Mathematics · Year 11

Active learning ideas

Ratio and Proportion (Advanced Problems)

Active learning works well for advanced ratio and proportion because students often misapply procedures when problems become complex. Hands-on tasks let them test their thinking in real time, spot mistakes through collaboration, and connect abstract steps to concrete outcomes like dividing sweets or adjusting builder hours.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Ratio, Proportion and Rates of Change

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Small Groups

Relay Race: Ratio Sharing Challenges

Divide class into teams of four. Each student solves one step of a multi-part ratio sharing problem, such as dividing £120 in 3:5:7 then adjusting for a new 2:3 split. Pass solutions along the line; teams check and correct as a group before racing to finish. Debrief key adjustments.

Analyze how to adjust ratios when new information is introduced.

Facilitation Tip

During the Relay Race, stand at the halfway point to listen for teams that confuse total parts with unit values and redirect with an example like ‘60 sweets shared 2:3—show me the unit.’

What to look for

Present students with a scenario: 'A class has a 3:2 ratio of boys to girls. If 5 more girls join the class, the new ratio becomes 1:1. What was the original number of students?' Ask students to show their working and identify the step where the ratio was adjusted.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 25 min · Pairs

Card Sort: Direct vs Inverse Proportion

Prepare cards with scenarios, graphs, and equations. In pairs, students sort into direct or inverse piles, justify choices, and create one example each. Circulate to prompt discussions on multipliers versus divisors. End with whole-class share-out.

Compare direct and inverse ratio problems, highlighting their key differences.

Facilitation Tip

While students sort cards, ask pairs to justify each placement using real-world language like ‘more builders, less time’ to anchor inverse proportion in experience.

What to look for

Pose the question: 'When would you use inverse proportion to solve a problem about sharing? Give an example.' Facilitate a class discussion comparing scenarios like sharing money (direct) versus sharing work (inverse).

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 45 min · individual then pairs then small groups

Problem Design Workshop: Multi-Step Ratios

Individuals brainstorm a real-world problem integrating sharing and inverse proportion, like fuel efficiency. Pairs refine and test solutions on each other. Groups present one polished problem for class solving. Provide templates for structure.

Design a multi-step problem that integrates different aspects of ratio and proportion.

Facilitation Tip

In the Problem Design Workshop, circulate with spare ratio strips so groups can physically break and recombine parts when new information arrives, making the adjustment visible.

What to look for

Provide students with two problems: 1) Share $120 in the ratio 2:3:7. 2) If 6 builders can build a wall in 10 days, how long would it take 4 builders? Ask students to write down the type of ratio problem each represents and the first step they would take to solve it.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 30 min · Small Groups

Scale Model: Ratio Adjustments

Provide recipes or maps. Small groups scale up or down, introduce changes like extra ingredients, and recalculate ratios. Use playdough or drawings for visuals. Compare results and discuss inverse elements like time adjustments.

Analyze how to adjust ratios when new information is introduced.

Facilitation Tip

For the Scale Model activity, provide unmarked rulers and ask groups to measure then redraw their scale plan when teammates argue over the new ratio steps.

What to look for

Present students with a scenario: 'A class has a 3:2 ratio of boys to girls. If 5 more girls join the class, the new ratio becomes 1:1. What was the original number of students?' Ask students to show their working and identify the step where the ratio was adjusted.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Teachers find success when they insist on annotation: students must label total parts, unit values, and adjustments before solving. Avoid rushing to answers; instead, model think-alouds that reveal why a ratio of 4:6 simplifies to 2:3 and how adding 2 boys changes the split. Research shows that drawing ratio bars and writing reciprocal multipliers reduces errors in inverse proportion tasks.

Students will move from guessing parts to calculating precisely, explain why a ratio changes instead of resets, and choose direct or inverse proportion correctly in multi-step problems. Success looks like clear annotations, peer corrections, and accurate final answers across all activities.

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Watch Out for These Misconceptions

• During Card Sort: Direct vs Inverse Proportion, watch for students who label all problems as direct because the numbers increase together, even when context requires inverse sharing.

  Have pairs re-read each card aloud and sketch a quick graph or timeline; for inverse problems like ‘more builders means fewer days,’ students should plot a curve to see the reciprocal relationship.

• During Relay Race: Ratio Sharing Challenges, watch for teams that add parts instead of finding a unit value when new shares are introduced mid-race.

  Pause the race and ask groups to write their current ratio on a mini-whiteboard, then model adding the new share as an extra strip before recalculating the total parts and unit value.

• During Problem Design Workshop: Multi-Step Ratios, watch for students who replace the original ratio entirely when new information arrives rather than adjusting it.

  Provide blank ratio templates and ask designers to mark the original ratio in one color, the change in another, then combine them by finding equivalents before recalculating shares.

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Methods used in this brief

• Collaborative Problem-Solving