Task 10
Multiples and Lowest Common Multiple
Use single-colour trains to model multiples, find where different sequences first meet, and connect this to an efficient method for finding the LCM.
Mathematical focus — and assigning number values
A multiple of a number is the result of multiplying that number by a whole number.
A common multiple is a number that is a multiple of two or more numbers.
The lowest common multiple (LCM) is the smallest positive number that is a multiple of each of those numbers.
For this activity, let white represent 1. The rods therefore take the numerical values white = 1, red = 2, light green = 3, pink = 4, yellow = 5, dark green = 6, black = 7, tan = 8, blue = 9 and orange = 10.
Teacher note: Earlier work deliberately keeps the rods relational and avoids fixing number values too soon. Here we assign values on purpose so that repeated single-colour trains connect directly to times tables, multiples and divisibility.
A train made from one colour can model the multiples of that rod's value. When trains of different colours finish at the same point, they represent a common multiple. The first point at which they line up represents the LCM.
We do — Find LCM(4, 5)
Put out six pink rods in one long train.
Read the joins as multiples of 4:
4, 8, 12, 16, 20, 24
Now place yellow rods alongside the pink train, one at a time. Say the cumulative value each time:
5, 10, 15, 20
At 20 the fourth yellow finishes exactly where the fifth pink finishes.
5p = 4y
5 × 4 = 4 × 5 = 20
So 20 is a multiple of both 4 and 5. There is no smaller positive number that is divisible by both 4 and 5.
LCM(4, 5) = 20
Connect to the formal method: start with the larger number and work through its multiples: 5, 10, 15, 20… At each stage ask, “Is this divisible by 4?” Stop at the first value for which the answer is yes.
We do — Why not just multiply? LCM(4, 6)
Make a reference train using six pink rods:
4, 8, 12, 16, 20, 24
Place dark green rods alongside it:
6, 12
The trains first line up at 12, so:
LCM(4, 6) = 12
Multiplying 4 × 6 gives 24. Is 24 a common multiple? Yes. Is it the lowest common multiple? No. The rods have already lined up at 12.
Teacher discussion: Multiplying two denominators will always give a common denominator, but it may be unnecessarily large. That creates larger numbers to work with and often more simplifying afterwards. Finding the LCM keeps later numerical — and eventually algebraic — fraction work as economical as possible.
Your turn — Three numbers
Use the rods to find:
LCM(3, 4, 6)
Build trains using light green, pink and dark green rods.
- Find the first length at which all three trains line up.
- Record the equivalent trains using colour letters.
- Record the multiples that helped you find the answer.
- Write the answer using LCM notation.
- Now explain how you could find the answer without the rods.
Show answers
The first common length is 12:
4g = 3p = 2d
LCM(3, 4, 6) = 12
Without rods, start with multiples of the largest number: 6, 12… Check each one for divisibility by both 3 and 4. Six is not divisible by 4; 12 is divisible by both.
Feedback / teacher prompts
- Which number's multiples did you choose to work through?
- What did you need to check each time?
- How do the rods support the formal method rather than replace it?
- At what point could the rods be removed?
Your turn — Find the LCM
For each pair, find the lowest common multiple.
Before you build: predict the answer, or work it out mentally if you can.
Then use the rods to model your answer. Build the multiples and show where the two trains first line up. A useful scaffold is to start with a train of six of the smaller-valued rod, then place the larger-valued rod alongside it one at a time. Add more rods if you need them.
Record the multiples you used and write the LCM.
- 2 and 3
- 3 and 5
- 2 and 4
- 3 and 6
- 6 and 8
- 6 and 9
Look again: These six questions were not chosen at random. Can you sort them into three groups? What is similar about the questions in each group?
Show answers
Answers: LCM(2,3)=6; LCM(3,5)=15; LCM(2,4)=4; LCM(3,6)=6; LCM(6,8)=24; LCM(6,9)=18.
Group 1 — the LCM is the product: 2 and 3; 3 and 5. The numbers have no common factor greater than 1.
Group 2 — one number is already a multiple of the other: 2 and 4; 3 and 6. The larger number is the LCM.
Group 3 — the numbers share a factor but neither is a multiple of the other: 6 and 8; 6 and 9. Multiplying gives a common multiple, but not the lowest one.
Feedback / teacher prompts
- Which questions could you solve without building?
- When was the LCM simply the larger number?
- When was the LCM the product?
- When was neither of those things true?
- What knowledge of times tables and divisibility makes the search efficient?
- Why should these patterns be noticed rather than taught as three separate rules?
What could you change?
- Give three numbers instead of two.
- Ask pupils to predict the LCM before building.
- Give an LCM and ask pupils to find possible pairs of numbers.
- Show two trains and ask pupils to identify the numbers and their LCM.
- Give an incorrect LCM and ask pupils to prove it is wrong.
- Withdraw the rods and use them only to check an answer.
Exit strategy: the general method is not “multiply the two numbers”. Work through the multiples of one number — usually the larger — and stop at the first one divisible by the other number or numbers.
Rehearse the teaching
For a task worth practising, work in pairs. One person teaches while the other responds as a pupil using the rods. Model enough to get the pupil started, then stop explaining and question their thinking. Swap roles.
Reflect: What was harder to explain than you expected? Which wording or question worked well?