Mathematics · Year 8

Active learning ideas

Inverse Proportion: Graphs and Equations

Active learning works here because inverse proportion relationships are counterintuitive. Students need to see, touch, and graph real data to move beyond abstract formulas. Plotting speed-time pairs or matching job scenarios makes the constant product k tangible and memorable.

• National Curriculum Attainment Targets
• KS3: Mathematics - Ratio, Proportion and Rates of Change
• KS3: Mathematics - Algebra

• 20–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Decision Matrix
• 30 min · Pairs

Pairs Plotting: Fixed Distance Speeds

Pairs measure time for a classmate to cover 20 metres at walking, jogging, and running paces. Record speed-time pairs, compute products to check constancy, plot the graph on paper, and draw the hyperbola. Discuss why it curves unlike direct proportion lines.

Differentiate between the graphical representations of direct and inverse proportion.

Facilitation Tip

During Pairs Plotting, circulate to ensure students use the same axes scales so their hyperbolas can be compared side-by-side in the closing discussion.

What to look for

Present students with two graphs, one linear and one hyperbolic. Ask them to label each graph with 'Direct Proportion' or 'Inverse Proportion' and write one sentence explaining their choice based on the graph's shape.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 02

• Decision Matrix
• 40 min · Small Groups

Small Groups: Scenario Matching Relay

Provide cards with inverse scenarios, data tables, graphs, and equations. Groups race to match sets correctly, calculate k for verification, and present one justification to the class. Extend by creating their own scenario.

Explain why the product of two variables remains constant in an inverse proportion.

Facilitation Tip

In the Scenario Matching Relay, set a timer so groups must justify their matches quickly, reinforcing the need to test the product xy for constancy.

What to look for

Provide students with a table of values showing an inverse proportion (e.g., speed and time for a fixed distance). Ask them to: 1. Calculate the constant of proportionality (k). 2. Write the equation for the relationship. 3. Predict the time taken if the speed was doubled.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 03

• Decision Matrix
• 25 min · Whole Class

Whole Class: Human Hyperbola

Mark axes on the floor with tape. Students hold cards with (x,y) pairs for y=30/x, stand at positions to form the curve. Class observes asymptotes, photographs for reference, and predicts missing points.

Construct an equation to model an inverse proportional relationship from given data.

Facilitation Tip

For the Human Hyperbola, choose a volunteer with clear movement cues so the shrinking distance from the center is visible to all.

What to look for

Pose the scenario: 'Four workers can complete a job in 12 days. How long would it take six workers to complete the same job?' Ask students to explain their reasoning, referencing the constant product principle and showing the equation they used to solve it.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 04

• Decision Matrix
• 20 min · Individual

Individual: Data to Equation Challenge

Give tables of inverse data. Students find k by multiplying pairs, write the equation, predict values, and sketch graphs. Share one prediction with a partner for checking.

Differentiate between the graphical representations of direct and inverse proportion.

Facilitation Tip

During the Data to Equation Challenge, provide calculators only after they have estimated k by hand to build number sense.

What to look for

Present students with two graphs, one linear and one hyperbolic. Ask them to label each graph with 'Direct Proportion' or 'Inverse Proportion' and write one sentence explaining their choice based on the graph's shape.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start with concrete data instead of formulas. Students plot speed and time for a fixed 24 km trip (speeds of 4, 6, 8, 12 km/h) and observe the curve. Teacher modeling is key: show how to calculate k from one pair (e.g., 6 km/h and 4 h gives k = 24) and verify it for the others. Avoid rushing to the equation y = k/x; let the pattern emerge from the plotted points and repeated checks of the product.

Students will recognize inverse proportion graphs as hyperbolas that approach but never touch the axes. They will calculate k from tables and use it to write equations and solve problems. Misconceptions about shape or product will be replaced by evidence from their own data and discussions.

---

Watch Out for These Misconceptions

• During Pairs Plotting, watch for students who connect their points with straight lines because they expect all graphs to be linear.

  Stop the class after 5 minutes and ask pairs to compare their graphs side-by-side; point out that the curve bends away from a straight line and ask them to redraw it as a smooth hyperbola.

• During the Scenario Matching Relay, watch for groups who match scenarios based on one pair of values decreasing without checking the product xy.

  Prompt them to calculate xy for each pair in the scenario cards and ask, 'Does this stay the same across all pairs?' Redirect them to the correct matches by having them list the products.

• During the Human Hyperbola, watch for students who think the person’s distance from the center decreases linearly.

  After the volunteer moves, ask the class to estimate the next position by halving the remaining distance, making the non-linear nature explicit.

---

Methods used in this brief

• Decision Matrix