--- title: "Equivalent fractions" subtitle: "Number Foundations · Fractions and decimals · Lesson 9" --- # Equivalent fractions **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Half a pizza cut into 2 slices or into 8: the same amount to eat. The numbers change; the amount does not. :::teacher-note **In one sentence:** Students match shaded amounts on fraction bars to find equivalent fractions, learn that multiplying or dividing the top and the bottom by the same number keeps the amount, and simplify by the highest common factor (AC9M7N04, node 1). **Students should be able to say:** I can find equivalent fractions by multiplying or dividing the top and the bottom by the same number. I can write a fraction in its simplest form using the highest common factor. I can explain with a picture why two fractions are equal. **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 Picture the bar split into 3 parts. What happens if each part is cut in half? :::keep-together **Which fraction is the same amount as $\tfrac{2}{3}$?** ☐ $\tfrac{4}{5}$ ☐ $\tfrac{3}{4}$ ☐ $\tfrac{2}{6}$ ☐ $\tfrac{4}{6}$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 4/6. Split each third into 2: 6 parts, and the 2 shaded thirds become 4 shaded sixths. 3/4 (add 1 to both) is more than 2/3. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Multiply or divide both :::keep-together ### Simplest form Here is how to find equivalent fractions, and the simplest one. 1. $\tfrac{2}{3} = \tfrac{2 \times 4}{3 \times 4} = \tfrac{8}{12}$ — Split each part into 4: multiply the top and the bottom by 4. 2. $\tfrac{12}{18} = \tfrac{12 \div 2}{18 \div 2} = \tfrac{6}{9}$ — Join parts: divide the top and the bottom by a common factor. 3. $\tfrac{6}{9} = \tfrac{6 \div 3}{9 \div 3} = \tfrac{2}{3}$ — 6 and 9 still share 3, so divide again. 4. $\tfrac{12}{18} = \tfrac{12 \div 6}{18 \div 6} = \tfrac{2}{3}$ — Or in one step: divide by the highest common factor, 6. A fraction is in **simplest form** (lowest terms) when the top and the bottom have no common factor but 1. Try one: simplify $\tfrac{12}{18}$. **Write $\tfrac{12}{18}$ in its simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2/3 The highest common factor of 12 and 18 is 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 2/3. ::: :::keep-together ### Find the mistake **Lily looks for a fraction equal to $\tfrac{2}{3}$. Click the line where her working goes wrong.** ``` 1 Find a fraction equal to 2/3. 2 Do the same to the top and the bottom. 3 Add 1 to the top and add 1 to the bottom. 4 So 2/3 = 3/4. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: adding changes the amount (3/4 is more than 2/3). Doing the same to the top and the bottom means multiplying or dividing: 2/3 = 4/6. ::: :::keep-together ### Why they are equal **Why is $\tfrac{2}{3} = \tfrac{8}{12}$?** ☐ 8 − 2 = 6 and 12 − 3 = 9 ☐ 8 and 12 are both even ☐ Each third is split into 4 twelfths, so 2 thirds are 8 twelfths ::: :::teacher-note **Answer:** Each third is split into 4 twelfths, so 2 thirds are 8 twelfths × 4 on the top and the bottom: the same amount, cut into 4 times as many parts. ::: :::typical ## One change at a time **Twelfths.** Each task changes one thing. **A missing number.** **Simplify.** **Fewest parts.** **Already simplest.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Equal fractions :::keep-together ### Simplify **Write $\tfrac{18}{24}$ in its simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3/4 HCF 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4. ::: :::keep-together ### Missing number **Fill the gap: $\tfrac{5}{8} = \tfrac{15}{?}$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 24 × 3: 8 × 3 = 24. ::: :::keep-together ### Compare **Which is bigger, $\tfrac{2}{3}$ or $\tfrac{3}{5}$? Write both with 15 parts.** ☐ 3/5 (it has bigger numbers) ☐ They are equal ☐ 2/3 (10/15 is more than 9/15) ::: :::teacher-note **Answer:** 2/3 (10/15 is more than 9/15) 2/3 = 10/15 and 3/5 = 9/15, so 2/3 is bigger. ::: :::keep-together ### Review: integers **Review: work out $-3 + 5$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2 Start at −3 and move 5 right: 2. ::: ## Make: neither in lowest terms Then: what is the simplest form of both your fractions? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Write 20/32 in its simplest form, and give one other fraction equal to it.** {{lines-3}} ::: :::teacher-note **Answer:** 5/8 (÷ 4). For example 10/16 or 15/24. :::