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# Modelling with money

Number Foundations · Modelling

<!-- Speaker note: Students model money problems (discounts, rises, markups, profit and sharing) by formulating what is known and unknown, choosing a representation, solving, and interpreting and justifying the answer in dollars (AC9M7N09, nodes 1 to 3). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** discounts, markups and profit: formulate the problem, choose fractions, percentages or ratios, solve, and say what the answer means in dollars.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A $200 bike is 20% off, and then a coupon takes a further 10% off the new price. Is that the same as 30% off?

<!-- Speaker note: Money problems are ratio and percentage problems with a context. The habit is the same three steps: 1. **Formulate:** what is known (price, discount, cost), what is unknown, and which representation (fraction, decimal, percentage, ratio) fits? 2. **Solve:** choose an efficient calculation, estimate first, and use a calculator where it helps. 3. **Interpret:** say what the answer means, in dollars, with a sensible amount of rounding, and **justify** why you chose that representation. Two traps: a percentage is always *of* something (the **original** amount for profit or discount), and percentages of different amounts cannot be added. -->

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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

**A $200 bike is 20% off, then a coupon takes 10% off the new price. What is the final price?**

- $144
- $160
- $140
- $150

<!-- Speaker note: What happens: $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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose $140 added 20% and 10%; $150 and $160 are other slips. -->

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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%2Fnumber%2Fmodelling%2Fstep-2&code=widget%3A+percent%0Aq%3A+200%0Aslider+p%3A+0%2C+50%2C+0%2C+10%0Amode%3A+decrease%0Aunits%3A+%24%0A&fresh=1}}

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

<!-- Speaker note: Now set 20%, then use 10% off that price. -->

<!-- Speaker note: Teacher: ask why 10% of $160 is less than 10% of $200. -->

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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?

* $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.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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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>

## Say what you know

**Known: a $60 jumper is 25% off. Unknown: the sale price. What is it, in dollars?**

* Answer: 45

<!-- Speaker note: 25% of $60 = $15, so $45. Or 60 × 0.75 = 45. -->

<!-- Speaker note: Teacher: have students say 'of what?' aloud for each percentage in the jumper and the scarf. -->

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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

**Ben finds the percentage profit on a scarf. Click the line where his working goes wrong.**

```
 1  A scarf costs $5 to make and sells for $8.
 2  Profit = 8 − 5 = $3
 3  Percentage profit = 3 ÷ 8 = 0.375 = 37.5%
 4  Answer: 37.5% profit.
```

* Answer: line 3

<!-- Speaker note: Line 3: profit is a percentage of the cost, the original amount: 3 ÷ 5 = 0.6 = 60%. Ben divided by the selling price. -->

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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>

## 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%
- They are the same
- Shop B: 30 is bigger than 3
- You cannot tell

* Answer: Shop A: $\tfrac{1}{3}$ is about 33%, more than 30%

<!-- Speaker note: $\tfrac{1}{3}$ of $90 = $30 off; 30% of $90 = $27 off. Writing both as percentages (33⅓% and 30%) makes the comparison immediate. -->

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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>

## New money problems



- **The game:** 
- **A pay rise:** 
- **Markup, then sale:** 
- **Write the price:** 
- **What does it assume?:** 

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

<!-- Speaker note: What this step is for: 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. For students who need more support: Use the bar for each step, and write 'of $…' next to every percentage. For early finishers: 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? 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: A student divides profit by the selling price (Ben's slip). Ask: What did the shop spend, and what is the profit compared with? Fix: Profit as a percentage is out of the cost, the original amount. / A student adds 40% and subtracts 20% to get +20%. Ask: Is the 20% off of $50 or of the new price? Fix: Do it in steps: × 1.4, then × 0.8. / A student gives an answer without units. Ask: 33.6 what? Fix: Write dollars to the cent: $33.60. -->

<!-- Speaker note: Teacher: for the markup then sale, ask whether the final price is more or less than $50, and why. -->

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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>

## 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?**

* Answer: 60

<!-- Speaker note: Ann gets $60, leaving $180. $180 ÷ 3 = $60 each. -->

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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>

## Saving up

**A bike costs $400. You have saved 65% of it. How many more dollars do you need?**

* Answer: 140

<!-- Speaker note: 65% of $400 = $260 saved. $400 − $260 = $140. Or 35% of $400. -->

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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>

## Bulk buying

**A pack of 12 pens costs $9, while one pen costs $1. What percentage cheaper is each pen in the pack?**

* Answer: 25

<!-- Speaker note: 12 singles cost $12; the pack costs $9, so each pen costs 75% as much. That is 25% cheaper. -->

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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: sharing

**Review: share $84 in the ratio $3 : 4$. What is the bigger share, in dollars?**

* Answer: 48

<!-- Speaker note: 3 + 4 = 7 parts. 84 ÷ 7 = 12. The bigger share is 4 × 12 = 48. -->

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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: a $75 sale

Then write one sentence for a shopper explaining your sale.

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

<!-- Speaker note: What this step is for: Choosing a price and a discount that give exactly $75 shows many models fit and each says something different about the shop. Hints, in order: The sale price is price × (1 − discount). / Try $100 with 25% off. -->

<!-- Speaker note: Teacher: share the pairs and ask which shop would you prefer, and why. -->

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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

**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.

<!-- 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: 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. -->