Working with Proportions

Overview

In this lesson 13 of 20, students practise setting up and solving proportion problems using ratio thinking and rates in a real-world context. They learn to choose an appropriate model, interpret the answer in context, and check that the result is reasonable.

Learning intentions

Students will:

• set up proportion problems using equivalent ratios or a unit-rate approach
• solve for an unknown quantity in a ratio/rate situation
• interpret solutions in context, including units and reasonableness
• reflect on whether their model fits the situation

Success criteria

Students can:

• write a correct proportion equation or equivalent-ratio table for a given scenario
• calculate the unknown quantity accurately
• explain what their answer means in the real situation and include units
• justify whether their answer is reasonable (e.g. larger ratio gives larger quantity)

Curriculum links

• Measurement and modelling: use mathematical modelling to solve practical problems involving ratios and rates, including financial contexts; formulate problems; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model
• Measurement: recognise and use rates to solve problems involving the comparison of two related quantities of different units of measure
• (Prior unit link) Number modelling and percentages are a prerequisite for understanding proportional change

Lesson structure (30 minutes)

1. 0–4 min · Warm-up (proportion check). Teacher writes two quick statements on the board (e.g. “2 bags for $6” and “5 shirts use 900 g of fabric”) and asks: “What stays consistent: the ratio or the total?” Students do a quick think and write one sentence for each.

2. 4–10 min · Mini teach (how to set up). Teacher demonstrates two set-ups using the same scenario:

• Equivalent-ratio table (line up quantities)
• Unit-rate method (find “per 1” then scale) Teacher emphasises consistent units and choosing a model that matches the scenario. Students copy a simple template into books: table/equation/unit-rate, then a “Reasonableness check” line.

3. 10–18 min · Guided practice (solve together). Teacher provides a scenario: “A smoothie recipe uses 3 oranges for 900 mL. How many oranges are needed for 1.2 L?” Teacher prompts students to:

• identify the two related quantities (oranges and volume)
• set up a proportion or unit-rate step
• calculate the unknown Students work in pairs, completing: table or equation, working, final answer with units, then one reasonableness sentence.

4. 18–24 min · Independent practice (two problems). Students choose Problem A or B from the board and then complete the second one (so both are attempted). Teacher circulates and checks setup quality first, then arithmetic.

• Problem A: “A phone plan charges $0.35 per text and has a $5 monthly fee. How much for 40 texts?” (Model: ratio/rate for text cost, then add fixed fee.)
• Problem B: “A map scale is 1 cm: 500 m. What distance is 3.6 cm on the ground?” (Model: convert using the scale ratio.)

5. 24–28 min · Share and critique (model appropriateness). Teacher selects one pair’s work (or one student’s solution) and asks:

• “Does your equation show the same ratio?”
• “What units did you use at the start and end?”
• “Would the answer make sense if the input doubled?” Students listen and offer one improvement suggestion using sentence starters: “I notice…”, “I think you should…”, “Your units show…”.

6. 28–30 min · Exit ticket (accuracy + meaning). Teacher gives one short proportional prompt: “A craft uses 4 beads for every 3 cm of string. How many beads for 9 cm?” Students submit final beads, working (brief), and one reasonableness check.

Resources

• Board or projector with two exemplar set-ups (table and unit-rate)
• Printed problem cards (or board text) for Problems A and B
• Student notebooks and pens
• Timer for pacing (optional)
• Exit ticket slips
• Scaffolds on paper: “Table / Equation / Unit rate” template and “Reasonableness check” sentence stems

Assessment

• Formative checks during guided practice: accuracy of setup, correct pairing of related quantities, and unit consistency
• Teacher observation during independent practice: students’ ability to select an appropriate model (table/equation/unit-rate) and justify reasonableness
• Exit ticket: solve proportion correctly and include a brief justification tied to the context (units + reasonableness)

Differentiation

• Support: Provide sentence starters for setup and interpretation (e.g. “The ratio stays constant because…”, “I used unit rate to find…”, “My answer is reasonable because…”). Offer a partially filled table for one of the problems.
• Support for unit handling: Encourage a “Units check” line in every response (input units → intermediate → final units).
• Extension: For students who finish early, ask: “Find a second method for the same problem and compare which is easier and why.”
• EAL/SEN considerations: Allow students to explain setup verbally to the teacher before writing; accept diagrams/tables as working; keep arithmetic steps minimal and focus on the model choice and units.