Additional Mathematics · Secondary 4

Active learning ideas

Maxima, Minima, and Rates of Change

Active learning helps students visualize reciprocal functions because the hyperbolic shape and asymptotic behavior are difficult to grasp through abstract discussion alone. By plotting points, manipulating graphs, and predicting outcomes, students build an intuitive understanding that leads to stronger retention of key features like domain restrictions and symmetry.

• MOE Syllabus Outcomes
• MOE Syllabus 4049: C1.7
• MOE Syllabus 4049: C1.10

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

Activity 01

• Problem-Based Learning
• 30 min · Pairs

Pairs Plotting: Reciprocal Tables

Pairs select values of x excluding zero, compute y for y = 1/x and y = 2/x, and plot both on shared graph paper. They draw asymptotes and label quadrants. Discuss how doubling k changes the curve.

How do we determine the nature of a stationary point?

Facilitation Tip

During Individual: Graph Matching, include one intentionally incorrect graph to prompt students to analyze why certain features do not match the function.

What to look for

Provide students with the function y = 6/x. Ask them to: 1. Sketch the graph, labeling the asymptotes. 2. Identify the coordinates of two points on the graph. 3. State the domain and range of the function.

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

Activity 02

• Problem-Based Learning
• 40 min · Small Groups

Small Groups: Desmos Exploration

Groups access Desmos to graph y = k/x for k = 1, 3, -1. They trace asymptotes, note domain, and slider-test transformations. Each group records three observations for class share.

What are connected rates of change?

What to look for

Display two graphs: y = 2/x and y = -3/x. Ask students: 'Which graph represents y = 2/x and why? What is the main difference in the shape and location of the branches compared to y = 2/x?'

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

Activity 03

• Problem-Based Learning
• 25 min · Whole Class

Whole Class: Prediction Relay

Project a base graph of y = 1/x. Call out changes like k=0.5 or add constant; students predict new features on mini-whiteboards. Reveal and correct as a class.

How can differentiation optimize real-world scenarios?

What to look for

Pose the question: 'Imagine you are designing a video game where the speed of an object is inversely proportional to its mass (speed = constant/mass). How would you explain to a player why an object with almost zero mass would have an infinitely fast speed, and why this is not possible in the real game?'

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

Activity 04

• Problem-Based Learning
• 20 min · Individual

Individual: Graph Matching

Provide printed graphs of y = k/x variants. Students match to equations, label asymptotes, and justify choices in writing.

How do we determine the nature of a stationary point?

What to look for

Provide students with the function y = 6/x. Ask them to: 1. Sketch the graph, labeling the asymptotes. 2. Identify the coordinates of two points on the graph. 3. State the domain and range of the function.

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

A few notes on teaching this unit

Experienced teachers begin with concrete examples, like y = 1/x and y = -2/x, before introducing general forms. They avoid starting with transformations, which can confuse students who haven’t yet internalized the basic shape. Research suggests spending time on point plotting first to build intuition before moving to technology-based explorations.

Students will accurately identify vertical and horizontal asymptotes, describe the domain and range, and explain how the constant k affects the graph’s steepness and quadrant placement. They will connect numerical patterns in tables to graphical behavior and articulate the difference between scaling and shifting transformations.

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Watch Out for These Misconceptions

• During Pairs Plotting: Reciprocal Tables, watch for students who assume the graph crosses the asymptote at x = 0.

  Ask students to calculate y for x = 0.1 and x = -0.1, then ask them to explain why division by zero is undefined and what this means for the graph’s behavior near the y-axis.

• During Small Groups: Desmos Exploration, watch for students who describe the graph as symmetric across the axes.

  Have students fold their printed graphs along the origin or use Desmos’ symmetry tool to observe point symmetry, and ask them to describe the difference between reflectional and rotational symmetry.

• During Whole Class: Prediction Relay, watch for students who think changing k shifts the graph horizontally.

  Ask students to sketch y = 2/x and y = 2/(x+1) on the same axes during the relay, then compare the effects of adding to the numerator versus the denominator to clarify the difference between scaling and shifting.

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Methods used in this brief

• Problem-Based Learning