Mathematics · Grade 6

Active learning ideas

Area of Triangles

Active learning works for this topic because students need to physically manipulate shapes to see the relationship between triangles and rectangles. Hands-on experiences help them move from abstract formulas to concrete understanding, making the concept memorable and reducing errors in application.

• Ontario Curriculum Expectations
• 6.G.A.1

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

Activity 01

• Gallery Walk
• 30 min · Pairs

Paper Cutting: Triangle Pairs to Rectangles

Students draw triangles on grid paper using given bases and heights, cut them out, and pair identical triangles to form rectangles. They calculate the rectangle area, divide by 2 for the triangle, and record findings. Pairs discuss why this works for any triangle.

Explain how the area of a triangle is related to the area of a rectangle with the same base and height.

Facilitation Tip

During Paper Cutting: Triangle Pairs to Rectangles, remind students to cut carefully along the height to ensure accurate rearrangement.

What to look for

Provide students with three different triangles (acute, obtuse, right-angled) drawn on grid paper, each with the base and height clearly labeled. Ask them to calculate the area of each triangle and write down the formula they used. Check for correct application of the formula and accurate calculation.

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

Activity 02

• Gallery Walk
• 40 min · Small Groups

Geoboard Challenge: Build and Measure

Provide geoboards, rubber bands, and rulers. Students construct triangles, label base and height, compute areas, then modify dimensions to predict area changes. They share results on class charts to spot patterns.

Construct a method for finding the area of any triangle.

Facilitation Tip

For Geoboard Challenge: Build and Measure, circulate to check that students understand height must be measured perpendicular to the base.

What to look for

Present students with a rectangle and a triangle that share the same base and height. Ask: 'How can you show, using drawings or words, that the area of the triangle is exactly half the area of the rectangle?' Facilitate a discussion where students share their methods, possibly involving cutting the rectangle or placing two triangles together.

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

Activity 03

• Stations Rotation
• 45 min · Small Groups

Stations Rotation: Triangle Decompositions

Set up stations: decompose irregular triangles into rectangles on dot paper, measure triangular book covers, use string for heights on wall triangles, and sort triangles by area. Groups rotate, documenting methods at each.

Analyze how changing the base or height affects a triangle's area.

Facilitation Tip

In Station Rotation: Triangle Decompositions, provide grid paper for students to sketch and verify their decompositions.

What to look for

Give each student a card with a triangle. One side of the card has a triangle with base = 10 cm and height = 6 cm. The other side has a triangle with base = 12 cm and height = 5 cm. Ask students to calculate the area of both triangles and write one sentence explaining which triangle has a larger area and why.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 04

• Gallery Walk
• 35 min · Whole Class

Whole Class: Area Prediction Relay

Display projected triangles with changing bases or heights. Teams predict areas, justify with formula, then verify by sketching rectangles. Correct predictions earn points; debrief misconceptions as a group.

Explain how the area of a triangle is related to the area of a rectangle with the same base and height.

Facilitation Tip

During Whole Class: Area Prediction Relay, encourage students to explain their reasoning aloud before revealing the answer.

What to look for

Provide students with three different triangles (acute, obtuse, right-angled) drawn on grid paper, each with the base and height clearly labeled. Ask them to calculate the area of each triangle and write down the formula they used. Check for correct application of the formula and accurate calculation.

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

A few notes on teaching this unit

Start with concrete experiences before moving to abstract formulas. Avoid rushing students into memorizing A = (1/2)bh without understanding why it works. Research suggests that students who physically manipulate shapes retain the concept longer. Use guided questions to prompt reflection, such as asking students to compare their triangle to a rectangle with the same base and height.

Successful learning looks like students confidently identifying base and height, applying the formula A = (1/2)bh correctly, and explaining why the area is half that of a rectangle with the same base and height. They should also justify their reasoning when comparing areas or predicting changes after modifications.

---

Watch Out for These Misconceptions

• During Paper Cutting: Triangle Pairs to Rectangles, watch for students who cut triangles incorrectly and fail to form a rectangle, leading to incorrect area calculations.

  Have students re-cut their triangles carefully along the height and remind them that the height must be perpendicular to the base for the pieces to align properly.

• During Geoboard Challenge: Build and Measure, watch for students who measure height along the slant of a side rather than perpendicular to the base.

  Ask students to draw the height as a dashed line on their geoboards and measure it with a ruler to confirm it is perpendicular.

• During Station Rotation: Triangle Decompositions, watch for students who confuse area with perimeter or misidentify the base and height.

  Provide a reference sheet with labeled examples of base and height, and have students justify their choices before calculating area.

---

Methods used in this brief

• Gallery Walk
• Stations Rotation