Mathematics · JC 2

Active learning ideas

Testing a Population Mean (Large Samples)

Active learning works because integration by substitution requires students to think flexibly about composite functions and their derivatives. By moving between algebraic manipulation and conceptual reasoning, students solidify their understanding of the chain rule in reverse, which is essential for recognizing when substitution applies.

• MOE Syllabus Outcomes
• 8865/01 Statistics: Hypothesis testing of population mean (large sample)
• 8865/01 Statistics: p-value interpretation

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Think-Pair-Share
• 30 min · Pairs

Pair Practice: u-Selection Challenge

Pairs receive cards with 10 composite integrals. They select u, write du, and outline substitution steps on mini-whiteboards. Switch cards with another pair after 5 minutes for peer review and revision. Conclude with class share-out of trickiest examples.

How do we conduct a z-test for a population mean?

Facilitation Tip

During the Pair Practice: u-Selection Challenge, circulate and ask each pair to explain their choice of u aloud, reinforcing the chain rule connection.

What to look for

Present students with three integrals: ∫ x(x^2 + 1)^3 dx, ∫ sin(x)cos(x) dx, and ∫ e^x dx. Ask them to identify which integral(s) can be solved using substitution and to briefly explain why for each.

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

Activity 02

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

Small Group Relay: Substitution Chain

Divide class into groups of 4. Each group lines up; first student solves first step of integral (choose u), tags next for du and substitution, and so on until back-substitution. First group to finish correctly wins. Repeat with 3-4 integrals.

What is a p-value and how is it used to make a decision?

Facilitation Tip

In the Small Group Relay: Substitution Chain, provide each group with a dry-erase board to write each step clearly, making errors visible and easier to address.

What to look for

Provide students with the integral ∫ 2x * sqrt(x^2 + 5) dx. Ask them to: 1. Identify the substitution u. 2. Calculate du. 3. Write the integral in terms of u. 4. State the final answer after back-substitution.

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

Activity 03

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

Whole Class: Integral Matching Game

Project 12 integrals and 12 substitution setups. Students stand and point to matches as you reveal hints. Discuss mismatches as a class, then have volunteers solve one fully on board. Use for definite integrals next round.

How do we state the conclusion in the context of the original problem?

Facilitation Tip

For the Whole Class: Integral Matching Game, assign each student a role, such as ‘u-identifier’ or ‘limit adjuster,’ to ensure everyone participates actively.

What to look for

Pose the question: 'When might integration by substitution NOT be the most efficient method for solving an integral?' Have students discuss in pairs and share scenarios where other integration techniques might be preferable.

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

Activity 04

• Think-Pair-Share
• 20 min · Individual

Individual: Guided Substitution Worksheet

Provide worksheets with partially worked integrals. Students fill gaps: choose u, compute du, integrate, back-substitute. Include 8 problems escalating in complexity, with space for self-check by differentiating answers.

How do we conduct a z-test for a population mean?

Facilitation Tip

With the Individual: Guided Substitution Worksheet, include a ‘think-aloud’ section where students write their reasoning for each step before performing calculations.

What to look for

Present students with three integrals: ∫ x(x^2 + 1)^3 dx, ∫ sin(x)cos(x) dx, and ∫ e^x dx. Ask them to identify which integral(s) can be solved using substitution and to briefly explain why for each.

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

A few notes on teaching this unit

Experienced teachers approach this topic by first grounding substitution in the chain rule, using visuals like function machines to show composition and decomposition. Avoid rushing to procedures—instead, emphasize the ‘why’ through concrete examples. Research suggests that students benefit from multiple representations, so pair algebraic steps with graphical or numerical checks to deepen understanding.

Successful learning looks like students confidently identifying the inner function for substitution, correctly computing du, and accurately back-substituting to the original variable. They should also recognize when substitution is appropriate and when other methods may be better.

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Watch Out for These Misconceptions

• During Pair Practice: u-Selection Challenge, watch for students who omit the derivative factor after substitution.

  Prompt partners to verbalize the chain rule link by asking, ‘What is the derivative of your chosen u? How does it relate to the integrand?’ This forces them to include du/dx explicitly.

• During Small Group Relay: Substitution Chain, watch for students who select u as the entire integrand instead of the inner function.

  Have groups compare their u choices on the board and discuss which one simplifies the integral most effectively. Peer teaching during the relay helps clarify nested function identification.

• During Whole Class: Integral Matching Game, watch for students who do not adjust limits for definite integrals.

  Use the matching game to highlight this error visually—assign one student to act as the ‘limit checker’ who must adjust limits before proceeding. This collaborative role reinforces the importance of limit changes.

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Methods used in this brief

• Think-Pair-Share