--- title: "Choosing a strategy" subtitle: "Number Foundations · Fractions and decimals · Lesson 15" --- # Choosing a strategy **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Two students work out 25% of 48. One writes a long multiplication; the other halves twice in their head. Both get 12. Which would you rather do in a shop? :::teacher-note **In one sentence:** Students compare strategies for calculations with fractions, decimals and percentages: changing the order (any order for multiplying), changing the form (0.75 = 3/4), and splitting (the distributive law), with an estimate to check (AC9M7N06, node 4). **Students should be able to say:** I can choose a quick strategy for a calculation and explain why it is quick. I can use any order for multiplying, a friendly fraction, or splitting a number. I can estimate to check that my answer is about right. **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 What fraction is 25%? :::keep-together **What is the quickest way to work out 25% of 48?** ☐ Work out 0.25 × 48 as a long multiplication ☐ Work out 48 × 25, then divide by 100 ☐ They are all as quick ☐ Halve 48, then halve again My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** All three give 12, but halving twice is quickest: 25% is a quarter, and 48 halves easily (24, then 12). ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Three moves :::keep-together ### Three moves Here are the three moves, each with a check. 1. $4 \times 2.7 \times 25 = (4 \times 25) \times 2.7 = 270$ — Change the order: 4 × 25 = 100. Check: 2.7 is about 3, and 100 × 3 = 300. 2. $0.75 \times 36 = \tfrac{3}{4} \times 36 = 27$ — Change the form: 36 ÷ 4 = 9, × 3 = 27. Check: a bit less than 36. 3. $12 \times 2.5 = 12 \times 2 + 12 \times 0.5 = 24 + 6 = 30$ — Split a number: 2.5 = 2 + 0.5. Check: between 12 × 2 and 12 × 3. Your working can be short notes. Then choose the reason for the key step. **Work out $4 \times 2.7 \times 25$ without a calculator. Write your steps.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $4 \times 25 = 100$ (multiplying can be done in any order, so pair the 4 and the 25); $100 \times 2.7 = 270$ (multiplying by 100 makes each digit 100 times bigger) 4 × 25 = 100, and 100 × 2.7 = 270. ::: :::keep-together ### Change the form **Work out $0.75 \times 36$ without a calculator. Write your steps.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $0.75 = \tfrac{3}{4}$ (a decimal can be written as a fraction); $\tfrac{1}{4}$ of 36 is 9, so $\tfrac{3}{4}$ of 36 is 27 (36 divides by 4, so the fraction is quick) 0.75 = 3/4. 36 ÷ 4 = 9, × 3 = 27. ::: :::keep-together ### Split **Which way works out $25 \times 3.2$ by splitting?** ☐ $25 \times 3 + 25 \times 0.2 = 75 + 5 = 80$ ☐ $25 \times 3 + 0.2 = 75.2$ ☐ $25 \times 3 \times 0.2 = 15$ ☐ $20 \times 3.2 + 5 \times 3 = 79$ ::: :::teacher-note **Answer:** $25 \times 3 + 25 \times 0.2 = 75 + 5 = 80$ 3.2 = 3 + 0.2, so 25 × 3.2 = 25 × 3 + 25 × 0.2 = 75 + 5 = 80 (the distributive law). ::: :::typical ## One change at a time **Three quarters.** Each task changes one thing. **A longer product.** **A fraction of a number.** **An awkward percentage.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Quick ways :::keep-together ### Reorder **Work out $8 \times 1.25$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 10 1.25 = 1 1/4, so 8 + 2 = 10. Or 8 × 1.25 = 8 × 5/4 = 10. ::: :::keep-together ### Friendly percentage **Find 20% of 35.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 20% is a fifth: 35 ÷ 5 = 7. ::: :::keep-together ### Near a round number **Work out $99 \times 0.4$. (Try $100 \times 0.4$ first.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 39.6 99 × 0.4 = 100 × 0.4 − 1 × 0.4 = 40 − 0.4 = 39.6. ::: :::keep-together ### Review: percentages **Review: a $40 bag is 15% off. What is the sale price, in dollars?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 34 10% is $4 and 5% is $2, so $6 off: $34. Or 40 × 0.85 = 34. ::: ## Make: 18 in your head Then find two more, and say which was quickest and why. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Work out 5 × 3.6 × 2 and 40% of 15, each in a quick way. Say which move you used.** {{lines-3}} ::: :::teacher-note **Answer:** 5 × 2 = 10, then × 3.6 = 36 (any order). 40% = 2/5: 15 ÷ 5 × 2 = 6 (change the form). :::