--- title: "Prime factors" subtitle: "Number Foundations · Squares, powers and primes · Lesson 7" --- # Prime factors **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Think about what 6, 4 and 9 split into. :::keep-together **Sam starts a factor tree for 36 with $6 \times 6$. Ali starts with $4 \times 9$. When both finish, will they have the same primes?** ☐ Yes, the same primes ☐ Only if they both start with 2 ☐ No: different starts give different primes My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** Yes. 6 × 6 → 2 × 3 × 2 × 3, and 4 × 9 → 2 × 2 × 3 × 3: both are 2² × 3². A number has only one prime factorisation. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## All the way down :::keep-together ### Split, then write with exponents Here is a tree for 60, written as working. 1. $60 = 6 \times 10$ — Start with any factor pair. 2. $6 = 2 \times 3, \quad 10 = 2 \times 5$ — Split every branch that is not prime. 3. $60 = 2 \times 2 \times 3 \times 5$ — Every branch is prime: collect them, smallest first. 4. $60 = 2^2 \times 3 \times 5$ — Write repeated primes with exponents. Check: 4 × 3 × 5 = 60. Now try one: which is the prime factorisation of 60? **What is the prime factorisation of 60?** ☐ $2 \times 3 \times 10$ ☐ $2 \times 30$ ☐ $4 \times 15$ ☐ $2^2 \times 3 \times 5$ ::: :::teacher-note **Answer:** $2^2 \times 3 \times 5$ 60 = 6 × 10 = 2 × 3 × 2 × 5 = 2² × 3 × 5. Check: 4 × 3 × 5 = 60. ::: :::keep-together ### Find the mistake **Leo writes 36 as a product of prime factors. Click the line where his reasoning goes wrong.** ``` 1 Write 36 as a product of prime factors. 2 36 = 4 × 9 3 4 and 9 are factors of 36, so the tree is finished. 4 36 = 4 × 9 ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: 4 and 9 are not prime, so the tree is not finished. 4 = 2 × 2 and 9 = 3 × 3, so 36 = 2² × 3². ::: :::keep-together ### Primes only **Why is $2 \times 2 \times 9$ not the prime factorisation of 36?** ☐ The 2s should be written first ☐ 9 is not prime: it splits into 3 × 3 ☐ It does not equal 36 ::: :::teacher-note **Answer:** 9 is not prime: it splits into 3 × 3 Every factor must be prime: 9 = 3 × 3, so 36 = 2² × 3². ::: :::typical ## One change at a time **Seventy-two.** Each task changes one thing. **Back to the number.** **A prime.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Prime factorisations :::keep-together ### Ninety **What is the prime factorisation of 90?** ☐ $2 \times 3 \times 15$ ☐ $2 \times 5 \times 9$ ☐ $2 \times 3^2 \times 5$ ☐ $9 \times 10$ ::: :::teacher-note **Answer:** $2 \times 3^2 \times 5$ 90 = 9 × 10 = 3 × 3 × 2 × 5 = 2 × 3² × 5. ::: :::keep-together ### Back to the number **Which number has the prime factorisation $2^3 \times 5$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 40 2³ = 8, and 8 × 5 = 40. ::: :::keep-together ### One hundred **What is the prime factorisation of 100?** ☐ $2 \times 5^2$ ☐ $4 \times 25$ ☐ $2^2 \times 5^2$ ☐ $10^2$ ::: :::teacher-note **Answer:** $2^2 \times 5^2$ 100 = 10 × 10 = 2 × 5 × 2 × 5 = 2² × 5². ::: :::keep-together ### Review: powers **Review: work out $3^4$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 81 3 × 3 × 3 × 3 = 81. ::: ## Make: a tree for 360 Then compare with someone who started differently. Check by expanding: $8 \times 9 \times 5$. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Write 84 as a product of prime factors, with exponents.** {{lines-3}} ::: :::teacher-note **Answer:** 84 = 2² × 3 × 7. :::