Same Relationship, Different Numbers

Overview

In this sixth lesson of Mapping Data and Change, students investigate how ratios can look different while describing the same relationship. They use multiplication and division to generate equivalent ratios, explain why both parts must be changed by the same factor, and connect ratio thinking to maps, scale and data contexts.

Learning intentions

• WALT generate equivalent ratios by multiplying or dividing both parts by the same factor.
• WALT identify whether two ratios describe the same relationship.
• WALT justify our mathematical thinking using representations, calculations and clear explanations.
• WALT work collaboratively to notice patterns and communicate conclusions.

Success criteria

• I can multiply or divide both parts of a ratio by the same non-zero factor.
• I can find a ratio equivalent to a given ratio.
• I can check whether two ratios are equivalent.
• I can explain why changing only one part does not preserve the relationship.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking and challenging questions for advanced learners.
• Mathematics and Statistics — Mathsteasers / Alignment: connecting challenge questions with relevant mathematical learning.
• Mathematics and Statistics — Mathsteasers / Additional resources for advanced learners: extending reasoning, justification and problem-solving.
• Mathematical practices: represent relationships, look for patterns, reason, and communicate mathematical ideas.

Lesson structure (45 minutes)

1. 0–5 min · Hook and notice. Open with the hook and map-scale image showing a map key of 1 cm: 5 km alongside 2 cm: 10 km. Ask, “Are these two scales different, or do they describe the same relationship?” Students think independently, discuss with a partner, and share what they notice without being told the answer.

2. 5–12 min · Build the idea. Use the worked-example slides to model a ratio table for 1: 5, 2: 10 and 3: 15. Teacher explicitly demonstrates multiplying both parts by the same factor and records: 1: 5 × 2 = 2: 10 1: 5 × 3 = 3: 15. Students use counters, drawings or a table to represent the relationship and explain why both quantities must change together.

3. 12–17 min · Check and challenge. Display the pairs 4: 6 and 8: 12, then 3: 7 and 6: 14 using the turn-and-talk prompts. Students decide whether each pair is equivalent, identify the factor used, and justify their answer to a partner. Teacher listens for the misconception that adding the same number to both parts always creates an equivalent ratio.

4. 17–32 min · Guided practice. Distribute the equivalent-ratios investigation worksheet. Students complete the ratio tables and questions independently for five minutes, then work in pairs to compare methods and solve the justification tasks. Teacher conferences with groups, asking, “What happened to both parts?” and “How can you prove the relationship stayed the same?” Students must show a table, scaling calculation or diagram, not just an answer.

5. 32–39 min · Reasoning gallery. Show the reasoning and discussion slides. Pairs choose one worksheet response to display or explain. Classmates test statements such as “6: 9 is equivalent to 2: 3” and “5: 8 is equivalent to 10: 16” by multiplying or dividing both parts. Teacher highlights precise language: same factor, both parts, same relationship.

6. 39–45 min · Plenary and exit check. Students complete the Success Criteria exit ticket slips. They record one equivalent ratio for 4: 7, decide whether 12: 18 and 2: 3 are equivalent, and explain how they know. Invite two students to share different valid strategies, then preview that the next lesson will use ratios to interpret change in maps and data.

Resources

• the complete equivalent-ratios slide deck
• the equivalent-ratios investigation worksheet
• the Success Criteria exit ticket slips
• Whiteboard and markers
• Ratio counters, linking cubes or coloured tiles
• Pencils and highlighters
• A3 paper or mini-whiteboards for pair explanations

Assessment

• During the hook and turn-and-talk, check whether students recognise that 1: 5 and 2: 10 describe the same relationship.
• Confer during the worksheet task, recording who multiplies or divides both parts by the same factor and who can justify equivalence.
• Use the exit ticket to identify students needing support with scaling, checking equivalence or explaining their reasoning.

Differentiation

• Support students with ratio tables, colour-coded counters and sentence starters: “Both ratios are equivalent because…” and “I multiplied/divided both parts by…”.
• Begin with multiplication by whole-number factors before introducing division; allow students to draw equal groups or use concrete materials.
• Pair students strategically and read worksheet instructions aloud for learners who need language or processing support. Use consistent vocabulary and visual examples for EAL learners.
• Challenge confident students to find several equivalent ratios, work backwards from a larger ratio, or explain why adding the same number to both parts does not generally preserve equivalence.