--- title: "Multiplying and dividing fractions" subtitle: "Number Foundations · Fractions and decimals · Lesson 13" --- # Multiplying and dividing fractions **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Half of a pizza is left, and you eat a third of what is left. How much of the whole pizza did you eat? Fractions of fractions are multiplications. :::teacher-note **In one sentence:** Students use a fraction area square to see a fraction of a fraction, multiply tops and bottoms, learn that multiplying by a fraction less than 1 makes smaller, and divide by counting how many fit (AC9M7N06, node 2). **Students should be able to say:** I can multiply two fractions and simplify the answer. I can divide a fraction by a whole number and by a fraction. I can say whether a product will be bigger or smaller than the number I started with. **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 of it as $\tfrac{1}{3}$ of $\tfrac{1}{2}$. :::keep-together **Is $\tfrac{1}{2} \times \tfrac{1}{3}$ bigger or smaller than $\tfrac{1}{2}$?** ☐ The same ☐ Bigger ☐ Smaller My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** Smaller: 1/2 × 1/3 = 1/6, a third of a half. Multiplying by a number less than 1 takes only part. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Multiply, and divide :::keep-together ### Multiply across; divide by flipping Here is how each kind works. 1. $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{2 \times 3}{3 \times 4} = \tfrac{6}{12} = \tfrac{1}{2}$ — Multiply the tops and the bottoms, then simplify. 2. $\tfrac{3}{4} \times 20 = \tfrac{60}{4} = 15$ — A whole number is over 1: 20 = 20/1. 15 is less than 20. 3. $\tfrac{1}{2} \div 3 = \tfrac{1}{6}$ — Share a half into 3: each share is a sixth (the same as × 1/3). 4. $3 \div \tfrac{1}{4} = 3 \times 4 = 12$ — How many quarters in 3? Dividing by 1/4 is multiplying by 4. 5. $\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{2}{3} \times 6 = 4$ — Divide by a fraction: multiply by it flipped (1/6 flipped is 6/1 = 6). 6. $0.4 \times 3 = 1.2$ — Decimals: 4 tenths × 3 = 12 tenths. Now try one: $\tfrac{2}{3} \times \tfrac{3}{5}$. **Work out $\tfrac{2}{3} \times \tfrac{3}{5}$ in simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2/5 (2 × 3)/(3 × 5) = 6/15 = 2/5. ::: :::keep-together ### Find the mistake **Jack compares $\tfrac{3}{4} \times 20$ with 20. Click the line where his reasoning goes wrong.** ``` 1 Which is bigger: 3/4 × 20, or 20? 2 3/4 × 20 means 3/4 of 20. 3 Multiplying always makes a number bigger. 4 So 3/4 × 20 is bigger than 20. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: multiplying by a number less than 1 makes smaller. 3/4 of 20 is 15, less than 20. ::: :::keep-together ### How many fit? **How many quarters are there in 3? Work out $3 \div \tfrac{1}{4}$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 4 quarters in each whole, so 12 in 3: 3 ÷ 1/4 = 3 × 4 = 12. ::: :::typical ## One change at a time **Three quarters of two fifths.** Each task changes one thing. **Of a whole number.** **Divide by a whole number.** **Divide by a fraction.** **Bigger or smaller?.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Multiply and divide :::keep-together ### Multiply **Work out $\tfrac{4}{5} \times \tfrac{5}{8}$ in simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/2 20/40 = 1/2. ::: :::keep-together ### Decimals **A bottle holds 0.6 L. How many litres do 4 bottles hold?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2.4 0.6 × 4 = 2.4 L. ::: :::keep-together ### Divide in context **A ribbon is $\tfrac{3}{4}$ m long. How many pieces $\tfrac{1}{8}$ m long can be cut from it?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 3/4 = 6/8: six eighths. (3/4 ÷ 1/8 = 3/4 × 8 = 6.) ::: :::keep-together ### Review: adding **Review: work out $\tfrac{1}{2} + \tfrac{1}{5}$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7/10 5/10 + 2/10 = 7/10. ::: ## Make: a quarter, another way Then find one where neither fraction has a 1 on top. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Work out 3/5 × 10/12, and 2 ÷ 1/3.** {{lines-3}} ::: :::teacher-note **Answer:** 30/60 = 1/2. 2 ÷ 1/3 = 6 (six thirds in 2). :::