Smart Budget Models

Overview

Students use mathematical modelling to solve practical problems that involve ratios, rates, percentages, and decimals in a financial context. They will formulate a model, calculate efficiently (including with digital tools if available), interpret results in the real-world situation, and critique whether the model is appropriate.

Learning intentions

• Students will solve financial problems using ratios and rates to compare options.
• Students will use percentages and decimals to calculate costs, discounts, fees, or tax-like totals.
• Students will formulate a modelling problem, show their working clearly, and interpret solutions in context.
• Students will review the appropriateness of their model (including reasonableness and units).

Success criteria

• I can convert between related quantities using ratios/rates and explain what the numbers mean.
• I can apply a percentage change using correct decimal calculations and round appropriately for money.
• I can state my solution in context (e.g., “Option A costs less because…”).
• I can check if my model makes sense by using units, substitution, and a reasonableness test.

Curriculum links

• Mathematics — Measurement and ratios/rates for practical problems using modelling (including financial contexts).
• Mathematics — Number: percentages and rational number modelling in real financial situations.
• Mathematics — Algebra: modelling applied problems, choosing a representation (tables/graphs where helpful) and interpreting solutions in context.
• Mathematics — Use and interpret rates to compare related quantities with different units.

Lesson structure (45 minutes)

1. 0–5 min · Hook (real decision). Teacher displays two phone-plan offers on the board (Plan A: $0.45 per minute, $9 monthly fee; Plan B: $0.25 per minute, $19 monthly fee) and asks: “Which plan is cheaper if you use 80 minutes?” Students think-pair-share a quick estimate and circle what information they need.

2. 5–12 min · Model set-up (direct teach). Teacher guides students to define variables: minutes used = (m), monthly cost = (C), and write two models:

• Plan A: (C_A = 9 + 0.45m)
• Plan B: (C_B = 19 + 0.25m) Students copy the models and complete a worked example for (m=80) with teacher support, focusing on correct units and money rounding.

3. 12–22 min · Guided problem (rates + comparison). Teacher introduces a second financial scenario: a savings offer with a ratio-style rate and a percentage fee: “A shop gives 10% off, then charges a 2.5% ‘service fee’ on the discounted price. An item costs $48.50 before discount. Find the final price.” Students calculate step-by-step using decimals: discounted price, then fee on discounted price, then final price, showing at least one intermediate line. Teacher circulates for common errors (percent of original vs percent of discounted).

4. 22–34 min · Independent modelling task (formulate → calculate → interpret). Teacher hands out a modelling card (or tasks on paper) with one choice problem and one constraint: “School fundraiser: You can buy meal tickets using two payment options: Option 1: $6 per ticket Option 2: Pay $30 for 6 tickets (bulk deal) After purchase, tickets are sold to families with a 15% profit added to the price you paid. (a) What is the profit price per ticket for each option? (b) If you buy 18 tickets, which option gives a cheaper cost, and what is the profit price you will charge per ticket?” Students complete (a) and (b) with a table for ticket cost, then apply 15% profit using a decimal (multiply by 1.15). They interpret results in context and label which option is cheaper and why.

5. 34–40 min · Model critique (appropriateness check). Teacher prompts: “Does your model match the real situation?” Include questions: “Did you apply percentage to the correct base? Did you use the right units (per ticket vs total)? Is your answer reasonable for money?” Students conduct a quick reasonableness check and highlight one improvement they would make (e.g., rounding at the end).

6. 40–45 min · Exit ticket (short, targeted). Teacher collects a 2-question exit ticket:

• “A student uses a model: ‘Final price = Original × 0.9 × 1.02’ for a 10% discount and 2% fee. Explain if their model is appropriate.”
• “Using your thinking, what does multiplying by 1.15 represent in context?” Students answer briefly in sentences.

Resources

• Board/projector with the phone-plan hook and example models
• Student task sheets (modelling card with the fundraiser problem)
• Calculator access (phones/tablets or class calculators where allowed)
• Rulers/graph paper or lined paper for tables and working
• Money rounding guidance card (e.g., round to nearest cent)
• Highlighters for marking intermediate values (optional)

Assessment

• Teacher observation during paired and independent work: correct setup of models, use of units, and percentage base.
• Marking of modelling task steps: clarity of variables, correct calculations, and contextual interpretation.
• Exit ticket checks for understanding of “percentage of the correct amount” and interpreting multiplicative factors (e.g., 1.15).

Differentiation

• Support: provide sentence starters (“Let (m) be…”, “First I find…”, “Then I apply…”, “This means…”), and a partially completed table for Option costs.
• Support: offer a worked mini-example for applying a percentage fee to a discounted price (teacher writes the structure, students fill numbers).
• Extension: ask students to generalise—“Write a formula for each option cost for (n) tickets, then explain how you would find the break-even point.”
• EAL/SEN: allow bilingual glossaries for key terms (discount, service fee, profit), and reduce cognitive load by separating the task into two clearly labelled sub-questions (cost first, then selling price).