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# Substituting into formulas

Algebra Foundations · Expressions

<!-- Speaker note: Students substitute values into formulas, writing in the hidden × signs and using the order of operations, including decimals, division and brackets (AC9M7A01, node 2). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** to use a formula, replace each letter with its value, write in the × signs, then work it out in the right order.

- I can replace each letter in a formula with its value, writing in the × signs.
- I can work out the result in the right order, including brackets and division.
- I can show my substitution as a line of working, with units.

<!-- Speaker note: Before class: Do the Modify boards yourself first: they show the substitution but hide the value, so students must finish the arithmetic. The new move is writing the hidden × signs. The common error is reading 2l with l = 5 as 25 (Leo's mistake). Insist on a line of working with × signs, every time, before any arithmetic. The Run board shows the full working, so students can compare it with their own line. -->

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## Why it matters

> A formula is a recipe for a number. Put in the values you know, and it gives you the one you want: a perimeter, a cost, a speed.

<!-- Speaker note: To **substitute** into a formula is to replace each letter with its value, then work out the answer. A number written next to a letter, such as $2l$, means multiply: $2 \times l$. -->

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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

**$P = 2l + 2w$. What is $P$ when $l = 5$ and $w = 3$?**

- 13
- 16
- 30
- 48

<!-- Speaker note: What happens: 16. 2l means 2 × l = 2 × 5 = 10, and 2w = 2 × 3 = 6. So P = 10 + 6 = 16. (48 comes from reading 2l as 25 and 2w as 23.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: show of hands. Ask a student who chose 48 how they got it, then write 2l = 2 × l on the board. -->

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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%2Fsubstituting%2Fstep-2&code=widget%3A+formula%0Aformula%3A+P+%3D+2l+%2B+2w%0Aslider+l%3A+1%2C+10%2C+2%0Aslider+w%3A+1%2C+6%2C+1%0Ashape%3A+rectangle%0Aunits%3A+cm%0A&fresh=1}}

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

<!-- Speaker note: Read the working under the rectangle. Where did the × signs come from? -->

<!-- Speaker note: Teacher: have students say the board's working line aloud: 'two times five plus two times three'. -->

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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?

* 16. 2l means 2 × l = 2 × 5 = 10, and 2w = 2 × 3 = 6. So P = 10 + 6 = 16. (48 comes from reading 2l as 25 and 2w as 23.)

<!-- 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>

## Area

Here is how to substitute, step by step.

$$P = 2l + 2w$$

1) $l = 5, \; w = 3$ — Write down the values you know.
2) $P = 2 \times 5 + 2 \times 3$ — Replace each letter with its value. Write in the × signs that were hidden in $2l$ and $2w$.
3) $P = 10 + 6$ — Multiply before adding.
4) $P = 16 \text{ cm}$ — Add, and give the units.

<!-- 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 P = 2l + 2w and say "Write the formula first." Line 1: write l = 5, w = 3 and say "Write down the values you know." Line 2: write P = 2 × 5 + 2 × 3 and say "Replace each letter with its value. Write in the × signs that were hidden in 2l and 2w." Line 3: write P = 10 + 6 and say "Multiply before adding." Line 4: write P = 16 cm and say "Add, and give the units." -->

<!-- Speaker note: Teacher: model the worked example live. Write the formula, the values, then the substituted line, before any arithmetic. -->

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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>

## Area

**The area of a rectangle is $A = lw$. Find $A$ when $l = 8$ and $w = 3$.**

* Answer: 24

<!-- Speaker note: A = 8 × 3 = 24. -->

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## A plumber

**A plumber charges $C = 5h + 20$ dollars. Find $C$ when $h = 3$.**

* Answer: 35

<!-- Speaker note: C = 5 × 3 + 20 = 15 + 20 = 35 dollars. -->

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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

**Leo works out a perimeter. Click the line where he goes wrong.**

```
 1  P = 2l + 2w, with l = 5 and w = 3.
 2  P = 25 + 23
 3  P = 48
```

* Answer: line 2

<!-- Speaker note: Line 2: 2l means 2 × l, so 2 × 5 = 10, not 25. With the × signs in: P = 2 × 5 + 2 × 3 = 10 + 6 = 16. -->

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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>

## Other formulas



- **A decimal:** Each task loads a new formula. The board shows the substitution, but not the answer: work it out yourself.
- **Three letters:** 
- **Divide:** 
- **Brackets:** 

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

<!-- Speaker note: What this step is for: One change at a time: a decimal value, three letters multiplied, a division, and brackets that are worked out first. For students who need more support: Give a template with three lines: formula, substituted (with × signs), answer with units. For early finishers: Find P = 2l + 2w and A = lw for l = 4.5 and w = 2. Find a rectangle where P and A are the same number. When students get stuck: A student writes digits side by side (25 for 2l with l = 5). Ask: What sign is hiding between the 2 and the l? Fix: Write the formula again with every × sign in, then substitute. / A student adds before multiplying (5 × 3 + 20 = 75). Ask: Which do you do first: multiply or add? Fix: Multiply and divide before adding and subtracting, unless brackets say otherwise. -->

<!-- Speaker note: Teacher: before brackets, ask what 5(h + 4) means in words. -->

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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>

## Tickets

**Find $T = 12n + 6$ when $n = 4$.**

* Answer: 54

<!-- Speaker note: T = 12 × 4 + 6 = 48 + 6 = 54. -->

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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>

## Brackets again

**Find $y = 3(x - 2)$ when $x = 7$.**

* Answer: 15

<!-- Speaker note: y = 3 × (7 − 2) = 3 × 5 = 15. -->

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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>

## A triangle

**Find $A = \frac{bh}{2}$ (the area of a triangle) when $b = 6$ and $h = 5$.**

* Answer: 15

<!-- Speaker note: A = 6 × 5 ÷ 2 = 30 ÷ 2 = 15. -->

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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>

## Review

**Review: in $V = 3t + 10$, which are the constants?**

- Only 10
- 3 and 10
- $V$ and $t$

* Answer: 3 and 10

<!-- Speaker note: 3 and 10 stay the same; V and t are variables that change. -->

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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: rectangles with the same area

Find a rectangle with area 24 cm². Then find another one. How many can you find with whole-number sides?

Stretch: which of your rectangles has the smallest perimeter? Use $P = 2l + 2w$ to check.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/algebra/substituting/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing values to hit a target reverses substitution: from the answer back to the letters. Hints, in order: A = l × w. Which two whole numbers multiply to 24? / Try l = 6. What must w be? -->

<!-- Speaker note: Teacher: collect every rectangle the class found with area 24. Is there a pattern? -->

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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

**Find C = 8h + 15 when h = 3, and V = lwh when l = 2, w = 5, h = 3.**

* C = 8 × 3 + 15 = 39. V = 2 × 5 × 3 = 30.

<!-- 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: finding an unknown, where the answer is known and one letter is not. Substitution is also how every solution is checked, in Equations and in Linear relations (tables from rules). -->