Put two prime factorisations side by side. The primes they share multiply to the highest common factor; all the primes (shared ones once) multiply to the lowest common multiple.
One bus leaves every 12 minutes, another every 18. Both leave at 8:00. When do they next leave together? Primes answer it without listing every time.
The highest common factor (HCF) of two numbers is the biggest number that divides both. The lowest common multiple (LCM) is the smallest number both divide into. You can find them by listing, but prime factorisations are quicker for big numbers:
HCF: multiply the primes the two numbers share.
LCM: multiply all the primes, counting each shared one once.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
The HCF of 12 and 18
List the factors of each, or think about their primes.
TryTry it on the board and compare the result with your prediction.
Two trees
When both trees are finished, a diagram sorts their primes.
Try it yourselfYour edits are saved in this browser.
Grow trees for 36 and 48. Which primes are in the middle of the diagram? Then use the same method on 12 and 18: was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Shared and all
Here is the method for 12 and 18.
Worked exampleBefore each step, say what you think comes next.
12=22×3,18=2×32Write both prime factorisations.
HCF=2×3=6Shared primes: each prime in both, to its lower power.
LCM=22×32=36All the primes, each to its higher power (shared ones once).
6×36=216=12×18A check: HCF × LCM = the two numbers multiplied.
Now try one: the HCF of 24 and 36.
Read Zoe's working. Is 24 a multiple of 6 and of 8?
Your working can be short notes. Then choose the reason for the key step.
Find the HCF of 24 and 36.
Zoe finds the LCM of 6 and 8. Click the line where her working goes wrong.
Bus A leaves every 12 minutes and bus B every 18 minutes. Both leave at 8:00. After how many minutes do they next leave together? Write each step with its reason.
Your working
A model solution
12=22×3 and 18=2×32prime factorisations, from factor trees
They leave together at common multiples of 12 and 18 each bus leaves at multiples of its own time
LCM =22×32=36the lowest common multiple: all the primes, shared ones once
Check your own working
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Finish both trees, then read the middle of the diagram.
Try it yourselfYour edits are saved in this browser.
Grow trees for 24 and 60. Write the HCF and the LCM.
Use the board. This ticks when you get there.
Find the LCM of 4 and 6.
24 apples and 36 pears are packed into identical bags, with nothing left over. What is the greatest number of bags?
PractiseA few on your own, to check you have it.
HCF and LCM
Find the LCM of 4 and 10.
Find the HCF of 18 and 30.
To add 61+81, both fractions need the same denominator. What is the smallest one that works?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
HCF 6, LCM 36
Check: grow trees for your two numbers on paper and compare their primes.
Need a hint?
6 = 2 × 3 and 36 = 2² × 3². Both numbers contain 2 × 3.
Another hint
One number takes the extra 2, the other the extra 3.
Find two numbers, not 6 and 36, whose HCF is 6 and whose LCM is 36. Write how you found them. Then type the smaller of your two numbers.
Your working
A model solution
6=2×3 and 36=22×32prime factorisations of the HCF and the LCM
Both numbers contain 2×3; one more 2 and one more 3 are shared out the HCF is what both share; the LCM has everything
22×3=12 and 2×32=18giving both extras to one number gives 6 and 36
12×18=216=6×36HCF × LCM is the two numbers multiplied
Check your own working
For teachers
Students grow two factor trees, sort the primes in a Venn diagram, and find the HCF (shared primes) and LCM (all the primes) of two numbers, then choose between them in context (AC9M7N02, node 3). About 45 minutes.
I can find the HCF and LCM of two numbers from their prime factorisations.
I can explain why the method works.
I can decide whether a problem needs the HCF or the LCM.
Before class
Do the Run board yourself first. Students need prime factorisation with exponents (previous lesson).
The one idea: shared primes give the HCF; all the primes, shared ones once, give the LCM. The common mistake is taking the product as the LCM (Zoe's mistake), which counts shared primes twice.
Listing factors and multiples is a fine check, but this lesson aims at the prime method, which the next unit uses for common denominators.
Running the lesson
Predict: students who chose 2 or 3 found a common factor but not the highest.
Run: ask what is in the middle of the diagram, and why it divides both numbers.
Investigate: for the buses, ask what would happen at 6 minutes (the HCF).
Modify: for the bags, ask how many apples and pears go in each bag.
Make: compare methods; some students list multiples of 6 up to 36.
The hard part (on purpose)
Deciding whether a problem needs the HCF or the LCM. Ask whether the answer should be smaller than the numbers (sharing) or bigger (lining up again) before any calculation.
When students get stuck
A student multiplies the two numbers for the LCM (Zoe's mistake). Ask: Is there a smaller number both divide into? Fix: Count each shared prime once: all the primes, not both lists.
A student gives a common factor that is not the highest. Ask: Does a bigger number also divide both? Fix: Multiply all the shared primes, not just one.
A student cannot tell whether a problem needs the HCF or the LCM. Ask: Is the answer smaller than the numbers (sharing out) or bigger (meeting again)? Fix: Sharing into equal groups: HCF. Events lining up: LCM.
Stretch and support
Stretch: Check that HCF × LCM = the two numbers multiplied for three pairs. Explain why with the Venn diagram.
Support: Use the board's Venn diagram for every pair, and list factors or multiples as a check.
Where it leads
Next: the fractions unit, where the LCM of two denominators is the lowest common denominator (AC9M7N06).
Where it leads: simplifying fractions in one step (dividing by the HCF), and algebraic HCFs when factorising.
What each step is for
the HCF of 12 and 18: Committing to an HCF shows whether they want the highest common factor or any common factor, and whether they confuse it with a multiple.
two trees: Finishing both trees sorts the primes into shared and not shared, and the board multiplies them into the HCF and the LCM.
shared and all: Shared primes give the HCF and all the primes the LCM; Zoe multiplies the numbers, which counts the shared 2 twice. Context decides which one a problem needs.
one change at a time: One change at a time: a new pair on the board, an LCM, an HCF in context, then numbers that share no primes.
HCF and LCM: HCF, LCM and a common denominator, mixed; the product as the LCM is the planted slip; the review keeps prime factorisation fresh.
HCF 6, LCM 36: Working backwards from an HCF and an LCM to the two numbers uses the primes in both directions, and the product check (HCF × LCM) ties them together.
Now you can
Find the HCF and LCM of two numbers from their prime factorisations.
Explain why the method works.
Decide whether a problem needs the HCF or the LCM.