Mathematics · Year 10

Active learning ideas

Graphing Quadratic Functions

Active learning works for graphing quadratic functions because students need to see the immediate impact of parameter changes on the shape and position of parabolas. Hands-on and visual tasks help them connect abstract coefficients to concrete geometric features, making non-linear relationships tangible.

• ACARA Content Descriptions
• AC9M10A06

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Carousel Brainstorm
• 35 min · Pairs

Parameter Sliders: Coefficient Exploration

Provide graphing software or calculators. Pairs input y = ax² + bx + c, vary a from -3 to 3 in steps of 0.5, sketch results, and note width and direction changes. Discuss how b and c shift the graph. Share one insight per pair with the class.

Analyze how changing the coefficient of the squared term affects the width and direction of a parabola.

Facilitation Tip

During Parameter Sliders, circulate to ensure students test both positive and negative values of a, b, and c to observe all transformations.

What to look for

Present students with three different quadratic equations (e.g., y = 2x^2, y = -x^2 + 3, y = (x-1)(x+5)). Ask them to identify the direction each parabola opens and whether it has a maximum or minimum value, justifying their answers based on the leading coefficient.

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Activity 02

• Carousel Brainstorm
• 40 min · Small Groups

Graph Matching Relay: Features Hunt

Prepare cards with equations, graphs, and feature labels. Small groups race to match sets by identifying intercepts, vertex, and axis, then verify by plotting points. Rotate roles: matcher, plotter, checker. Debrief mismatches.

Explain the significance of the turning point in real-world optimization problems.

Facilitation Tip

For Graph Matching Relay, provide only one graph or equation at a time to prevent students from comparing answers prematurely.

What to look for

Provide students with a graph of a parabola showing its vertex and intercepts. Ask them to write down the coordinates of the vertex and the x-intercepts, and to determine the equation of the axis of symmetry.

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Activity 03

• Carousel Brainstorm
• 30 min · Small Groups

String Parabola Builders: Physical Sketching

Groups stretch string over thumbtacks pinned to vertex and intercepts on poster board to form parabolas. Measure axis of symmetry, label features, then derive equations. Compare to algebraic sketches.

Construct a parabola given its equation in different forms (vertex, intercept, general).

Facilitation Tip

When students build String Parabolas, ask them to measure the distance from the vertex to the focus to introduce the concept of focal length.

What to look for

In pairs, students are given a quadratic equation in general form. One student sketches the graph identifying key features, while the other checks their work. They then swap roles with a different equation, providing constructive feedback on accuracy and clarity.

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Activity 04

• Carousel Brainstorm
• 45 min · individual then pairs

Optimisation Challenge: Vertex Applications

Whole class brainstorms real scenarios like fencing max area. Individuals graph given quadratics, locate vertices, solve for optima. Pairs peer-review calculations and graphs.

Analyze how changing the coefficient of the squared term affects the width and direction of a parabola.

Facilitation Tip

In Optimisation Challenge, require students to present their solutions with clear labels for the vertex and constraints to reinforce precision.

What to look for

Present students with three different quadratic equations (e.g., y = 2x^2, y = -x^2 + 3, y = (x-1)(x+5)). Ask them to identify the direction each parabola opens and whether it has a maximum or minimum value, justifying their answers based on the leading coefficient.

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

A few notes on teaching this unit

Teach this topic by starting with concrete examples before moving to abstract forms. Use dynamic graphing tools to show how small changes in coefficients affect the parabola’s shape and position. Avoid rushing through form conversions—instead, emphasize why each form is useful and when to use it. Research shows that students retain concepts better when they manipulate parameters themselves and explain their observations aloud.

Successful learning looks like students confidently identifying key features of parabolas, converting between forms, and explaining transformations using precise mathematical language. They should sketch accurate graphs and justify their reasoning with evidence from their work.

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Watch Out for These Misconceptions

• During Parameter Sliders, watch for students assuming all parabolas open upwards.

  Use the sliders to toggle the sign of a (e.g., from 1 to -1) and ask students to predict the direction change before observing the graph. Have them record their predictions and explanations in a table.

• During Graph Matching Relay, watch for students confusing the vertex with the y-intercept.

  Provide equations and graphs where the vertex is not on the y-axis. Ask students to identify both the vertex and y-intercept on their graphs, then justify their choices in pairs using the equation features.

• During String Parabola Builders, watch for students assuming the axis of symmetry passes through the origin.

  Have students measure the distance from the string’s vertex to the y-axis and record the equation of the axis of symmetry (x = h) for their physical models. Compare results across groups to highlight variability.

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Methods used in this brief

• Carousel Brainstorm