--- title: "The balance method" subtitle: "Algebra Foundations · Equations · Lesson 9" --- # The balance method **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Engineers, nurses working out doses and anyone splitting a bill rearrange equations this way: keep it balanced, and every step stays true. :::teacher-note **In one sentence:** Students solve linear equations by doing the same to both sides (the balance method), including x on both sides, and verify every answer in the first equation (AC9M7A03, node 3). **Students should be able to say:** I can solve an equation by doing the same thing to both sides, and show each step. I can solve an equation with x on both sides. I can check my answer by putting it back into the first equation, every time. **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 The left side has two bags of $x$ and 3 blocks. The right side has 11 blocks. Picture taking 3 blocks from each side. :::keep-together **$2x + 3 = 11$. You take 3 from **both** sides. What equation is left?** ☐ $2x = 8$ ☐ $2x = 11$ ☐ $2x + 3 = 8$ ☐ $x = 8$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 2x = 8. The left loses its 3 blocks and the right loses 3 too: 11 − 3 = 8. The bags of x stay. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Write it as algebra :::keep-together ### In order Each line keeps the equation balanced. 1. $3x + 5 = 20$ — Start with the equation. 2. $3x + 5 - 5 = 20 - 5$ — Take 5 from **both** sides. 3. $3x = 15$ — Now three lots of $x$ are 15. 4. $\frac{3x}{3} = \frac{15}{3}$ — Divide **both** sides by 3. 5. $x = 5$ — $x$ is on its own. 6. $3(5) + 5 = 20 \checkmark$ — Check in the **first** equation, not the last line. **Put the steps for solving $4x + 3 = 23$ in order.** A. `4x = 20` B. `x = 5` C. `4x + 3 − 3 = 23 − 3` D. `4x = 26` E. `Check: 4 × 5 + 3 = 23` F. `4x + 3 = 23` Correct order (letters): {{field-24}} ::: :::teacher-note **Answer:** see the program below Take 3 from both sides (4x = 20), divide both sides by 4 (x = 5), then check in the first equation. ```maths 4x + 3 = 23 4x + 3 − 3 = 23 − 3 4x = 20 x = 5 Check: 4 × 5 + 3 = 23 ``` ::: :::keep-together ### Find the mistake **Leo solves $2x + 3 = 11$, but his check fails. Click the first line that goes wrong.** ``` 1 2x + 3 = 11 2 2x = 11 3 x = 5.5 4 Check: 2 × 5.5 + 3 = 14, not 11 ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: Leo took 3 from the left side only. The right side has to lose 3 too: 2x = 8, so x = 4. Check: 2 × 4 + 3 = 11. ::: :::keep-together ### Which step first? **To solve $3x + 5 = 20$, which first step keeps whole numbers?** ☐ Divide both sides by 3 ☐ Take 5 from both sides ::: :::teacher-note **Answer:** Take 5 from both sides Both are allowed, but dividing first gives x + 5/3 = 20/3. Taking 5 first gives 3x = 15, then x = 5. ::: :::typical ## Harder scales **Bigger numbers.** Each task loads a new scale. Same moves, bigger numbers. **Bags on both sides.** Now there are bags of $x$ on both sides. Backtracking cannot solve this; the balance method can. **Bags and blocks on both sides.** Bags and blocks on both sides. Get the bags on one side first. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Solve and check :::keep-together ### Two steps **Solve $5x + 4 = 39$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 5x + 4 − 4 = 39 − 4, so 5x = 35 and x = 7. Check: 5 × 7 + 4 = 39. ::: :::keep-together ### Bags on both sides **Solve $9x = 4x + 35$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 Take 4x from both sides: 5x = 35. Divide by 5: x = 7. Check: 9 × 7 = 63 and 4 × 7 + 35 = 63. ::: :::keep-together ### Bags and blocks on both sides **Solve $6x + 2 = 4x + 12$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 Take 4x from both sides: 2x + 2 = 12. Take 2: 2x = 10. Divide by 2: x = 5. Check: 6 × 5 + 2 = 32 and 4 × 5 + 12 = 32. ::: ## Make: your own balance Give it to a friend to solve with the balance method, then check their answer in your first equation. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Solve 4x + 3 = 2x + 11. Show each step and check your answer in the first equation.** {{lines-3}} ::: :::teacher-note **Answer:** Take 2x from both sides: 2x + 3 = 11. Take 3: 2x = 8. Divide by 2: x = 4. Check: 4 × 4 + 3 = 19 and 2 × 4 + 11 = 19. :::