Split a number into factor pairs until every branch is a prime: 36 = 2 × 2 × 3 × 3 = 2² × 3². Every tree for 36 ends with the same primes.
Every whole number above 1 is built from primes in exactly one way, like a code. 60 is 2² × 3 × 5, and no other number has that code.
A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, … (1 is not prime.) Every whole number above 1 that is not prime can be split into primes. Writing it that way, with exponents, is its prime factorisation:
36=2×2×3×3=22×32
A factor tree finds it: split the number into any factor pair, then split each branch that is not prime, until every branch ends in a prime.
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.
Two trees for 36
Think about what 6, 4 and 9 split into.
TryTry it on the board and compare the result with your prediction.
Grow a tree for 36
Prime branches are circled. The board writes the product with exponents when the tree is finished.
Try it yourselfYour edits are saved in this browser.
Grow a factor tree for 36 until every branch is a prime. Then press Start again and try a different first split. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
All the way down
Here is a tree for 60, written as working.
Worked exampleBefore each step, say what you think comes next.
60=6×10Start with any factor pair.
6=2×3,10=2×5Split every branch that is not prime.
60=2×2×3×5Every branch is prime: collect them, smallest first.
Now try one: which is the prime factorisation of 60?
Read Leo's working. Is every branch a prime?
Leo writes 36 as a product of prime factors. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Start with 8 × 9, or 2 × 36.
Try it yourselfYour edits are saved in this browser.
Grow a tree for 72. Write its prime factorisation.
Use the board. This ticks when you get there.
Which number has the prime factorisation 22×5×7?
PractiseA few on your own, to check you have it.
Prime factorisations
Which number has the prime factorisation 23×5?
Review: work out 34.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A tree for 360
Then compare with someone who started differently. Check by expanding: 8×9×5.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Start with 36 × 10, or 6 × 60, or 2 × 180.
Another hint
Keep going until every branch is circled.
Grow a tree for 360, starting with any factor pair you like.
Use the board. This ticks when you get there.
For teachers
Students grow factor trees on the board until every branch is prime, write the prime factorisation in exponent notation, check it by expanding, and see that every tree for a number ends with the same primes (AC9M7N02, node 2). About 40 minutes.
I can use a factor tree to write a number as a product of primes.
I can write the prime factorisation with exponents and check it by expanding.
I can explain why different trees for a number give the same primes.
Before class
Do the Run board yourself first, twice, with different first splits. Students need factor pairs, primes (Year 6) and exponent notation (previous lesson).
The one idea: split until every branch is prime; the primes are the same whichever way you split (uniqueness, informally). The common mistake is stopping at composite branches (Leo's mistake).
This is the hinge for HCF and LCM in the next lesson: the method depends on there being one factorisation to compare.
Running the lesson
Predict: take a show of hands; then have half the class start 36 with 6 × 6 and half with 4 × 9.
Run: collect the final products. Are they the same?
Investigate: insist on checking by expanding every time.
Modify: for 13, ask what a factor tree for a prime looks like.
Make: compare the 360 trees across the room: different shapes, same primes.
The hard part (on purpose)
Knowing when a tree is finished. Ask students to check each end of a branch for factors other than 1 and itself, rather than circling the primes for them.
When students get stuck
A student stops at 4 or 9 (Leo's mistake). Ask: Is every circled number prime? Can 4 be split? Fix: Keep splitting until each branch has only 1 and itself as factors.
A student writes 1 in the factorisation. Ask: Is 1 prime? Does × 1 change anything? Fix: Leave 1 out: it is not prime.
A student writes 2 × 2 × 3 × 3 but not 2² × 3². Ask: How many 2s? How many 3s? Fix: Count each prime and write the count as its exponent.
Stretch and support
Stretch: Which numbers below 100 have only one prime in their factorisation (like 32 = 2⁵)? What do their trees look like?
Support: Use the board for every factorisation, and keep a list of primes to 30 visible.
Where it leads
Next: HCF and LCM (node 3), comparing two factorisations.
Where it leads: simplifying fractions and finding common denominators by primes, and simplifying surds (Year 10).
What each step is for
two trees for 36: Committing to whether two different first splits give the same primes surfaces the belief that a prime factorisation depends on the tree.
grow a tree for 36: Growing a tree for 36 with any first split ends in 2² × 3², which tests the prediction.
all the way down: A tree is finished only when every branch is prime; Leo stops at composite branches. Any route gives the same primes, so a factorisation can be checked by expanding.
one change at a time: One change at a time: a bigger tree, expanding a factorisation back to its number, then a prime as the edge case (and why 1 is left out).
prime factorisations: Finding, reading and checking factorisations, mixed; stopping at a composite is the planted slip; the review keeps powers fresh.
a tree for 360: A larger number, with a first split of the student's choosing, shows the factorisation is the same whatever the route.
Now you can
Use a factor tree to write a number as a product of primes.
Write the prime factorisation with exponents and check it by expanding.
Explain why different trees for a number give the same primes.