Percentages of amounts

Classroom resources

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Students will be able to

• I can find a percentage of an amount, mentally and with a decimal.
• I can increase or decrease an amount by a percentage, including with one multiplier.
• I can write one amount as a percentage of another.

Timing

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

Before class

Do the Run board yourself first. Students need percentages as decimals (Fractions, decimals and percentages) and multiplying by a fraction less than 1 (previous lesson).

The one idea: a percentage is hundredths, and "of" means multiply. For a decrease, multiply by what is left (Grace multiplies by what is taken off). Varied structures matter: of, off, on, and out of.

Calculators are fine for the decimals; the mental route (10%, 5%, 1%) builds number sense and checks calculator answers.

Running the lesson

1. Predict: students who chose $12 stopped at the discount; $65 took 15 dollars off.
2. Run: ask what percentage the blue part is, and how that gives a one-step multiplier.
3. Investigate: compare the mental route and the decimal route for 35% of 60.
4. Modify: for up then down, take a vote before working it out.
5. Make: collect pairs on the board; order them by sale price.

The hard part (on purpose)

Choosing the multiplier for a decrease. Ask what percentage is left after the discount before students touch the calculator.

When students get stuck

Answers to the checks

1. 15% off: A jacket costs $80. It is 15% off. What do you pay?
   $68. 15% of $80 is $12, and $80 − $12 = $68. $12 is the discount, not the price; $65 takes 15 dollars off, not 15%.
2. take it off: The bar is the $80 price. Set the discount to 15%. What do you pay? Was your prediction right?
   
   15% of $80 = $12, so you pay $68: 85% of $80.
3. Find it, then use it: Find 35% of 60.
   21
   10% of 60 is 6, so 30% is 18, and 5% is 3: 21. Or 0.35 × 60 = 21.
4. Find the mistake: Grace finds the sale price of a $50 game that is 20% off. Click the line where her working goes wrong.
   line 3
   Line 3: 50 × 0.2 is the discount ($10), not the price. What is left is 80%: 50 × 0.8 = $40.
5. Out of: Ari scored 18 out of 24. What percentage is that?
   75
   18 ÷ 24 = 0.75 = 75%.
6. GST: A TV is $250 before GST. Add 10% GST. What is the price?
   
   10% of $250 = $25, so $275: 250 × 1.1.
7. Increase: Increase 60 by 25%.
   75
   25% of 60 = 15, so 75. Or 60 × 1.25 = 75.
8. Decrease: Decrease 45 by 40%.
   27
   60% is left: 0.6 × 45 = 27.
9. What percentage?: In a class of 30, 12 students walk to school. What percentage walk?
   40
   12 ÷ 30 = 0.4 = 40%.
10. Up, then down: Edge case: a $100 price goes up 10%, then the new price goes down 10%. What is it now?
    $99
    $100 × 1.1 = $110, then $110 × 0.9 = $99. The decrease is 10% of a bigger amount.
11. A friendly percentage: Find 12.5% of 64. (12.5% is $\tfrac{1}{8}$.)
    8
    12.5% = 1/8, and 64 ÷ 8 = 8.
12. A multiplier: What do you multiply by to decrease an amount by 30%?
    0.7
    100% − 30% = 70% = 0.7.
13. Growth: A plant 120 mm tall grows 5% taller. How tall is it now, in mm?
    126
    5% of 120 = 6, so 126 mm. Or 120 × 1.05 = 126.
14. Review: multiplying fractions: Review: work out $\tfrac{2}{3} \times \tfrac{3}{4}$ in simplest form.
    1/2
    6/12 = 1/2.
15. save $15: Set a price and a discount that save exactly $15.
    
    For example 10% of $150, 15% of $100, 25% of $60, 30% of $50, 50% of $30 or 75% of $20.

Solutions

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

one change at a time › GST

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

one change at a time › Increase

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

one change at a time › Decrease

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

one change at a time › What percentage?

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

one change at a time › Up, then down

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

save $15

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

Stretch and support

Stretch: A shop increases a price by 25%, then offers 20% off the new price. Is the final price more, less or the same as the start? Explain with multipliers.

Support: Use the bar for every problem; start from 10% and build.

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 "Percentages of amounts".

The idea: Operations with percentages: I can Students calculate percentages of quantities and solve problems involving percentage increase, decrease, and finding the whole given a percentage.

What understanding looks like:
- Correctly finds a percentage of a quantity using an efficient method.
- Solves a percentage increase or decrease problem correctly.
- Selects an appropriate representation (fraction, decimal, or percentage) for efficient computation.

Mistakes students often make here:
- A student multiplies by the percentage taken off for a decrease (Grace's mistake).
- A student takes off the number, not the percentage (80 − 15 = 65).
- A student gives a decimal for 'what percentage' (0.75).

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 (Multiplication and division of fractions and decimals), 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 "Percentages of amounts".

Exit ticket

A $40 shirt is 35% off. Find the sale price.

35% of $40 = $14, so $26. Or 40 × 0.65 = 26.

What each step is for

• 15% off: Committing to a sale price shows whether they find the discount and take it off, stop at the discount, or add it.
• take it off: Sliding the percentage to 15 shows the discount as part of the bar and the price as what is left: 85% of the whole.
• of, off and out of: A percentage of an amount is a multiplication; Grace multiplies by the part taken off instead of what is left. Finding a percentage divides first.
• one change at a time: One change at a time: an increase on the board, an increase by a multiplier, a decrease, a percentage of a group, then increase then decrease.
• percentages: A friendly percentage, a multiplier, an increase and a percentage of a group, mixed; the multiplier for a decrease is the planted slip; the review keeps multiplying fractions fresh.
• save $15: Finding a price and a discount that save exactly $15 shows that many pairs work, and that the saving is percentage × price.

Where it leads

Next: Choosing a strategy (node 4), choosing between fractions, decimals and percentages.

Where it leads: financial maths (profit, GST, discounts in AC9M7N09), and percentage change with repeated multipliers (Year 9 and 10).