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).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min
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.
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.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student stops at 4 or 9 (Leo's mistake). | Is every circled number prime? Can 4 be split? | Keep splitting until each branch has only 1 and itself as factors. |
| A student writes 1 in the factorisation. | Is 1 prime? Does × 1 change anything? | Leave 1 out: it is not prime. |
| A student writes 2 × 2 × 3 × 3 but not 2² × 3². | How many 2s? How many 3s? | Count each prime and write the count as its exponent. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
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.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Prime factors". The idea: Prime factorisation via factor trees: I can express any natural number as a product of prime factors using a factor tree, and write the result using exponent notation. What understanding looks like: - Produces a correct factor tree and writes the result in exponent notation. - Verifies the factorisation by expanding and multiplying. - Understands that the prime factorisation of a number is unique. Mistakes students often make here: - A student stops at 4 or 9 (Leo's mistake). - A student writes 1 in the factorisation. - A student writes 2 × 2 × 3 × 3 but not 2² × 3². How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - If I seem to be missing the idea that comes before this one (Exponent notation for repeated multiplication), check it quickly, then come back. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Prime factors".
Write 84 as a product of prime factors, with exponents.
84 = 2² × 3 × 7.
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).