Percentage change

Classroom resources

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Students will be able to

• I can use a multiplier for a percentage increase or decrease.
• I can work out the result of two changes, one after the other.
• I can explain why up 20% then down 20% is not back to the start.

Timing

Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min

Before class

Do the Run board yourself first. Students need percentages of amounts and increases and decreases (Number Foundations).

The one idea: each change multiplies the current amount, so successive changes multiply. The common mistake is taking the second percentage of the original amount (Sam's mistake).

The board draws the three amounts as bars with each multiplier between them; the Mark-up task uses the percentage bar.

Wider: population or prices changing by the same percentage each year (repeated multipliers: informal exponential growth).

Running the lesson

1. Predict: ask '20% of what?' for the second change.
2. Run: ask whether any pair of equal ups and downs gets back to $100.
3. Investigate: ask Sam what 10% of $220 is.
4. Modify: ask whether two 20% discounts are better or worse for the shopper than one of 40%.
5. Make: ask why the order does not matter (multiplying in either order).

The hard part (on purpose)

Asking 'a percentage of what?'. Ask students to name the amount each percentage is taken of before they calculate.

When students get stuck

Answers to the checks

1. up then down: A $100 price goes up 10%, then the new price goes down 10%. What is the final price?
   $99. Up 10% makes $110. Then 10% of $110 is $11, so the price falls to $99.
2. two changes: Set up 10%, then down 10%. Was your prediction right? Then try 20% each way.
   
   1.1 × 0.9 = 0.99: $99, a 1% fall overall. 20% each way gives 1.2 × 0.8 = 0.96: $96.
3. Multipliers: What do you multiply by for a 15% increase?
   1.15
   100% + 15% = 115% = 1.15.
4. Find the mistake: Sam works out the jacket's final price. Click the line where his working goes wrong.
   line 3
   Line 3: the 10% decrease is 10% of the new price, $220, which is $22. The jacket costs $198.
5. Reading a multiplier: A price is multiplied by 0.85. Has it gone up or down, and by how much?
   Down 15%
   0.85 = 85%: 15% has been taken off.
6. Undo it: Go up 25%, then find the decrease that brings it back to exactly $100.
   
   1.25 × 0.8 = 1: down 20% undoes up 25%.
7. Two discounts: Set two discounts of 20% in a row. Is that the same as 40% off?
   
   0.8 × 0.8 = 0.64: 36% off, not 40%.
8. Up 50, down 50: A price goes up 50% and then down 50%. What is the overall percentage change? (A decrease is negative.)
   -25
   1.5 × 0.5 = 0.75: a 25% decrease.
9. Mark-up: A shop buys a hat for $40 and marks it up 30%. Set the board. What is the selling price?
   
   40 × 1.3 = $52.
10. Increase: Increase $80 by 15%.
    92
    80 × 1.15 = $92.
11. Decrease multiplier: What do you multiply by for a 35% decrease?
    0.65
    100% − 35% = 65% = 0.65.
12. Two steps: A $250 bike goes up 20%, then has a 10% discount. What does it cost now, in dollars?
    270
    250 × 1.2 = 300, then 300 × 0.9 = $270.
13. Review: percentage of: Review: what is 15% of 60?
    9
    0.15 × 60 = 9.
14. GST: A price is $80 before GST. GST adds 10%. What is the price with GST, in dollars?
    88
    80 × 1.1 = $88.
15. an overall change: Find two changes that make an overall decrease of exactly 28%.
    
    For example down 10% then down 20% (0.9 × 0.8 = 0.72), or up 20% then down 40% (1.2 × 0.6 = 0.72).

Solutions

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

back to the start, and discounts › Undo it

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

back to the start, and discounts › Two discounts

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

back to the start, and discounts › Up 50, down 50

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

back to the start, and discounts › Mark-up

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

an overall change

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

Stretch and support

Stretch: A town's population grows 10% a year for 3 years. Is that a 30% increase? Find the true overall change.

Support: Write each multiplier under its arrow, and multiply with a calculator.

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 8 student in Australia. Help me understand ONE idea only, from the lesson "Percentage change".

The idea: Modelling percentage increase and decrease: I can Students model situations involving percentage increase and decrease, including mark-ups, discounts, and GST, and explain why the result is an increase or decrease.

What understanding looks like:
- Correctly models a percentage increase or decrease problem.
- Explains why the change is an increase or decrease.
- Uses an efficient multiplier strategy where appropriate.

Mistakes students often make here:
- A student takes the second percentage of the original.
- A student multiplies by 0.15 for a 15% increase.
- A student adds the percentages (−10 and −20 gives −30).

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.
- 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 "Percentage change".

Exit ticket

A $60 item is discounted 25%, then the discounted price goes up 25%. What is the final price?

60 × 0.75 × 1.25 = $56.25.

What each step is for

• up then down: Committing to the final price after up 10% then down 10% shows whether they take the second percentage of the new amount.
• two changes: Trying pairs of changes on the bars shows that equal percentages up and down do not cancel.
• multipliers: Each change is a multiplier on the current amount (Sam takes 10% of the old price); the overall multiplier gives the overall change.
• back to the start, and discounts: Finding a change that undoes another, comparing two discounts with one, and a mark-up show the multiplier model at work.
• percentage change: Multipliers both ways and a two-step change, mixed; taking the second percentage of the old amount is the planted slip; the percentage-of review keeps the basics fresh.
• an overall change: Searching for two changes with a given overall multiplier uses the product of multipliers as the model.

Where it leads

This starts AC9M8N05.

Where it leads: money models (next lesson), compound interest and exponential growth (Years 9 and 10).