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# Equations as a balance

Algebra Foundations · Equations

<!-- Speaker note: Students treat an equation as a balance, find the value that balances it, and check a value by substitution (AC9M7A03, node 1). About 45 minutes (Predict and Run 10 min · Investigate 15 min · Modify 12 min · Make 8 min). -->

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## Today

**We are learning to** an equation says two things are equal. Find the value that makes it true, then check it.

- 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.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. The surprise is the last Modify task: $2x = 7$ never balances with a whole number, and that is the point. Be ready to ask "what would $x$ have to be?" and accept 3.5 without teaching a method yet: solving step by step is the next two lessons. Teacher: Say 2x aloud as 'two times x', never 'two x', for the whole lesson. It prevents the 24-for-8 slip in the second task. -->

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## Why it matters

> 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.

<!-- Speaker note: An **expression** is a calculation with a letter in it, such as $2x + 3$. An **equation** puts an = sign between two expressions: it says they are equal, like a balance scale with the same weight on each side. The **solution** is the value of the letter that makes it true. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Which value of $x$ makes $2x + 3 = 11$ true?**

- $x = 7$
- $x = 4$
- $x = 8$
- $x = 5.5$

<!-- Speaker note: What happens: x = 4. Two lots of 4 is 8, and 8 + 3 = 11, the same as the right side. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students lock in a value before touching the board. Ask two students with different predictions to explain them. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Fequation-balance%2Fstep-2&code=widget%3A+balance%0Aleft%3A+2x+%2B+3%0Aright%3A+11%0Amin%3A+0%0Amax%3A+10%0Astart%3A+1%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/algebra/equation-balance/?full=step-2#step-2)

<!-- Speaker note: Change $x$ with the slider, or the − and + buttons. Watch both sides and stop when the scale balances. -->

<!-- Speaker note: Teacher: the board shows each side worked out for the current x. Point at the line under each pan: that is substitution, done for them. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* x = 4. Two lots of 4 is 8, and 8 + 3 = 11, the same as the right side.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Check by substitution

To check a value, put it in place of the letter and work out each side.

$$2x + 3 = 11$$

1) $2(4) + 3$ — Put 4 in place of $x$ on the left. The brackets show that $2x$ means 2 times $x$.
2) $= 8 + 3 = 11$ — Work out the left side.
3) $11 = 11$ — The left side equals the right side.
4) $\text{So } x = 4 \text{ is a solution.}$ — If the two sides were different, it would not be.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 2x + 3 = 11 and say "Is x = 4 a solution? Start with the equation." Line 1: write 2(4) + 3 and say "Put 4 in place of x on the left. The brackets show that 2x means 2 times x." Line 2: write = 8 + 3 = 11 and say "Work out the left side." Line 3: write 11 = 11 and say "The left side equals the right side." Line 4: write So x = 4 is a solution. and say "If the two sides were different, it would not be." -->

<!-- Speaker note: Teacher: walk through the worked example together, one step at a time. Before each click, ask what comes next. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Check by substitution

**Is $x = 5$ a solution of $3x - 2 = 13$? Start by working out the left side, $3x - 2$, when $x = 5$.**

* Answer: 13

<!-- Speaker note: 3 × 5 − 2 = 15 − 2 = 13. The right side is 13 too, so x = 5 is a solution. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.
```

* Answer: line 2

<!-- Speaker note: 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. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Expression or equation?

**Which one is an equation?**

- $2x + 3$
- $2x + 3 = 11$
- $x + 7 - 2$
- $5x$

* Answer: $2x + 3 = 11$

<!-- Speaker note: 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. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## 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 $x = 4$?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/algebra/equation-balance/?full=step-4#step-4)

<!-- Speaker note: What this step is for: The idea holds for any equation, even with the letter on both sides, and some equations have no whole-number solution at all. Hints, in order: Each bag holds the same number of blocks. How many blocks are on the right, shared between four bags? / Watch both sides change as you move $x$. / Try each value from 0 upwards and compare the two totals. For students who need more support: Use the board for the Investigate questions too: set up 3x − 2 and 13 by hand on paper as bags and blocks, then count. For early finishers: Write an equation with x on both sides whose solution is 6, then one whose solution is not a whole number. When students get stuck: A student says x = 4 gives 2x = 24. Ask: What does 2x mean? Say it as a times sentence. Fix: 2x means 2 × x. Write the multiplication sign in, or use brackets: 2(4) = 8. / The student moves the slider at random and stops when it balances, without noticing why. Ask: Before you move it, which way will it tip? Why? Fix: Ask them to read both values under the pans aloud before each move. / A student writes an expression, such as 2x + 3, for the Make task. Ask: What is your expression equal to? Where is the = sign? Fix: An equation needs two sides. Work out 2x + 3 when x = 6, then write = and that number. -->

<!-- Speaker note: Teacher: the last task has no whole-number answer. Let students try every value before discussing. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Work it out

**Work out $5x - 4$ when $x = 3$.**

* Answer: 11

<!-- Speaker note: 5 × 3 − 4 = 15 − 4 = 11. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which value?

**Which value of $x$ solves $3x + 2 = 20$? Try 5, 6 and 7.**

* Answer: 6

<!-- Speaker note: 3 × 6 + 2 = 18 + 2 = 20, the same as the right side. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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

* Answer: No: the left side is 9 and the right side is 10

<!-- Speaker note: 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.) -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## 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.

<!-- Speaker note: What this step is for: Building an equation from its solution shows that the solution is the value that makes both sides equal. Hints, in order: Start with what $x = 6$ gives you: $2x$ is 12. / Then add the same number to both sides: $2x + 1 = 13$. -->

<!-- Speaker note: Teacher: share a few student equations and have the class check them by substitution. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Is x = 7 a solution of 3x − 5 = 16? Show how you know.**

* Yes. 3 × 7 − 5 = 21 − 5 = 16, which equals the right side.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: solving by backtracking, then the balance method (do the same to both sides). Both lean on the habit built here: check every answer by substitution. Where it leads in class: missing totals in recipes or budgets ("4 tickets and a $5 booking fee cost $41"), and formulas in science where one value is unknown. -->