Substituting into formulas

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

• Slides Download Copy address for Prepare Present from the browser, or export to PowerPoint or PDF.
• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: show of hands. Ask a student who chose 48 how they got it, then write 2l = 2 × l on the board.
2. Run: have students say the board's working line aloud: 'two times five plus two times three'.
3. Investigate: model the worked example live. Write the formula, the values, then the substituted line, before any arithmetic.
4. Modify: before brackets, ask what 5(h + 4) means in words.
5. Make: collect every rectangle the class found with area 24. Is there a pattern?

When students get stuck

Answers to the checks

1. a perimeter: $P = 2l + 2w$. What is $P$ when $l = 5$ and $w = 3$?
   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.)
2. check your value: Set $l = 5$ and $w = 3$. Was your prediction right?
   
3. Area: The area of a rectangle is $A = lw$. Find $A$ when $l = 8$ and $w = 3$.
   24
   A = 8 × 3 = 24.
4. A plumber: A plumber charges $C = 5h + 20$ dollars. Find $C$ when $h = 3$.
   35
   C = 5 × 3 + 20 = 15 + 20 = 35 dollars.
5. Find the mistake: Leo works out a perimeter. Click the line where he goes wrong.
   line 2
   Line 2: 2l means 2 × l, so 2 × 5 = 10, not 25. With the × signs in: P = 2 × 5 + 2 × 3 = 10 + 6 = 16.
6. A decimal: Find $P = 2l + 2w$ when $l = 2.5$ and $w = 4$. Set the board to check your substitution.
   13
   P = 2 × 2.5 + 2 × 4 = 5 + 8 = 13 cm.
7. Three letters: The volume of a box is $V = lwh$. Find $V$ when $l = 2$, $w = 3$ and $h = 4$.
   24
   V = 2 × 3 × 4 = 24 cubic units.
8. Divide: Average speed is $s = d / t$ (distance ÷ time). Find $s$ in km/h when $d = 120$ km and $t = 2$ hours.
   60
   s = 120 ÷ 2 = 60 km/h.
9. Brackets: A hire company charges $C = 5(h + 4)$ dollars. Find $C$ when $h = 2$.
   30
   C = 5 × (2 + 4) = 5 × 6 = 30 dollars.
10. Tickets: Find $T = 12n + 6$ when $n = 4$.
    54
    T = 12 × 4 + 6 = 48 + 6 = 54.
11. Brackets again: Find $y = 3(x - 2)$ when $x = 7$.
    15
    y = 3 × (7 − 2) = 3 × 5 = 15.
12. A triangle: Find $A = \frac{bh}{2}$ (the area of a triangle) when $b = 6$ and $h = 5$.
    15
    A = 6 × 5 ÷ 2 = 30 ÷ 2 = 15.
13. Review: Review: in $V = 3t + 10$, which are the constants?
    3 and 10
    3 and 10 stay the same; V and t are variables that change.
14. rectangles with the same area: Make a rectangle with an area of 24 cm².
    
    Your l × w = 24. Other pairs work too: 3 × 8, 4 × 6, 2 × 12.

Solutions

One possible solution for each task. Other correct programs are fine.

other formulas › A decimal

Open task: any program that does what the step asks.

other formulas › Three letters

Open task: any program that does what the step asks.

other formulas › Divide

Open task: any program that does what the step asks.

other formulas › Brackets

Open task: any program that does what the step asks.

rectangles with the same area

Open task: any program that does what the step asks.

Stretch and support

Stretch: 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.

Support: Give a template with three lines: formula, substituted (with × signs), answer with units.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Substituting into formulas".

The idea: Substituting values into formulas: I can Students substitute given numerical values into a formula and calculate the result.

What understanding looks like:
- Correctly substitutes values into a formula and evaluates.
- Applies correct order of operations during substitution.
- Does not multiply the coefficient by the variable label (e.g. does not treat 3A as 3×'the letter A' rather than 3 times the value of A).

Mistakes students often make here:
- A student writes digits side by side (25 for 2l with l = 5).
- A student adds before multiplying (5 × 3 + 20 = 75).

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Variables as generalised quantities), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Substituting into formulas".

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.

What each step is for

• a perimeter: Committing to a value exposes reading 2l with l = 5 as 25, the digits side by side.
• check your value: The board writes every substitution out with the × signs, so the student can compare it with their own working.
• three steps: Substituting is three steps: write the formula, replace each letter (with × signs in), then work out in order: multiply before adding.
• other formulas: One change at a time: a decimal value, three letters multiplied, a division, and brackets that are worked out first.
• substitute: Substituting into new formulas with the order of operations, mixed; the review is naming variables and constants from the last lesson.
• rectangles with the same area: Choosing values to hit a target reverses substitution: from the answer back to the letters.

Where it 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).