Mathematics · Year 10

Active learning ideas

Graphs of Trigonometric Functions

Active learning works for graphs of trigonometric functions because students must physically or digitally manipulate graphs to see patterns. Plotting points, matching shapes, and moving between representations helps students build mental models of periodicity and asymptotes that static notes cannot provide.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Geometry and Measures

• 20–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Gallery Walk
• 30 min · Pairs

Graph Matching: Trig Functions

Prepare cards with equations, graphs, and descriptions of sin, cos, tan. In pairs, students match sets and justify choices by noting amplitude, period, and key points. Follow with a class share-out to verify matches.

Compare the key features of the sine, cosine, and tangent graphs.

Facilitation Tip

During Graph Matching: Trig Functions, circulate and ask pairs to justify why they matched a specific graph, reinforcing precise vocabulary like amplitude and period.

What to look for

Provide students with printed graphs of y = sin(x), y = cos(x), and y = tan(x) for x from 0 to 360 degrees. Ask them to label the period of each graph and identify one x-intercept for each.

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

Activity 02

• Gallery Walk
• 45 min · Small Groups

Desmos Exploration: Phase Shifts

Using Desmos, small groups input y=sin(x), y=cos(x), y=tan(x) and adjust sliders for amplitude and period. They sketch predictions first, then compare outputs and note changes in key features.

Predict the values of sin(x), cos(x), and tan(x) for angles beyond 90 degrees using their graphs.

Facilitation Tip

For Desmos Exploration: Phase Shifts, pause the class at key moments to discuss how horizontal shifts affect the starting point of the graph.

What to look for

On an exit ticket, ask students to sketch a rough graph of y = cos(x) for 0 to 360 degrees. Then, ask them to state the value of cos(90) and cos(270) based on their sketch and explain why the tangent graph has vertical asymptotes.

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

Activity 03

• Gallery Walk
• 20 min · Whole Class

Human Graph: Sine Wave

Mark axes on the floor with tape. Whole class students hold cards with x,y values for y=sin(x) from 0 to 360 degrees, forming the graph. Discuss shape, then repeat for cosine.

Explain the periodic nature of trigonometric functions.

Facilitation Tip

When running Human Graph: Sine Wave, remind students that their bodies represent points on the graph, so spacing must be consistent with the function's period.

What to look for

Pose the question: 'How are the sine and cosine graphs similar, and how are they different?' Encourage students to discuss features like starting points, amplitude, and period. Then, ask how the tangent graph fundamentally differs from the other two.

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

Activity 04

• Gallery Walk
• 25 min · Pairs

Key Features Sort: Comparison Cards

Distribute cards listing features like 'period 180 degrees' or 'asymptotes at 90 degrees'. Individuals or pairs sort into sin, cos, tan columns, then explain reasoning to the group.

Compare the key features of the sine, cosine, and tangent graphs.

Facilitation Tip

Use Key Features Sort: Comparison Cards to prompt students to verbalize why sine and cosine are not identical before shifting into the tangent card sort.

What to look for

Provide students with printed graphs of y = sin(x), y = cos(x), and y = tan(x) for x from 0 to 360 degrees. Ask them to label the period of each graph and identify one x-intercept for each.

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

A few notes on teaching this unit

Experienced teachers approach this topic by connecting graphs to the unit circle first, then using technology to animate transformations. Avoid rushing to formulas; instead, build intuition through multiple representations. Research suggests students grasp periodicity better when they observe how sine and cosine waves repeat every 360 degrees, while tangent's 180-degree cycle emerges from its definition in terms of sine and cosine.

Successful learning looks like students confidently sketch sine, cosine, and tangent graphs, accurately labeling amplitude, period, and key points. They should explain differences between the functions and predict values using the unit circle without relying on memorized rules alone.

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Watch Out for These Misconceptions

• During Graph Matching: Trig Functions, watch for students who claim sine and cosine graphs are identical but shifted.

  Hand pairs two unlabeled graphs and ask them to compare starting points and zeros. Have them mark where each graph crosses the x-axis and identify the y-value at x=0 to highlight differences before matching.

• During Desmos Exploration: Phase Shifts, watch for students who assume tangent has the same period as sine and cosine.

  Use the graphing tool to zoom in on one period of tangent and ask students to count repetitions from 0 to 360 degrees. Guide them to observe that tangent repeats twice in this interval while sine and cosine repeat once.

• During Human Graph: Sine Wave, watch for students who assume all trigonometric functions are defined for all angles.

  After forming the sine wave, pause at odd multiples of 90 degrees and ask students to consider the value of tangent at those points. Direct their attention to the asymptotes in the graph cards to connect the physical representation to the discontinuities.

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Methods used in this brief

• Gallery Walk