Mathematics · Year 10

Active learning ideas

Solving Quadratic Equations Graphically

Plotting quadratic equations lets students see how changing coefficients transform the parabola’s shape and roots. Active tasks build spatial reasoning and connect symbolic rules to concrete images, which research shows strengthens retention and conceptual fluency for Year 10 learners.

• ACARA Content Descriptions
• AC9M10A06

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

Activity 01

• Think-Pair-Share
• 30 min · Pairs

Pair Graph Matching: Quadratic Intercepts

Provide cards with quadratic equations and their graphs. Pairs match them by identifying x-intercepts as roots, then test by substituting values. Groups share one match and explain discriminant clues.

Explain how to identify the roots of a function from its graphical representation.

Facilitation Tip

During Pair Graph Matching, circulate and ask each pair to justify why a particular graph matches their quadratic so students vocalize the connection between equations and intercepts.

What to look for

Provide students with a printed graph of a parabola. Ask them to: 1. Write down the approximate x-intercepts. 2. State the corresponding quadratic equation if the roots are integers. 3. Explain what these intercepts represent in terms of the equation.

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

Activity 02

• Think-Pair-Share
• 40 min · Small Groups

Small Group: Dynamic Graph Sliders

Use Desmos or graphing calculators for groups to vary a, b, c in y = ax² + bx + c. Observe changes in x-intercepts, record discriminant values, and hypothesize patterns. Present findings to class.

Analyze the relationship between the number of x-intercepts and the discriminant of a quadratic equation.

Facilitation Tip

In Small Group Dynamic Graph Sliders, give each group only one device to encourage collaborative parameter exploration and shared observation.

What to look for

On an index card, have students draw a parabola with two distinct x-intercepts. Below the graph, they should write: 1. The approximate values of the roots. 2. A statement about the discriminant (e.g., positive, negative, zero). 3. One sentence comparing the certainty of this graphical solution to an algebraic one.

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

Activity 03

• Think-Pair-Share
• 25 min · Whole Class

Whole Class: Root Prediction Relay

Display graphs one by one. Students write predicted roots and discriminant sign on mini-whiteboards, hold up answers. Discuss discrepancies, then reveal algebraic solutions for comparison.

Critique the accuracy of graphical solutions compared to algebraic solutions.

Facilitation Tip

For the Root Prediction Relay, stagger the order of equations so quicker finishers get a more complex case to analyze before sharing back with the class.

What to look for

Present students with two quadratic equations: y = x² - 4 and y = x² + 4. Ask them to: 1. Sketch the graphs of both equations. 2. Identify the x-intercepts for each graph. 3. Discuss why one graph has x-intercepts and the other does not, relating this to the discriminant.

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

Activity 04

• Think-Pair-Share
• 20 min · Individual

Individual: Sketch and Solve Challenge

Students sketch graphs for given quadratics, mark roots, and estimate discriminant. Check with algebra or software, reflect on accuracy in journals.

Explain how to identify the roots of a function from its graphical representation.

What to look for

Provide students with a printed graph of a parabola. Ask them to: 1. Write down the approximate x-intercepts. 2. State the corresponding quadratic equation if the roots are integers. 3. Explain what these intercepts represent in terms of the equation.

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

A few notes on teaching this unit

Start with a quick sketch of y = x² – 4 versus y = x² + 4 on the board to surface prior knowledge. Emphasize that graphs reveal the number of real roots but not always their exact values; follow up each visual task with a 60-second algebraic check so students compare strengths and limits in real time. Avoid rushing to the discriminant formula before students have internalized its meaning through repeated graphical evidence.

By the end of the activities, students will confidently link each x-intercept to a root, explain how the discriminant predicts intercepts, and choose when to use graphical versus algebraic methods for solving.

---

Watch Out for These Misconceptions

• During Pair Graph Matching, watch for students who assume every quadratic graph crosses the x-axis twice.

  Prompt each pair to find the matching graph for y = x² + 1 and explain why it has no x-intercepts, using the visual trace of the curve above the axis.

• During Pair Graph Matching, watch for students who treat graphical roots as exact solutions.

  Have each pair record the approximate intercepts, then switch to their calculators to solve the equation algebraically and compare the decimal estimates to the precise values.

• During Whole Class Root Prediction Relay, watch for students who confuse the y-intercept with the roots.

  Before teams sketch, ask them to label the y-intercept with its coordinates and circle the x-intercepts, explicitly naming where y equals zero.

---

Methods used in this brief

• Think-Pair-Share