--- title: "Changing the subject" subtitle: "Algebra Foundations · Brackets · Lesson 12" --- # Changing the subject **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** P = 2l + 2w gives the fence around a garden. But you have 20 m of fence and a 6 m side. How wide can it be? Turn the formula around. :::teacher-note **In one sentence:** Students rearrange formulas to change the subject by undoing the steps in reverse order on both sides, and check each rearrangement by substituting numbers, on a formula board that works the original forwards (AC9M8A01, node 3). **Students should be able to say:** I can make a given letter the subject of a formula by undoing the steps in reverse order. I can do the same to the whole of both sides, so I divide every term. I can check a rearranged formula by substituting numbers. **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 Test each option with numbers: if $l = 6$ and $w = 4$, then $P = 20$. Which option gives back 4? :::keep-together **Which formula gives $w$ from $P = 2l + 2w$?** ☐ $w = P \div 2 - 2l$ ☐ $w = P - 2l \div 2$ ☐ $w = (P - 2l) \div 2$ ☐ $w = (P + 2l) \div 2$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** w = (P − 2l) ÷ 2. To get w on its own, undo + 2l (take 2l from both sides: P − 2l = 2w), then undo × 2 (halve both sides). With P = 20 and l = 6: (20 − 12) ÷ 2 = 4. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Undo in reverse order :::keep-together ### Undo the last step first $x$ was multiplied by 3, then 5 was added. Undo in reverse order: take away 5, then divide by 3. 1. $y = 3x + 5$ — $x$ was multiplied by 3, then 5 was added. 2. $y - 5 = 3x$ — Undo the last step first: take 5 from both sides. 3. $\frac{y - 5}{3} = x$ — Undo $\times 3$: divide the whole of both sides by 3. 4. $x = \frac{y - 5}{3}$ — Write the subject on the left. 5. $y = 11: x = \frac{11 - 5}{3} = 2$ — Check by substituting: $3 \times 2 + 5 = 11$. ✓ Try one: make $x$ the subject of $y = 2x + 7$. **Make $x$ the subject of $y = 2x + 7$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** x = (y - 7)/2 y − 7 = 2x, then halve both sides: x = (y − 7)/2. Check: y = 17 gives x = 5, and 2 × 5 + 7 = 17. ::: :::keep-together ### Find the mistake **Jo makes $x$ the subject of $y = 4x - 3$. Click the line where her working goes wrong.** ``` 1 Make x the subject of y = 4x − 3. 2 Divide both sides by 4: y ÷ 4 = x − 3. 3 Add 3 to both sides: y ÷ 4 + 3 = x. 4 So x = y/4 + 3. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: dividing the right side by 4 divides both terms, so (4x − 3) ÷ 4 = x − 3/4, not x − 3. Undo − 3 first: y + 3 = 4x, so x = (y + 3)/4. ::: :::keep-together ### Your working **Make $x$ the subject of $y = 3x + 5$. Write one step per line.** Answer: {{field-15}} ::: :::teacher-note **Answer:** y = 3x + 5 → y - 5 = 3x → x = (y - 5)/3 Undo + 5: y − 5 = 3x. Undo × 3: x = (y − 5)/3. ::: :::typical ## One change at a time **Check on the board.** Each task changes one thing. **One step.** No board from here: undo the steps, then check with numbers. **A minus sign.** **Undo a divide.** **A fraction.** **The subject is subtracted.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Rearrange and check :::keep-together ### A formula in context **A plumber charges $c = 50h + 80$ dollars for $h$ hours. Make $h$ the subject.** Answer: {{field-15}} ::: :::teacher-note **Answer:** h = (c - 80)/50 c − 80 = 50h, so h = (c − 80)/50. Check: h = 2 gives c = 180, and (180 − 80)/50 = 2. ::: :::keep-together ### Full working **Make $w$ the subject of $P = 2l + 2w$. Write one step per line.** Answer: {{field-15}} ::: :::teacher-note **Answer:** P = 2l + 2w → P - 2l = 2w → w = (P - 2l)/2 Undo + 2l: P − 2l = 2w. Halve both sides: w = (P − 2l)/2. ::: :::keep-together ### Use it **Use your formula: what is $w$ when $P = 30$ and $l = 9$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 (30 − 18) ÷ 2 = 6. Check: 2 × 9 + 2 × 6 = 30. ::: :::keep-together ### Review: factorising **Review: factorise $6x - 15$ fully.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3(2x - 5) HCF 3: 3(2x − 5). ::: ## Make: work it backwards Write your formula for $n$ first. Then use it, and only then move the slider. Stretch: make $l$ the subject of $P = 2l + 2w$, then make $r$ the subject of $C = 2\pi r$. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Make x the subject of y = 6x − 2, then check it with x = 3.** {{lines-3}} ::: :::teacher-note **Answer:** x = (y + 2)/6. Check: x = 3 gives y = 16, and (16 + 2)/6 = 3. :::