Mathematics · Year 12

Active learning ideas

Introduction to Limits

Introducing limits is foundational for calculus, and active learning helps students build intuition before formalizing concepts. Engaging with functions visually and numerically allows students to grapple with the idea of 'approaching' a value, a concept that can be abstract at first.

• ACARA Content Descriptions
• AC9MFM01

• 30–45 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Think-Pair-Share
• 45 min · Pairs

Graphical Limit Exploration: Zooming In

Students use graphing software or calculators to examine the behavior of functions near specific x-values. They 'zoom in' repeatedly on the graph to observe the y-values approaching a particular number, reinforcing the idea of getting closer and closer.

Explain how the concept of a limit allows us to define change at a single point.

Facilitation Tip

During the Think-Pair-Share, encourage students to first focus on their individual graphical observations before discussing with a partner to ensure everyone has a starting point.

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

Activity 02

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

Numerical Limit Investigation: Tables of Values

Working in small groups, students create tables of values for a function, inputting x-values that approach a specific number from both the left and the right. They then analyze the resulting y-values to infer the limit.

Predict the behavior of a function as it approaches a specific value or infinity.

Facilitation Tip

In the Inquiry Circle, guide students to formulate questions that can be investigated using the provided graphs, rather than simply asking for definitions.

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

Activity 03

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

Limit Existence Scenarios

Present students with various graphs exhibiting different limit behaviors (existing, not existing due to jump, oscillation, asymptote). Students identify the limit at specified points or explain why it doesn't exist, fostering critical analysis.

Analyze graphical representations to determine if a limit exists at a given point.

Facilitation Tip

When students are working on the 'Tables of Values' activity, prompt them to consider the values from both the left and the right side of the target x-value to build their understanding of convergence.

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

A few notes on teaching this unit

This topic benefits from a constructivist approach where students build understanding through exploration. Start with activities that allow students to 'see' and 'feel' the behavior of functions near a point, like graphical zooms or tables, before introducing formal notation. Resist the urge to jump straight to epsilon-delta definitions; focus on building a strong conceptual foundation first.

Students will be able to articulate that the limit of a function as x approaches a value is about the function's behavior near that value, not necessarily at the value itself. They will also be able to identify scenarios where a limit exists and where it does not, justifying their reasoning with graphical or numerical evidence.

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Watch Out for These Misconceptions

• During 'Graphical Limit Exploration: Zooming In', watch for students who assume the limit must be the y-value of any point on the graph at the target x, even if it's a hole.

  Redirect students to focus on the y-values the function is *approaching* as they zoom in, using the graph to show that the function gets arbitrarily close to a specific y-value, even if that point is not included on the graph.

• During 'Limit Existence Scenarios', students might think a limit doesn't exist only if the graph goes to infinity.

  Use the graphs with jumps or breaks to prompt discussion: 'Why doesn't this function approach a single y-value here, even though it doesn't go to infinity?' This encourages them to consider other reasons for non-existence.

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Methods used in this brief

• Think-Pair-Share