Overview

This 45-minute Year 8 lesson develops students’ understanding of percentage as a model for real-life discounts and links the model to rational-number calculations and interpretation in context. It is intended as Lesson 1 (conceptual) within a 2–3 lesson sequence on percentages and rates.

Learning intentions

Students will:

• recognise that a percentage represents a fraction out of 100 and connect it to equivalent decimals
• convert between percentages, fractions, and decimals using visual models (hundred grids and fraction strips)
• model everyday discount statements (e.g. “30% off”) using rational-number thinking
• explain what a percentage discount means in terms of “paying” and “saving” rather than memorising steps

Success criteria

Students can:

• convert 10% increments and non-10% examples (e.g. 25%, 35%) between percent, fraction, and decimal
• locate approximate values for percentages on hundred grids and justify the fraction/decimal match
• interpret “x% off” correctly as a reduction to a new sale price (not just a subtraction trick)
• use mathematical language to explain their reasoning clearly to a partner

Curriculum links

• Number — AC9M8N5: use mathematical modelling to solve practical problems involving rational numbers and percentages; formulate problems, choose efficient calculation strategies, use digital tools where appropriate; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model
• Number — AC9M8N4: use the 4 operations with integers and rational numbers, choosing efficient strategies and digital tools where appropriate (applied in later practice to discounts)
• Digital Literacy: use a calculator/digital tool to check conversions and interpret results (teacher-guided)

Lesson structure (45 minutes)

1. 0–5 min · Hook (real-life meaning). Teacher writes: “30% OFF a $20 price”. Students predict: What is the sale price and what is the saving? (No calculating yet—just reasoning.)

2. 5–15 min · Direct teach (percentage as out of 100). Teacher demonstrates a hundred grid: shading 30 squares out of 100 and states “30% means 30 out of 100.” Students shade 30% on their own hundred grid and record the matching fraction (30/100) and decimal (0.30).

3. 15–25 min · Guided practice (fraction–decimal–percent conversions). Teacher models converting three examples using the same visual logic:

• 25% = 25/100 = 0.25
• 40% = 40/100 = 0.40
• 35% = 35/100 = 0.35 Students work in pairs to complete a table for 6 values (include at least two that are not multiples of 10, e.g. 35%, 15%). Teacher circulates for misconceptions (e.g. swapping 0.35 with 3.5).

4. 25–35 min · Model the discount (interpretation of “off”). Teacher focuses on meaning: “30% off” means paying 70% of the original. Demonstrate with the grid: 30% removed leaves 70 squares shaded. Students complete two prompts:

• “20% off” a $50 item: find what % they pay, then estimate sale price using proportional thinking (students can use a calculator to check, not to replace meaning)
• “15% off” a $80 item: same process Students write a short explanation: “Because off means subtract from 100%, I pay …%.”

5. 35–43 min · Independent application (quick practice set). Students complete 5 conversions and 3 interpretation items on a worksheet (e.g. Convert 0.48 to a percent; Interpret 25% off as paying 75%; Convert 0.6 to 60%). A calculator is available for checking decimals.

6. 43–45 min · Exit ticket (formative). Students answer:

• “35% of 100 is 35. Write 35% as a fraction and a decimal.”
• “What does ‘25% off’ mean in terms of what you pay?”

Resources

• Hundred grids (printable) and markers/pencils
• Fraction strips (tenths and hundredths if available)
• Percentage–fraction–decimal conversion table (teacher slide/handout)
• Discount meaning worksheet (8–10 short items)
• Calculators (phones not required; teacher-managed if used)
• Projector/whiteboard with examples
• “30% off” word cards for discussion

Assessment

• Teacher checks pair work during shading and table completion for correct fraction/decimal links
• Targeted questioning during the discount model (“What are we shading—saved or paid?”)
• Exit ticket checks: conversion accuracy and interpretation of “off”

Differentiation

• Support: provide partially completed hundred grids and sentence starters such as “30% off means I pay ___%.” Offer a conversion bank (e.g. tenths and quarters already worked)
• Support for misconceptions: explicitly compare “% off” vs “% of” using two contrasting examples on the board
• Extension: ask students to justify whether 35% off a price is the same as paying 0.65 of the original, and to estimate without calculating
• EAL/SEN: allow oral explanation to a partner and provide word banks (percent, off, pay, save, out of 100)

Unit Pricing Rates

Overview

This lesson focuses on rates and unit pricing as a practical modelling tool for comparing products. Students will connect rates to comparison using rational-number reasoning and interpret results in context.

Learning intentions

Students will:

• recognise a rate as a comparison of two quantities with different units
• calculate unit prices (e.g. cost per kilogram, cost per litre) to compare items fairly
• formulate a modelling approach for a given comparison scenario and communicate the conclusion

Success criteria

Students can:

• calculate a unit rate by dividing total cost by quantity
• compare two options using the same unit and justify which is better value
• interpret their unit price conclusion in a sentence relating back to the scenario

Curriculum links

• Measurement — AC9M8M05: recognise and use rates to solve problems involving the comparison of 2 related quantities of different units of measure
• Number — AC9M8N5: use mathematical modelling to solve practical problems involving ratios/percentages in financial contexts (applied here through purchasing decisions)
• Digital Literacy: use digital tools to check unit-rate calculations

Lesson structure (45 minutes)

1. 0–8 min · Hook (fair comparison). Teacher shows two product labels: e.g. 1.5 L for $3.30 and 2 L for $4.40. Students decide “which is cheaper?” and explain their reasoning.

2. 8–18 min · Direct teach (what is a rate/unit price?). Teacher defines rate/unit price as “cost per 1 unit” and models the method for 1 product (e.g. $3.30 ÷ 1.5 L = $2.20 per L). Students repeat on mini-whiteboards.

3. 18–30 min · Guided practice (two examples). Students in pairs calculate unit prices for two scenarios provided (one cost per kg, one cost per litre). Teacher checks steps and emphasises consistent units.

4. 30–40 min · Application (realistic modelling task). Students solve a short comparison task: given 3 items, choose the best value and state the unit price for each. They must include a final recommendation sentence.

5. 40–45 min · Exit ticket. “A product costs $9 for 3 kg. What is the cost per kg? Which is cheaper if another product costs $13 for 5 kg?”

Resources

• Product label cards (cost and quantity in different units)
• Calculator (teacher-managed or devices)
• Unit-rate worksheet with grid for working
• Mini-whiteboards (optional but useful)

Assessment

• Formative checks during pair calculations for correct division and unit naming
• Exit ticket for accurate unit rate and comparison judgement

Differentiation

• Support: provide a template table: Cost | Quantity | Unit price | Conclusion
• Extension: include a scenario with conversion needed (e.g. 500 g vs 1.2 kg) so students must convert to the same unit before comparing
• EAL/SEN: sentence frames for conclusion: “Item A is better value because its unit price is … per ….”

Ratio and Rate Modelling

Overview

This lesson extends into modelling ratios and rates within financial contexts, requiring students to formulate an approach, choose efficient strategies, and review whether the model fits the situation.

Learning intentions

Students will:

• model a practical situation using a ratio/rate approach
• calculate using efficient strategies (including digital tools for checking)
• interpret and communicate solutions in context, including brief review of reasonableness

Success criteria

Students can:

• set up a rate/ratio model that matches the question (units make sense)
• calculate the unknown quantity using a correct strategy
• explain why their answer makes sense for the situation (reasonableness check)

Curriculum links

• Measurement — AC9M8M07: use mathematical modelling to solve practical problems involving ratios and rates, including financial contexts; formulate problems; interpret and communicate solutions; review appropriateness of the model
• Number — AC9M8N5: use mathematical modelling for practical rational-number/financial situations; interpret solutions in terms of the situation
• Digital Literacy: use digital tools to check calculations

Lesson structure (45 minutes)

1. 0–7 min · Hook (rate reasoning). Teacher asks: “Transport costs $60 for 10 rides. What is the cost per ride? Then what would 14 rides cost?” Students predict first.

2. 7–18 min · Direct teach (model set-up and units). Teacher shows how to label “per ride” (or per item) and how unit consistency guides the calculation. Students underline units on their sheet.

3. 18–30 min · Guided modelling task (teacher + students). Students follow a model-steps checklist: (a) identify quantities and units (b) choose rate/ratio (c) calculate (d) state final answer in context (e) quick reasonableness check. Teacher models one full scenario, then students do a second with support.

4. 30–41 min · Independent task (scenario cards). Students choose one scenario card (e.g. phone plan cost per GB, ticket cost per person with a rate, discount bundled with a per-unit cost). They solve and write a short “Is this reasonable?” justification.

5. 41–45 min · Exit ticket. “Write the rate/ratio model you used (with units) and check: Does your answer seem larger/smaller than the given data, and why?”

Resources

• Scenario cards (financial contexts using rates/ratios)
• Rate/ratio modelling checklist
• Calculators and a class spreadsheet template (optional)
• Student worksheet for model, calculation, explanation

Assessment

• Teacher observes model set-up and unit correctness
• Exit ticket checks reasoning and appropriateness review

Differentiation

• Support: provide a worked example of setting up the rate and a sentence frame for “This is reasonable because…”
• Extension: require students to compare two possible models and decide which fits best (e.g. where a rate changes after a threshold)
• EAL/SEN: focus on visual unit mapping (matching “per 1” labels to the calculation) and oral explanation before writing