Additional Mathematics · Secondary 3

Active learning ideas

Rates of Change

Graphing quadratic functions is a visual and tactile topic where students benefit from building curves themselves rather than memorizing steps. Active learning works here because plotting points in order, comparing shapes, and discussing shifts deepen understanding of symmetry and transformations more than passive reading or listening.

• MOE Syllabus Outcomes
• C1.10 Connected rates of change
• C1.11 Formulation of equations from word problems

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

Activity 01

• Think-Pair-Share
• 25 min · Pairs

Pairs Plotting: Vertex First Challenge

Pairs select a quadratic equation and calculate the vertex coordinates first. They build a table with four x-values symmetric around the vertex, plot points on graph paper, and sketch the parabola. Partners check each other's axis of symmetry and discuss any asymmetries.

How do we relate the rates of change of two connected variables?

Facilitation Tip

During Pairs Plotting, circulate and ask each pair to explain why they chose their x-values around the vertex before they plot.

What to look for

Provide students with the equation y = 2x^2 - 4x + 1. Ask them to calculate the x-coordinate of the vertex and identify the y-intercept. Then, have them plot these two points and sketch the axis of symmetry.

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

Activity 02

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

Small Groups: Transformation Match-Up

Provide cards with quadratic equations and their graphs. Groups match them by predicting shifts from changes in coefficients, then verify by plotting one example per equation on shared paper. Discuss why certain changes stretch or reflect the parabola.

What is the role of the chain rule in solving related rates problems?

Facilitation Tip

For Transformation Match-Up, provide colored pencils so groups can trace and compare graphs side-by-side.

What to look for

Give students two equations: y = x^2 + 3 and y = x^2 - 3. Ask them to describe in one sentence how the graphs of these two equations will differ in position and to identify the y-intercept for each.

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

Activity 03

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

Whole Class: Prediction Parade

Display an equation on the board. Students predict and sketch the graph individually on slips of paper, then parade to tape sketches under the correct description (e.g., 'vertex at (2, -1)'). Reveal the accurate plot and tally predictions.

How can calculus model physical phenomena like expanding volumes?

Facilitation Tip

In Prediction Parade, pause after each student shares their guess to ask the class to justify or challenge the prediction using the equation.

What to look for

Pose the question: 'Why is it important to include the vertex in your table of values when graphing a quadratic function?' Facilitate a brief class discussion, encouraging students to reference the shape and key features of a parabola.

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

Activity 04

• Think-Pair-Share
• 20 min · Individual

Individual: Table Tweaks

Give students a poorly designed table for a quadratic. They revise it to include the vertex and symmetric points, replot the graph, and note improvements in accuracy. Submit before-and-after graphs with justifications.

How do we relate the rates of change of two connected variables?

Facilitation Tip

During Table Tweaks, have students swap their tables with a partner to verify the symmetry before plotting.

What to look for

Provide students with the equation y = 2x^2 - 4x + 1. Ask them to calculate the x-coordinate of the vertex and identify the y-intercept. Then, have them plot these two points and sketch the axis of symmetry.

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

A few notes on teaching this unit

Start with concrete examples before abstract formulas. Have students graph y = x^2 by hand first, then y = -x^2, so they see the effect of ‘a’ immediately. Avoid teaching vertex form too early; let students discover the vertex formula through repeated calculations in tables. Research shows that students retain transformations better when they manipulate graphs physically, so provide grid paper and rulers for accuracy.

By the end of these activities, students should confidently identify the vertex, axis of symmetry, and intercepts from any quadratic equation. They should also explain how changes to coefficients affect the graph’s position and shape, using precise vocabulary like ‘vertical shift’ and ‘direction of opening’ in their discussions.

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Watch Out for These Misconceptions

• During Pairs Plotting, watch for students who assume all parabolas open upwards.

  Have these pairs graph y = x^2 and y = -x^2 side-by-side, then ask them to compare the two curves and explain how the coefficient a determines the direction of opening based on their observations.

• During Pairs Plotting, watch for students who place the vertex at x=0 without calculation.

  Prompt them to calculate the vertex using -b/(2a) for their given equation, then justify the x-coordinate’s position by comparing it to the linear term in the equation.

• During Transformation Match-Up, watch for students who think changing the constant term shifts the graph horizontally.

  Have them plot y = x^2 + 2 and y = (x + 2)^2 next to each other, then ask which graph moved vertically and which moved horizontally, using their plotted points to correct the misunderstanding.

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Methods used in this brief

• Think-Pair-Share