Mathematics · Year 10

Active learning ideas

Graphing Functions: Linear and Quadratic

Active learning helps students grasp the dynamic nature of gradients and areas under curves. By manipulating graphs and discussing real-world contexts, students move beyond abstract rules to see how functions behave and change.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Algebra

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

Activity 01

• Inquiry Circle
• 40 min · Small Groups

Inquiry Circle: The Racing Car

Groups use a motion sensor or a video of a car accelerating to create a distance-time graph. They must draw tangents at different points to estimate the speed at those moments and discuss how the speed is changing.

Analyze how changes in the parameters of a linear function affect its graph.

Facilitation Tip

During the Racing Car investigation, circulate and ask groups to explain why the gradient changes as the curve steepens, pushing them to connect the visual with the mathematical idea.

What to look for

Provide students with graphs of several linear and quadratic functions. Ask them to label the roots and turning point (if applicable) on each graph and write the equation of the line for two linear examples. This checks their ability to identify key features.

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Activity 02

• Think-Pair-Share
• 20 min · Pairs

Think-Pair-Share: Area Under the Curve

Students are given a velocity-time graph and asked to estimate the total distance using one large trapezium versus four smaller ones. They discuss in pairs why more trapeziums lead to a more accurate answer.

Interpret the significance of the roots and turning point of a quadratic graph.

Facilitation Tip

For the Think-Pair-Share on area under the curve, assign each pair a different curve so the gallery of answers helps the whole class see the range of possible approaches.

What to look for

Give each student a card with a function, either linear (e.g., y = 2x - 3) or quadratic (e.g., y = x² - 4). Ask them to: 1. Sketch the graph. 2. Identify one key feature (y-intercept for linear, roots for quadratic). 3. Write one sentence explaining what that feature represents.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 03

• Gallery Walk
• 30 min · Pairs

Gallery Walk: Real-World Gradients

Stations feature different non-linear graphs (e.g., cooling of a cup of tea, bacterial growth). Students move in pairs to estimate the rate of change at specific points and explain what that rate means in the context of the problem.

Compare the graphical properties of linear and quadratic functions.

Facilitation Tip

In the Gallery Walk, place graphs with clear real-world contexts at each station so students can focus on interpreting units and meanings rather than just calculating.

What to look for

Pose the question: 'How does changing the 'c' value in y = mx + c affect the graph, and how does changing the 'a' value in y = ax² + bx + c affect its graph differently?' Facilitate a class discussion where students use their knowledge of graphs to explain the transformations.

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

A few notes on teaching this unit

Teach gradients by having students physically draw tangents on printed curves, then measure and compare slopes. Emphasize that the tangent line is a local approximation, not the whole curve. For areas, start with simple shapes like rectangles and trapezoids before moving to curves, and always connect calculations back to what the area represents in context.

Students will confidently draw tangents to find gradients at points on curves and use the trapezium rule to approximate areas under curves. They will explain what these values represent in practical terms, such as acceleration or distance traveled.

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Watch Out for These Misconceptions

• During the Collaborative Investigation: The Racing Car, watch for students drawing the same tangent line at every point on the curve.

  Ask students to place their rulers at a specific point on the curve and explain why the tangent line they drew only touches the curve at that one point. Have them compare slopes at different points to see the gradient changes.

• During the Gallery Walk: Real-World Gradients, watch for students ignoring the units on the axes when interpreting gradients or areas.

  Guide students to first identify the units on each axis, then ask them to write the units of the gradient (e.g., m/s²) and the area (e.g., m) before explaining what those units mean in context.

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Methods used in this brief

• Inquiry Circle
• Think-Pair-Share
• Gallery Walk