--- title: "Equations as a balance" subtitle: "Algebra Foundations · Equations · Lesson 7" --- # Equations as a balance **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Shops, recipes and sports tables all hide an equation: a total that has to match. Finding the missing number, and checking it, is what solving means. :::teacher-note **In one sentence:** Students treat an equation as a balance, find the value that balances it, and check a value by substitution (AC9M7A03, node 1). **Students should be able to say:** I can check whether a value is a solution by putting it in place of the letter and working out both sides. I can tell an expression from an equation. I can explain that a solution makes both sides of an equation 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 $$2x + 3 = 11$$ Each bag on the left holds $x$ blocks, and there are two bags and 3 loose blocks. The right side has 11 blocks. How many blocks must be in each bag for the scale to balance? :::keep-together **Which value of $x$ makes $2x + 3 = 11$ true?** ☐ $x = 7$ ☐ $x = 4$ ☐ $x = 8$ ☐ $x = 5.5$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** x = 4. Two lots of 4 is 8, and 8 + 3 = 11, the same as the right side. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Check a solution :::keep-together ### Check by substitution To check a value, put it in place of the letter and work out each side. 1. $2x + 3 = 11$ — Is $x = 4$ a solution? Start with the equation. 2. $2(4) + 3$ — Put 4 in place of $x$ on the left. The brackets show that $2x$ means 2 times $x$. 3. $= 8 + 3 = 11$ — Work out the left side. 4. $11 = 11$ — The left side equals the right side. 5. $\text{So } x = 4 \text{ is a solution.}$ — If the two sides were different, it would not be. **Is $x = 5$ a solution of $3x - 2 = 13$? Start by working out the left side, $3x - 2$, when $x = 5$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 13 3 × 5 − 2 = 15 − 2 = 13. The right side is 13 too, so x = 5 is a solution. ::: :::keep-together ### Find the mistake **Sam decides that $x = 4$ is not a solution of $2x + 3 = 11$, but it balanced the scale. Click the line where his working goes wrong.** ``` 1 Is x = 4 a solution of 2x + 3 = 11? 2 2x + 3 = 24 + 3 3 = 27 4 27 is not 11, so x = 4 is not a solution. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: Sam wrote 2 and 4 side by side to get 24. 2x means 2 × x, so it is 8, and 8 + 3 = 11. x = 4 is a solution after all. ::: :::keep-together ### Expression or equation? **Which one is an equation?** ☐ $2x + 3$ ☐ $2x + 3 = 11$ ☐ $x + 7 - 2$ ☐ $5x$ ::: :::teacher-note **Answer:** $2x + 3 = 11$ An equation has an = sign: it says two things are equal. The others are expressions: they have a value, but they do not say that anything is equal. ::: :::typical ## New scales **Four bags.** Each task loads a new scale. Find the value of $x$ that balances it. **$x$ on both sides.** Now both sides change when $x$ changes. The solution still makes the two sides equal. **A scale that will not balance.** Try every value. What happens between $x = 3$ and $x = 4$? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Check a value :::keep-together ### Work it out **Work out $5x - 4$ when $x = 3$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 11 5 × 3 − 4 = 15 − 4 = 11. ::: :::keep-together ### Which value? **Which value of $x$ solves $3x + 2 = 20$? Try 5, 6 and 7.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 3 × 6 + 2 = 18 + 2 = 20, the same as the right side. ::: :::keep-together ### Both sides **Is $x = 2$ a solution of $4x + 1 = 3x + 4$?** ☐ No: the left side is 9 and the right side is 10 ☐ No: the left side is 42 and the right side is 34 ☐ Yes: both sides are 9 ::: :::teacher-note **Answer:** No: the left side is 9 and the right side is 10 4 × 2 + 1 = 9 and 3 × 2 + 4 = 10. The sides are different, so x = 2 is not a solution. (x = 3 is: both sides are 13.) ::: ## Make: your own equation Make an equation that a friend could solve. Check it by substitution before you press **Check answer**. Stretch: write one with $x$ on both sides. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Is x = 7 a solution of 3x − 5 = 16? Show how you know.** {{lines-3}} ::: :::teacher-note **Answer:** Yes. 3 × 7 − 5 = 21 − 5 = 16, which equals the right side. :::