--- title: "Powers" subtitle: "Number Foundations · Squares, powers and primes · Lesson 6" --- # Powers **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Fold a sheet of paper in half, then again, and again. After 7 folds it is 2⁷ = 128 layers thick. Powers grow fast, and the short way to write them is exponent notation. :::teacher-note **In one sentence:** Students grow factor trees for powers of 2 to see repeated equal factors, read and write exponent notation (base and exponent), evaluate powers, and meet powers of 10 (AC9M7N02, node 1). **Students should be able to say:** I can write a repeated multiplication as a power, and expand a power. I can say what the base and the exponent mean. I can work out powers, including powers of 10. **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 How many 2s are multiplied in $2^3$? :::keep-together **What is $2^3$?** ☐ 5 ☐ 6 ☐ 8 ☐ 23 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 8. 2³ = 2 × 2 × 2 = 8: three 2s multiplied. 6 is 2 × 3. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Base and exponent :::keep-together ### Read the power Here is how to go between the two forms. 1. $2 \times 2 \times 2 \times 2 \times 2 = 2^5$ — Count the equal factors: five 2s, so the exponent is 5. 2. $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$ — Expand and multiply: 4, 8, 16, 32. 3. $3 \times 3 \times 5 = 3^2 \times 5$ — Different bases get their own powers. 4. $10^4 = 10\,000$ — Powers of 10: the exponent counts the zeros. Now try one: $3^4$. **Work out $3^4$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 81 3 × 3 = 9, × 3 = 27, × 3 = 81. ::: :::keep-together ### Find the mistake **Mia works out $2^5$. Click the line where her working goes wrong.** ``` 1 Work out 2⁵. 2 The base is 2 and the exponent is 5. 3 2⁵ = 2 × 5 4 = 10 ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: 2⁵ is five 2s multiplied, 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5. ::: :::keep-together ### Match them **Match each power to what it means.** A. 3 × 3 = 9 B. 10 × 10 × 10 = 1000 C. 2 × 2 × 2 = 8 D. just 5 {{columns 60% 40%}} {{rows 1}} | Part | Letter | |---|---| | `$10^3$` | | | `$3^2$` | | | `$2^3$` | | | `$5^1$` | | ::: :::teacher-note **Answer:** $10^3$ → B. 10 × 10 × 10 = 1000; $3^2$ → A. 3 × 3 = 9; $2^3$ → C. 2 × 2 × 2 = 8; $5^1$ → D. just 5 The exponent counts the factors. 10³ is 1 followed by three zeros; 5¹ is one 5. ::: :::typical ## One change at a time **Thirty-two.** Each task changes one thing. **Write the power.** **Two bases.** **Swap them.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Powers :::keep-together ### Evaluate **Work out $4^3$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 64 4 × 4 × 4 = 64. ::: :::keep-together ### Powers of 10 **Work out $10^5$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 100000 1 followed by five zeros: 100 000. ::: :::keep-together ### Write it **Write $2 \times 2 \times 5 \times 5 \times 5$ with exponents.** ☐ $4 \times 15$ ☐ $2^3 \times 5^2$ ☐ $10^5$ ☐ $2^2 \times 5^3$ ::: :::teacher-note **Answer:** $2^2 \times 5^3$ Two 2s and three 5s: 2² × 5³ = 4 × 125 = 500. ::: :::keep-together ### Review: roots **Review: work out $\sqrt{144}$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 12 × 12 = 144. ::: ## Make: 64 three ways Then write 64 as a power of 2, a power of 4 and a power of 8. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Work out 3³ and write 5 × 5 × 5 × 5 as a power.** {{lines-3}} ::: :::teacher-note **Answer:** 3³ = 27; 5⁴. :::