Understanding Ratios

Overview

In this lesson 12 of 20, students build on earlier work with multiplication/division and rates by using ratios to describe relationships between quantities. They will model real situations, represent ratios in words and using numbers, and interpret what a given ratio means in context.

Learning intentions

Students will:

• understand that a ratio compares two quantities of the same kind and can describe relationships between parts and wholes
• use ratio language (for example, “for every…”) to describe situations
• calculate an unknown quantity using equivalent ratios in practical contexts (non-financial and financial examples if available)
• interpret and check whether their ratio model makes sense for the situation

Success criteria

Students can:

• correctly write a ratio to match a given description or diagram
• explain what each part of a ratio means using “for every” wording
• use equivalent ratios to find missing quantities and state the answer with units
• justify whether their answer is reasonable by referring back to the original context

Curriculum links

• Measurement — AC9M8M07: use mathematical modelling to solve practical problems involving ratios and rates, formulate problems, interpret solutions in terms of the situation, and review the appropriateness of the model
• Measurement — AC9M8M05: recognise and use rates to solve problems involving comparison of two related quantities of different units of measure (as a link to ratio thinking and to check “per” meanings)

Lesson structure (30 minutes)

1. 0–3 min · Warm-up (ratio language). Teacher displays two short statements: “Team A has 3 red pens and 2 blue pens” and “A drink mix uses 1 cup concentrate for every 4 cups water.” Students write the ratio in words (“red to blue”; “concentrate to water”) and circle which quantity is first.

2. 3–10 min · Direct teach (what ratios represent). Teacher models a simple situation with drawings (for example, 2 green counters for every 3 yellow counters) and writes: “green: yellow = 2: 3.” Students complete a guided table: given a ratio, they list at least two possible pairs of numbers that keep the ratio equivalent (for example, 2:3, 4:6, 6:9).

3. 10–18 min · Modelling task 1 (counters to numbers). Teacher shows a real-world context card: “A badge maker uses the pattern: 3 stars for every 2 circles.” (Use a small set like 15 stars needed.) Students:

• write the ratio as stars: circles
• set up equivalent ratio steps to find circles (teacher prompts scaling: “How many times larger is your stars amount than 3?”)
• record calculations clearly and label the final answer with the quantity name.

4. 18–25 min · Modelling task 2 (interpretation and reasonableness). Teacher presents a second context with a potential mismatch to prompt model checking: “A recipe uses 1 cup syrup for every 3 cups yoghurt. The packet says 4 cups syrup—how much yoghurt?” Students first find the yoghurt quantity, then answer: “Does your model fit the original description?” by checking the ‘for every’ relationship using their result.

5. 25–30 min · Exit ticket (communication). Teacher gives an exit ticket with one ratio description (for example, “At the pool, there are 2 lifejackets for every 5 swimmers.”) and asks students to:

• write the ratio using numbers
• interpret it in one sentence using “for every…”
• find one missing quantity using scaling (choose a whole-number answer).

Resources

• Counter sets (green and yellow) or paper cut-outs
• Ratio cards with short context statements
• Whiteboard/markers or slides for modelling ratio diagrams
• Student worksheets for two modelling tasks and an exit ticket
• Sentence starters: “The ratio of ___ to ___ is ___: ___. For every ___, there are ___.”
• Optional: digit-based calculator allowed for final checking (not for reasoning)

Assessment

• Teacher observation during warm-up: correct ratio language and ordering (“first” quantity)
• Marking of modelling tasks: correct ratio representation and correct scaling using equivalent ratios
• Exit ticket check: ability to interpret ratio in context and verify reasonableness (e.g., “for every” sentence)

Differentiation

• Support:
• Provide a partially completed equivalent-ratio table (scale factors 1, 2, 3) for modelling tasks
• Sentence starters for ratio writing and interpretation
• Use concrete counters first, then move to numbers
• Extension:
• Challenge students to create a new equivalent ratio pair from their own answer and explain it
• Add a “spot the error” prompt: one incorrect ratio interpretation for students to critique
• Language/EAL:
• Emphasise ratio wording (“for every”; “to”; “per” if needed) and offer a word bank
• SEN:
• Reduce computations by choosing contexts where scaling factor is straightforward (whole numbers) while maintaining conceptual focus