Mathematics · Grade 8

Active learning ideas

Surface Area of Cylinders

Active learning helps students grasp the surface area of cylinders because folding paper into nets makes the abstract formula concrete. When students physically manipulate shapes, they see how the formula connects to the parts of the cylinder they can touch and measure themselves.

• Ontario Curriculum Expectations
• Ontario Curriculum Mathematics 2020, Grade 8, Algebra C1.1: identify and compare a variety of patterns, including those that are found in real-life contexts, and distinguish between linear and non-linear patterns
• Ontario Curriculum Mathematics 2020, Grade 8, Algebra C1.2: create and translate patterns using various representations, including tables of values, graphs, and equations
• Ontario Curriculum Mathematics 2020, Grade 8, Algebra C1.3: determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in patterns

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

Activity 01

• Experiential Learning
• 35 min · Small Groups

Net Building Lab: Unfold and Measure

Provide templates for cylinder nets on cardstock. Students cut, assemble with tape, measure radius and height, then calculate surface area using the formula. Compare total paper area to calculated value and discuss discrepancies.

Explain how the net of a cylinder relates to its surface area formula.

Facilitation Tip

During the Net Building Lab, circulate with scissors and tape to help students align edges precisely when folding their nets.

What to look for

Provide students with diagrams of two different cylindrical cans, each with labeled radius and height. Ask them to calculate the surface area for both and write one sentence comparing which can uses more material.

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Activity 02

• Experiential Learning
• 45 min · Small Groups

Optimization Relay: Design Efficient Cans

Teams receive a fixed volume requirement. Each pair tests three radius-height combinations, calculates surface areas, and passes optimal design to next pair for graphing. Class votes on best low-material design.

Analyze how changes in radius or height affect the surface area of a cylinder.

Facilitation Tip

For the Optimization Relay, set a 10-minute timer so teams must justify their design choices before moving to the next can.

What to look for

Give students a cylinder with a radius of 5 cm and a height of 10 cm. Ask them to: 1. Sketch its net. 2. Write the formula for its surface area. 3. Calculate the total surface area.

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Activity 03

• Experiential Learning
• 40 min · Pairs

Dimension Slider: Interactive Exploration

Use grid paper or online tools for students to vary radius and height systematically. Record surface areas in tables, plot graphs, and identify patterns like minimal surface area points. Share findings in a gallery walk.

Design a cylindrical container to minimize material usage for a given volume.

Facilitation Tip

In the Dimension Slider activity, ask students to verbalize how changing one slider affects the net’s area before recording results.

What to look for

Pose the question: 'If you double the radius of a cylinder but keep the height the same, does the surface area double? Explain your reasoning using the formula and by referring to the cylinder's net.'

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Activity 04

• Experiential Learning
• 30 min · Pairs

Real-World Audit: Classroom Cylinders

Students measure surface areas of classroom items like cups or cans using string for circumference. Calculate, label, and create a display comparing actual versus predicted material usage.

Explain how the net of a cylinder relates to its surface area formula.

Facilitation Tip

During the Real-World Audit, assign each group a different classroom cylinder to measure and present their findings to the class.

What to look for

Provide students with diagrams of two different cylindrical cans, each with labeled radius and height. Ask them to calculate the surface area for both and write one sentence comparing which can uses more material.

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

A few notes on teaching this unit

Start with nets because they bridge prior knowledge of circles and rectangles to the new cylinder shape. Avoid rushing to the formula—instead, let students derive it through measurement and discussion. Research shows that when students articulate their own formulas, they retain concepts longer and make fewer calculation errors.

Students will confidently explain how the net’s rectangle and circles relate to the cylinder’s surface area formula. They will measure real objects, calculate accurately, and justify their reasoning using both the formula and the net’s structure.

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Watch Out for These Misconceptions

• During the Net Building Lab, watch for students who cut the net but forget to include the two circular bases in their calculations.

  Have students lay the folded net flat and count the parts aloud together: two circles and one rectangle. Ask them to label each part with its area formula before calculating.

• During the Net Building Lab, watch for confusion between the rectangle’s width and height.

  Ask students to trace the edge of the cylinder’s base onto paper to find the rectangle’s width, then measure the cylinder’s height to label the rectangle’s length. Compare their traced rectangle to the actual cylinder.

• During the Dimension Slider activity, watch for students who accidentally use diameter in place of radius in the formula.

  Have peers check each other’s slider settings against the formula’s components. Ask them to explain why radius is used twice in the formula while diameter is not.

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Methods used in this brief

• Experiential Learning