is not prime and use divisibility tests.
Calculator: no calculator. Nothing needs one. Spotting factors and applying divisibility rules by hand is the skill being built.
What students will learn
Know
- A factor divides a number exactly; a multiple is the number multiplied by a whole number.
- A prime has exactly two factors; a composite has more than two; 1 is neither.
- The divisibility tests for 2, 3, 4, 5, 6, 8, 9 and 10.
Understand
- Factors come in pairs, and each pair is a rectangle with that area.
- Divisibility tests work because of place value: they look at the part of a number that decides the remainder.
- Primes are the building blocks of every whole number above 1.
Do
- List all factors of a number in pairs.
- Decide whether a number is prime and justify it.
- Apply divisibility tests to find factors quickly.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: factors, multiples and primes in familiar and unfamiliar numbers (questions 1 to 6).
- Criterion B. Investigating patterns: the rectangle figure. Ask which numbers up to 60 have exactly one rectangle and look for a pattern in the primes.
- Criterion C. Communicating: justifying why a number is or is not prime, with a factor pair (questions 1 and 9).
- Criterion D. Applying mathematics in real-life contexts: arranging objects in equal rows and groups (the hook and the extra questions).
The one thing to do with the figure
Ask for the count first. “How many rectangles can 12 squares make?” Take answers, then move the slider. Then ask which numbers in the 50s have only one rectangle.
Finish on 36: one of its rectangles is a square, and that is why square numbers have an odd number of factors. Name it, because it comes back with square roots and with prime factors.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the rectangle figure, with predictions first. |
| 10 min | Key idea; the divisibility figure. Students find the numbers between 100 and 200 that pass the most tests (120 and 180 pass seven each). |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. Hands up for each letter. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 42 and 7; 144; 5³ = 125. |
|---|---|
| Worked example | 91 = 7 × 13, composite. |
| Faded example | 89 is prime. |
| 1. Spot the prime | C, 29. A, B and D are all multiples of 3 (digit sums 3, 9 and 12). |
| 2. Counting factors | 6: 1, 2, 3, 6, 9, 18. |
| 3. Next prime | 23. 21 = 3 × 7 and 22 = 2 × 11. |
| 4. Next multiple | 102. 6 × 17. |
| 5. Divisible by 6 | C. A passes the test for 2 only, B and D pass the test for 3 only. |
| 6. Prime riddle | 37. Primes 31, 37, 41, 43, 47; digit sums 4, 10, 5, 7, 11. |
| 7. Missing digit | 7. 7 + 7 + 4 = 18, and 774 ÷ 9 = 86. |
| 8. Six factors | 12. Factors 1, 2, 3, 4, 6, 12; no smaller number has six. |
| 9. Is 1 prime? | B. A and D reason from “divisible by 1 and itself” without counting factors. C confuses parity with primality. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Treats numbers ending in 1, 7, 9 as prime by looks. | Ask for a factor pair; 21 = 3 × 7 and 27 = 3 × 9. |
| 3 or 4 in question 2 | Counted pairs, or left out 1 and the number. | List the factors in a row, then circle each one. |
| 21 or 22 in question 3 | Not testing for a factor. | Ask for a factor pair for each. |
| 96 or 100 in question 4 | Used the nearest multiple, or the number itself. | List multiples of 6 around 100. |
| A, B or D in question 5 | Used only one of the tests for 2 and 3. | Ask which tests a number passes: 2 and 3 are both needed. |
| 46 in question 6 | Forgot to check that the number is prime. | Ask them to find a factor of 46. |
Questions worth asking
- Is 2 prime? Is it odd? Yes it is prime; no it is not odd. It is the only even prime.
- Why isn't 1 prime? Because it has only one factor, not two. Listen for “exactly two factors”.
- Why does the digit-sum test work? Because 10, 100 and 1000 each leave remainder 1 when divided by 3 or 9, so only the digits matter. This is the full proof for stronger students.
- Is every multiple of 4 a multiple of 2? Yes: 4 = 2 × 2, so anything divisible by 4 is divisible by 2.
Differentiation
A way in for anyone stuck
- Give squared paper so each factor pair can be drawn as a rectangle.
- Give a list of primes to 50 for the first half of the lesson.
- Use one divisibility test at a time before combining them for 6.
Stretch
- Question 7, then: why does a number with digit sum 9 always give a multiple of 9, however long it is? (Each place leaves remainder 1.)
- Find the smallest number with exactly 8 factors. (24.)
Students learning in English as an additional language
“Factor” and “multiple” are often mixed up. Pair each with its example: 3 is a factor of 12; 12 is a multiple of 3.
| Word | Plain meaning | Example |
|---|---|---|
| factor | a whole number that divides another exactly | the factors of 6 are 1, 2, 3, 6 |
| multiple | what you get when you multiply by a whole number | the multiples of 6 are 6, 12, 18 |
| prime number | a number with exactly two factors, 1 and itself | 2, 3, 5, 7, 11 |
| composite number | a number with more than two factors | 12 is composite |
| divisible | can be divided exactly, with no remainder | 12 is divisible by 4 |
| digit sum | the sum of the digits of a number | the digit sum of 138 is 12 |
Sentence starters for explaining aloud or in writing:
- … is prime because its only factors are …
- … is composite because … × … = …
- … is divisible by 3 because its digit sum is …
Two things not to say
“A prime is a number that cannot be divided.” Every number divides by 1 and itself. A prime cannot be divided by anything else.
“Check up to half the number.” Only the square root is needed, which is a useful link to the last lesson.
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.
- FactorsList all the factors of 30.
1, 2, 3, 5, 6, 10, 15, 30. - Earlier workWhat is the square root of 81?
9.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- GroupsA teacher has 36 students and wants equal groups of at least 2 and at most 12. Which group sizes work?
2, 3, 4, 6, 9 and 12. - RhythmA drum sounds every 4 counts and a bell every 6 counts. On which counts up to 24 do they sound together?
12 and 24.
3Challenge
Reasoning, generalising and working backwards.
- GoldbachFind all the ways to write 30 as the sum of two primes.
7 + 23, 11 + 19 and 13 + 17. - ProofShow that 100a + 10b + c is a multiple of 3 exactly when a + b + c is.
100a + 10b + c = 99a + 9b + (a + b + c), and 99a and 9b are multiples of 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use counters or squared paper for the rectangles.