Mathematics · 12th Grade

Active learning ideas

Continuity and Discontinuities

Active learning works for continuity and discontinuities because students often mistake visual recognition for conceptual understanding. Moving beyond graphs to the three formal conditions forces them to articulate why a function fails to be continuous at a point, not just identify the hole or jump.

• Common Core State Standards
• CCSS.Math.Content.HSF.IF.B.4

• 15–25 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 25 min · Pairs

Three-Condition Checklist Activity

Students receive six functions (graphical and algebraic) and systematically verify each continuity condition at a specified point, recording pass or fail for each. Pairs compare results and reconcile any disagreements by tracing back to the specific condition checked differently, building precision before the class debrief.

Analyze the conditions required for a function to be continuous at a point.

Facilitation Tip

In the Three-Condition Checklist Activity, require students to write the exact limit value and function value for each point rather than just checking boxes.

What to look for

Provide students with three function graphs, each exhibiting a different type of discontinuity. Ask them to label each discontinuity type (removable, jump, infinite) and write one sentence justifying their classification for each graph.

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

Activity 02

• Gallery Walk
• 20 min · Small Groups

Gallery Walk: Classify the Discontinuity

Post graphs around the room, each containing one discontinuity. Students annotate the type (removable, jump, or infinite) and explain what would need to change to repair each one -- or why a jump discontinuity cannot be fixed by redefining a single point.

Differentiate between the three main types of discontinuities and their graphical implications.

Facilitation Tip

During the Gallery Walk, position the discontinuity classification chart at a central location so students must bring their justifications to compare with peers before finalizing answers.

What to look for

Present students with a piecewise function. Ask them to: 1. Check the three conditions for continuity at the point where the function definition changes. 2. Classify the discontinuity if one exists.

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

Activity 03

• Collaborative Problem-Solving
• 25 min · Small Groups

Piecewise Construction Challenge

Small groups build a piecewise function that contains exactly one of each discontinuity type. Groups then trade functions with another group for verification -- the receiving group identifies each discontinuity, classifies it, and confirms it matches the builder's intent.

Justify why a function might be discontinuous in a real-world model.

Facilitation Tip

For the Piecewise Construction Challenge, provide a rubric that explicitly scores how well students satisfy all three continuity conditions at the boundary point, not just the final graph appearance.

What to look for

Pose the question: 'Why is it important for engineers or economists to understand and identify discontinuities in their mathematical models?' Facilitate a class discussion where students share examples and explain the real-world consequences of these discontinuities.

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

Activity 04

• Think-Pair-Share
• 15 min · Pairs

Think-Pair-Share: Real-World Discontinuities

Students brainstorm real contexts that produce each discontinuity type: tiered pricing or tax brackets (jump), a function redefined at an isolated point (removable), and a physical quantity that blows up at a boundary (infinite). Pairs justify their classifications and share examples the class may not have considered.

Analyze the conditions required for a function to be continuous at a point.

What to look for

Provide students with three function graphs, each exhibiting a different type of discontinuity. Ask them to label each discontinuity type (removable, jump, infinite) and write one sentence justifying their classification for each graph.

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

A few notes on teaching this unit

Teach this topic by making the formal definition the anchor for all discussions. Avoid letting students rely on visual shortcuts; instead, insist they compute limits and function values at every point under consideration. Research shows that students who practice articulating the three conditions in multiple contexts develop a stronger conceptual foundation than those who only classify discontinuities graphically.

Successful learning looks like students specifying exact points and intervals when discussing continuity, classifying discontinuities correctly with justifications tied to the formal definition, and using precise language such as 'continuous at x = a' versus 'continuous on [a, b]' to describe function behavior.

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Watch Out for These Misconceptions

• During the Three-Condition Checklist Activity, watch for students labeling a function as continuous everywhere except at the hole.

  Have them specify an interval around the hole and explicitly write 'not continuous on [a, b]' where a and b are points flanking the hole, emphasizing that continuity is not a global property.

• During the Gallery Walk: Classify the Discontinuity, watch for students claiming a removable discontinuity can be fixed with any value.

  Direct them to the Gallery Walk chart that requires them to state the limit value and show that the assigned value must match it to satisfy the third condition.

• During the Piecewise Construction Challenge, watch for students describing functions as either continuous or discontinuous everywhere.

  Ask them to write the exact interval where the function is continuous and the exact point of discontinuity, reinforcing the point-by-point and interval-based language.

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Methods used in this brief

• Collaborative Problem-Solving
• Gallery Walk
• Think-Pair-Share