Modelling with money

Classroom resources

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Students will be able to

• I can say what is known and unknown in a money problem and choose how to model it.
• I can solve problems with fractions, percentages and ratios, including percentages in two steps.
• I can write my answer in dollars and explain why I chose that representation.

Timing

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

Before class

Do the Run board yourself first. Students need percentages of amounts (Percentages), sharing in a ratio, and rounding for money.

Three nodes in one lesson: formulate (known, unknown, representation), solve and interpret in dollars, and justify the choice. The two traps are using the wrong base (profit as a percentage of the selling price, Ben) and adding percentages that are of different amounts (20% then 10% is not 30%).

Calculators are fine. The estimate comes first, and the answer is written to the cent with units.

Running the lesson

1. Predict: students who chose $140 added 20% and 10%; $150 and $160 are other slips.
2. Run: ask why 10% of $160 is less than 10% of $200.
3. Investigate: have students say 'of what?' aloud for each percentage in the jumper and the scarf.
4. Modify: for the markup then sale, ask whether the final price is more or less than $50, and why.
5. Make: share the pairs and ask which shop would you prefer, and why.

The hard part (on purpose)

Knowing what the percentage is of. Ask 'of which amount?' every time a percentage appears, especially the second one in a two-step problem.

When students get stuck

Answers to the checks

1. two discounts: A $200 bike is 20% off, then a coupon takes 10% off the new price. What is the final price?
   $144. 20% off $200 is $160, then 10% off $160 is $144. $140 would be 30% off $200; the 10% is of a smaller amount.
2. 30% off $200: Set the discount to 30%. What is the sale price? Is it the same as your predicted price for two discounts?
   
   30% off $200 is $140. Two discounts of 20% then 10% leave $144, because the second is of $160, not $200.
3. Say what you know: Known: a $60 jumper is 25% off. Unknown: the sale price. What is it, in dollars?
   45
   25% of $60 = $15, so $45. Or 60 × 0.75 = 45.
4. Find the mistake: Ben finds the percentage profit on a scarf. Click the line where his working goes wrong.
   line 3
   Line 3: profit is a percentage of the cost, the original amount: 3 ÷ 5 = 0.6 = 60%. Ben divided by the selling price.
5. Which representation?: Shop A takes $\tfrac{1}{3}$ off a $90 jacket. Shop B takes 30% off. Which is the better deal, and why?
   Shop A: $\tfrac{1}{3}$ is about 33%, more than 30%
   $\tfrac{1}{3}$ of $90 = $30 off; 30% of $90 = $27 off. Writing both as percentages (33⅓% and 30%) makes the comparison immediate.
6. The game: A game costs $120 and has 15% off. What is the sale price?
   
   15% of $120 is $18, so $102. Or 120 × 0.85 = 102.
7. A pay rise: You earn $20 an hour and get a 5% pay rise. What is your new hourly rate, in dollars?
   21
   5% of $20 = $1, so $21. Or 20 × 1.05 = 21.
8. Markup, then sale: A $50 top is marked up by 40% and then sold at 20% off the new price. Estimate first: will the final price be more or less than $50? Then work it out, in dollars.
   56
   50 × 1.4 = $70, then 70 × 0.8 = $56. The two percentages are of different amounts.
9. Write the price: The calculator shows a sale price of 33.6. How do you write it in a shop?
   $33.60
   Dollars to the cent: $33.60.
10. What does it assume?: You model the top with 50 × 1.4 × 0.8. What does the model assume about the 20% off?
    It is 20% of the marked-up price, not of $50
    Writing × 0.8 after × 1.4 assumes the sale is 20% of the marked-up price. Stating assumptions like this is part of the model.
11. Fractions of a prize: A $240 prize is shared: $\tfrac{1}{4}$ to Ann, and the rest equally among 3 others. How much does each of the other three get, in dollars?
    60
    Ann gets $60, leaving $180. $180 ÷ 3 = $60 each.
12. Saving up: A bike costs $400. You have saved 65% of it. How many more dollars do you need?
    140
    65% of $400 = $260 saved. $400 − $260 = $140. Or 35% of $400.
13. Bulk buying: A pack of 12 pens costs $9, while one pen costs $1. What percentage cheaper is each pen in the pack?
    25
    12 singles cost $12; the pack costs $9, so each pen costs 75% as much. That is 25% cheaper.
14. Review: sharing: Review: share $84 in the ratio $3 : 4$. What is the bigger share, in dollars?
    48
    3 + 4 = 7 parts. 84 ÷ 7 = 12. The bigger share is 4 × 12 = 48.
15. a $75 sale: Choose a price and a discount so the sale price is exactly $75.
    
    For example $100 less 25%, or $150 less 50%. The sale price is the original times (1 − the discount).

Solutions

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

new money problems › The game

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

new money problems › A pay rise

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

new money problems › Markup, then sale

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

new money problems › Write the price

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

new money problems › What does it assume?

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

a $75 sale

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

Stretch and support

Stretch: A shop marks up its cost by 25%, then offers a 20% sale on the new price. Does it make a profit? Does the shop ever lose money with a 25% markup followed by a bigger sale?

Support: Use the bar for each step, and write 'of $…' next to every percentage.

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 the ideas in ONE lesson only, from the lesson "Modelling with money".

The ideas:
- Formulating a model from a practical problem: I can Students identify the mathematical structure of a practical problem and choose an appropriate representation (fraction, decimal, percentage, or ratio) to model it.
- Solving and interpreting in context: I can Students solve a mathematical model and interpret the result in terms of the original situation, checking reasonableness.
- Justifying representation choices: I can Students justify why they chose a particular representation and calculation strategy, and communicate this reasoning clearly.

What understanding looks like:
- Correctly identifies the operation and representation needed for a given practical problem.
- States any assumptions made in setting up the model.
- Does not start calculating before identifying the model structure.
- Expresses the solution in contextually appropriate terms (correct units, reasonable precision).
- Checks the reasonableness of the solution using estimation.
- Does not present a raw number without contextual interpretation.
- Justifies representation choices with reference to the problem context and computation efficiency.
- Communicates the solution process, not just the answer.
- Can identify a more efficient representation when presented with an inefficient one.

Mistakes students often make here:
- A student divides profit by the selling price (Ben's slip).
- A student adds 40% and subtracts 20% to get +20%.
- A student gives an answer without units.

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 "Modelling with money".

Exit ticket

A $80 jacket is marked up 25% then sold at 25% off the new price. Is the final price more than, less than or equal to $80? Work it out.

80 × 1.25 = 100, then 100 × 0.75 = 75: less than $80.

What each step is for

• two discounts: Committing shows whether they add the percentages (30% off, $140), apply them in turn ($144), or take both off the original.
• 30% off $200: Setting 30% off $200 gives $140, the single-step answer to compare with the two-step $144.
• formulate, solve, interpret: Profit percentage is out of the cost, the original; Ben divides by the selling price. A fraction is easier to compare when both discounts are in the same form.
• new money problems: A discount on the board, a pay rise, a markup then a discount, and the answer written the way a shop would: each asks for the model, then the units.
• money models: A fraction of a prize, a saving goal as a percentage, a bulk saving, and a ratio review.
• a $75 sale: Choosing a price and a discount that give exactly $75 shows many models fit and each says something different about the shop.

Where it leads

Next: Modelling with ratios, with mixes, scale drawings and models in measurement.

Where it leads: GST, simple and compound interest, profit and loss, and financial literacy in Years 8 to 10.