Mathematics · Grade 6

Active learning ideas

Area of Trapezoids and Rhombuses

Active learning works well for area of trapezoids and rhombuses because hands-on decomposition builds spatial reasoning and corrects formula misconceptions. When students physically cut, rearrange, or measure shapes, they internalize why formulas use specific dimensions rather than memorizing them blindly.

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

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Jigsaw
• 30 min · Pairs

Grid Paper Decomposition: Trapezoids

Give students trapezoid outlines on grid paper. They cut along the midline to form a rectangle, measure bases and height, then calculate area two ways and compare. Pairs share rearrangements with the class.

Construct a method for finding the area of a trapezoid by decomposing it.

Facilitation Tip

During Grid Paper Decomposition, have students trace trapezoid outlines with different base lengths to compare how the midline rectangle changes.

What to look for

Provide students with grid paper drawings of a trapezoid and a rhombus. Ask them to decompose each shape into simpler figures, label the dimensions, and calculate the area, showing their work for each step.

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Activity 02

• Jigsaw
• 40 min · Small Groups

Geoboard Stations: Rhombus Diagonals

Set up geoboards with elastics for rhombuses. Students measure diagonals, compute area using the formula, and verify by counting squares or base-height method. Rotate stations to try different rhombuses.

Explain how the area formula for a rhombus can be derived from its diagonals.

Facilitation Tip

At Geoboard Stations, ask students to stretch rubber bands to form rhombuses with diagonals of varying lengths before measuring their areas.

What to look for

Pose the question: 'How is finding the area of a rhombus similar to finding the area of a parallelogram, and how is it different?' Guide students to discuss the role of diagonals versus base and height.

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Activity 03

• Jigsaw
• 35 min · Small Groups

Quadrilateral Match-Up: Area Formulas

Provide cut-out quadrilaterals including trapezoids and rhombuses. Students match each to its decomposition diagram and formula, then justify with measurements. Discuss as whole class.

Differentiate between the area formulas for various quadrilaterals.

Facilitation Tip

For Quadrilateral Match-Up, require students to justify matches by calculating areas using both decomposed and formula methods.

What to look for

Give students a trapezoid with bases 8 cm and 12 cm, and height 5 cm. Ask them to write the formula they used (either decomposed or formula-based) and calculate the area. Then, provide a rhombus with diagonals 10 cm and 6 cm and ask for its area.

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Activity 04

• Jigsaw
• 25 min · Whole Class

Formula Derivation Relay: Class Challenge

Divide class into teams. Each member adds a step to derive trapezoid or rhombus area by decomposing on chart paper, passes to next teammate. First accurate relay wins.

Construct a method for finding the area of a trapezoid by decomposing it.

Facilitation Tip

In Formula Derivation Relay, rotate groups every 3 minutes so all students experience cutting, rearranging, and calculating.

What to look for

Provide students with grid paper drawings of a trapezoid and a rhombus. Ask them to decompose each shape into simpler figures, label the dimensions, and calculate the area, showing their work for each step.

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

A few notes on teaching this unit

Teach this topic by letting students struggle first with incorrect assumptions, then providing tools to test their ideas. Avoid giving formulas upfront. Research shows students retain area formulas better when they derive them through decomposition and peer discussion rather than direct instruction. Focus on guiding questions that prompt students to notice relationships between shapes.

Successful learning looks like students confidently decomposing trapezoids into a midline rectangle and two triangles, or splitting rhombuses into four right triangles along diagonals. They should articulate how the dimensions of these simpler shapes relate to the original figure’s area formula.

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Watch Out for These Misconceptions

• During Grid Paper Decomposition, watch for students averaging all four sides instead of just the two parallel bases.

  Ask students to cut along the midline they drew and rearrange the two resulting triangles to form a rectangle, then measure its length and width to see how the bases relate to the area.

• During Geoboard Stations, watch for students assuming a rhombus’s area equals side length squared, like a square.

  Have students measure both diagonals with the geoboard’s peg spacing, then compare this product to their calculated area using the formula.

• During Grid Paper Decomposition, watch for students using a slanted leg as the height for a trapezoid.

  Ask students to draw a dashed perpendicular line from the top base to the bottom base, then measure this vertical distance to confirm it matches the given height.

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Methods used in this brief

• Jigsaw