At a glance. One lesson of about 60 minutes. Students use prime factorization to find the greatest common factor and lowest common multiple, and decide which one a context needs.
Calculator: no calculator. Nothing needs one. Reading primes off a factor tree is the skill, and a calculator would not show why the answers work.
What students will learn
Know
- The definitions of GCF and LCM.
- GCF is the product of the shared primes; LCM uses every prime at its greatest count.
- GCF × LCM = a × b for two numbers.
Understand
- Common factors are divisors of both numbers; common multiples are numbers both divide into.
- Which one a problem needs depends on whether a quantity is being shared out or lined up.
- The prime factorization of a number shows its factors and multiples.
Do
- Find the GCF and LCM by listing and by prime factors.
- Choose GCF or LCM for a context.
- Use GCF × LCM = a × b to find a missing number.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: GCF and LCM for pairs of numbers (questions 1 to 5, 7).
- Criterion B. Investigating patterns: the Venn figure. Ask students to look at several pairs and discover GCF × LCM = a × b.
- Criterion C. Communicating: justifying a choice between GCF and LCM in words (questions 6 and 9).
- Criterion D. Applying mathematics in real-life contexts: schedules and cutting materials (questions 5 and 6).
The one thing to do with the figure
Ask for a prediction about the pattern. “For 12 and 18 the GCF is 6 and the LCM is 36. What happens if you multiply them?” Let students find 216 and notice it equals 12 × 18, then test it on three more pairs of their own.
On the timeline figure, ask why every common multiple is a multiple of the LCM, so the meetings are evenly spaced. That is the structure behind “when do they next meet”.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, prediction and the Venn figure. |
| 10 min | Key idea; the timeline figure. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 2² × 3; 1, 2, 3, 6, 9, 18; 4, 8, 12, 16, 20. |
|---|---|
| Worked example | GCF 12, LCM 180. |
| Faded example | GCF 8, LCM 280. |
| 1. GCF of 12 and 18 | C, 6. A and B are common factors but not the greatest; D is the LCM. |
| 2. GCF by listing | 4. Shared factors 1, 2, 4. |
| 3. LCM by listing | 12. 4 × 3 = 6 × 2. |
| 4. GCF from primes | 12. 2² × 3. |
| 5. Buses | 90 minutes. LCM of 18 and 30. |
| 6. GCF or LCM? | A, GCF 24 cm. B: LCM, wrongly chosen; the pieces must divide both lengths. C: a common factor, not the greatest. D: right value, wrong name. |
| 7. GCF of larger numbers | 42. 2 × 3 × 7. |
| 8. Missing number | 24. 6 × 72 = 432, and 432 ÷ 18 = 24. |
| 9. LCM is not the product | B. A and D use the product; C names the GCF. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or B in question 1 | A common factor, but not the greatest. | List all common factors and ask which is greatest. |
| 24 in question 3 | Multiplied the numbers. | List multiples of the larger one until one divides by the smaller. |
| 6 in question 5 | Found the GCF for a multiple question. | Ask whether the answer should be bigger or smaller than 30. |
| B or C in question 6 | Chose the wrong word or a non-greatest factor. | Ask: should the pieces be longer than the ribbon? |
| 14 in question 7 | Missed a shared prime. | Write both in index form and compare exponents. |
| 4 or 432 in question 8 | Divided 72 by 18, or stopped at the product. | Test with 4 and 6: 2 × 12 = 24 = 4 × 6. |
Questions worth asking
- Is the GCF ever bigger than the smaller number? No: a factor of a number cannot be bigger than it.
- Why do the buses meet every 12 minutes, not just once? Because every multiple of 12 is a multiple of both 4 and 6.
- How do you know when a problem needs GCF and when LCM? GCF: splitting into equal pieces. LCM: events that repeat until they line up.
- When is the LCM just the product? When the numbers share no prime, as with 3 and 5.
Differentiation
A way in for anyone stuck
- Let students list factors and multiples before they use primes.
- Colour the primes on a tree: shared primes in one colour.
- Give the question with small numbers first, such as the ribbons 6 cm and 9 cm.
Stretch
- Question 8, then: find every pair of numbers with GCF 6 and LCM 72. (6 and 72, and 18 and 24: both multiply to 432.)
- Show that two consecutive whole numbers always have GCF 1. (A common factor must divide their difference, 1.)
Students learning in English as an additional language
“Common” here means shared, not usual. Show the overlap of the Venn diagram when you say it.
| Word | Plain meaning | Example |
|---|---|---|
| common factor | a factor that two numbers share | 4 is a common factor of 8 and 12 |
| common multiple | a number that is a multiple of both | 12 and 24 are common multiples of 4 and 6 |
| greatest common factor (GCF) | the biggest common factor; also called the highest common factor (HCF) | the GCF of 8 and 12 is 4 |
| lowest common multiple (LCM) | the smallest common multiple | the LCM of 4 and 6 is 12 |
| Venn diagram | overlapping circles that sort what is shared | shared primes sit in the overlap |
Sentence starters for explaining aloud or in writing:
- The GCF of … and … is … because they share …
- The LCM is … because both … and … go into it.
- I need the … because the problem asks for …
Two things not to say
“LCM is multiply, GCF is divide.” Neither is a single operation. Say what each one is: the largest shared factor and the smallest shared multiple.
“Just use the bigger answer.” GCF and LCM answer different questions; the size of the answer follows from the question.
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.
- GCFWhat is the GCF of 16 and 40?
8. - Earlier workWrite 90 as a product of primes in index form.
2 × 3² × 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- BellsTwo bells ring together at noon. One rings every 12 minutes, the other every 20. When do they next ring together?
60 minutes later. - TilesA floor is 60 cm by 84 cm. What is the largest square tile, with whole-number sides, that tiles it exactly with none cut?
12 cm, the GCF.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningExplain why two consecutive whole numbers always have GCF 1.
A common factor divides their difference, which is 1, so it is 1. - Missing numbersFind two numbers with GCF 5 and LCM 60.
15 and 20 (or 5 and 60): 5 × 60 = 300 = 15 × 20.
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? Draw two overlapping circles on the board and let students place prime cards.