Comparing Quantities

Overview

In this fifth lesson of Mapping Data and Change, students are introduced to ratio as a way of comparing two quantities. They use familiar contexts, concrete objects, diagrams and ratio notation, building on prior work with equal groups, multiplication and division.

Learning intentions

• WALT compare two quantities using a ratio.
• WALT represent a ratio with objects, diagrams and notation.
• WALT explain what each number in a ratio tells us.
• WALT connect ratios to familiar situations and data.

Success criteria

• I can describe which two quantities are being compared.
• I can make or draw a representation of a ratio.
• I can write a ratio using the correct order and notation.
• I can explain my ratio in a complete sentence.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking and challenge through non-routine mathematical questions.
• Mathematics and Statistics — Mathsteasers / Alignment: connecting ratio thinking with familiar number and data contexts.
• Mathematics and Statistics — Additional resources for advanced learners: extending reasoning through multiple representations and explanation.
• Te Mātaiaho mathematical practices: representing, reasoning, communicating and connecting mathematical ideas.

Lesson structure (45 minutes)

1. 0–5 min · Hook and notice. Open with the hook and visual comparison slides showing a basket containing 6 red counters and 3 blue counters. Ask, “How could we compare the red counters with the blue counters without saying ‘there are 3 more’?” Students discuss possible comparisons with a partner and share ideas. Record both “3 more” and “twice as many” as useful comparisons.

2. 5–13 min · Explicit teaching. Use the ratio teaching slides to introduce ratio as a comparison of two quantities. Build 6 red counters and 3 blue counters where all students can see them, and model the language: “The ratio of red to blue is 6 to 3.” Show the notation 6:3, read it aloud, and stress that order matters: red comes first, so red is named first. Students copy one example and rehearse the sentence with a partner.

3. 13–20 min · Represent and explain. Distribute the ratio representation worksheet. Model the first question: students draw or arrange objects, write the ratio, and complete a sentence explaining it. Check understanding by asking, “Which quantity is first? How do you know?” Students complete two guided examples independently, then compare answers with a partner.

4. 20–32 min · Collaborative ratio investigation. Return to the investigation instruction slides and display four familiar contexts, such as fruit in a lunchbox, coloured tiles in a pattern, boys and girls in a sports group, and books on two shelves. In groups of five, students choose or are assigned a context, identify the two quantities, represent them with counters or a labelled drawing, and record the ratio in words and notation on the worksheet. Each group prepares one explanation beginning, “The ratio of ___ to ___ is ___ because…”.

5. 32–39 min · Share and reason. Invite groups to present one representation. Ask the class to check the order of the quantities and whether the notation matches the diagram. Include a reasoning prompt: “Could 2:1 describe the same comparison as 6:3? Explain.” Guide students to notice that both show twice as many of the first quantity, while keeping the focus on comparison rather than formal simplification.

6. 39–45 min · Plenary and exit check. Use the plenary and exit-question slide to display: “There are 4 green apples and 8 red apples. What is the ratio of green to red? Show it with a diagram and explain what it means.” Students complete the final box on the ratio representation worksheet and hand it in as they leave. Invite two students to explain different representations.

Resources

• the complete ratio comparison slide deck
• the ratio representation worksheet
• Two-coloured counters, linking cubes or small classroom objects
• Mini-whiteboards and pens
• Chart paper or whiteboard
• Pencils and coloured pencils
• Group roles cards or a visible list of roles
• Prepared context images or simple classroom data

Assessment

• During teaching, ask students to identify the first and second quantities before they write notation. Listen for correct language and order.
• During group work, check whether students’ objects or diagrams match their written ratio and prompt them to justify their comparison.
• Use the exit response to assess whether students can represent, write and explain a ratio independently. Note whether errors relate to order, notation or interpretation.

Differentiation

• Support students with pre-counted objects, clearly labelled “first quantity” and “second quantity” spaces, and the sentence frame: “The ratio of ___ to ___ is ___ because there are ___ for every ___.”
• Pair students strategically and allow them to rehearse explanations orally before recording them. Use counters and drawings before moving to notation.
• For students ready for challenge, ask them to find two different diagrams that represent the same comparison, or create a classroom context for a given ratio such as 3:2.
• Provide enlarged print, reduced writing load and access to concrete materials for students with additional learning needs. Explicitly teach and display the words ratio, compare, quantity, to and notation for multilingual learners.