Change the shape. Preserve the space. Defend the conclusion.
Opening chapter · Three lessons, approximately 50 minutes each.
Use graph paper. This site does not save or submit responses.
By the end of these lessons, you should be able to decompose composite regions, explain the area of a parallelogram, and derive the area of a triangle. You will also distinguish a numerical result from the claim it supports.
Begin with 1.1 — Area and Evidence →
Area: the space covered by a flat region, measured in square units. Base: a chosen side. Height: the perpendicular distance to the opposite side or vertex. Perpendicular: meeting at a right angle. Decompose: split into simpler pieces.
Mara noticed a large section of wooden wall beside the harbor window. One board had come loose. Its edges slanted, but the top and bottom were parallel.
“We need to cover this before rain gets in,” Tess said.
Jun measured the bottom. “Six meters. The sloping side is five. Thirty square meters of covering?”
Eli held a carpenter’s square against the panel. “That’s not the distance straight across between the parallel edges.”
Mara reached for a paper copy of the shape. “What if we move a piece?”
Without looking back: a 10 m × 8 m rectangle has a 4 m × 3 m corner excluded. Find the remaining area and explain your units.
The panel is a parallelogram with base 6 m, perpendicular height 4 m, and sloping side 5 m. Use the button to move the left triangle to the right.
Schematic illustration. The moving piece is a cut-and-slide model; dimensions describe the original panel.
Parallelogram area = base × perpendicular height. The sloping side usually is not the height. A parallelogram with base 8 cm and perpendicular height 3 cm has area 24 cm².
1. 36 cm². 2. 35 m²; the sloping side is unused. 3. 6 cm. 4. Both have area 15 square units; drawings vary.
On graph paper: explain to Jun why the panel requires 24 m² of covering before allowances for overlap or waste. Include a labeled sketch.
Mara slid the paper triangle into place. “Same pieces. Different outline.”
Behind the loose board was a folded sheet, covered in triangular sketches. Tess read its heading: Window braces — check before ordering.
The sheet showed a triangular brace with a base of 8 cm and a perpendicular height of 5 cm.
“Forty square centimeters,” Jun said.
“For what shape?” Eli asked.
Tess traced the triangle onto another sheet. “Let’s make a second one.”
A parallelogram has base 11 cm and perpendicular height 4 cm. Find its area. What does perpendicular mean?
Two congruent copies of this right triangle fit together into a rectangle. Toggle the second copy, then explain how the rectangle helps measure one triangle.
Triangle area = ½ × base × perpendicular height. For any triangle, two congruent copies can be arranged into a parallelogram. One triangle occupies half that parallelogram’s area. For obtuse triangles, the height may fall outside the triangle.
Example: base 10 m and perpendicular height 3 m give area ½ × 10 × 3 = 15 m².
1. 42 cm². 2. 27 m². 3. 5 cm. 4. No. Moving the top vertex along a line parallel to the base preserves the height and area while changing the shape.
Draw and label a triangle with area 18 square units. Show how a second congruent copy supports your result.
“Twenty square centimeters,” Tess said. “Forty for the pair.”
Eli turned the sheet toward the window. On its reverse was a rough sketch of the workshop’s back corner—and an arrow pointing to a cabinet.
The room was still a question. Now they knew where to look next.
Complete these on graph paper before opening the solutions. Include units and reasoning. These tasks check the opening lessons; surface area and the rest of the Grade 6 course will follow.
1. 126 − 15 = 111 m²; also 9 × 9 + 5 × 6 = 111 m². 2. 78 cm²; use perpendicular height, not sloping side. 3. 60 cm²; two congruent triangles form a parallelogram of area 120 cm². 4. 9 cm. 5. No: 6 × 10 and 5 × 12 rectangles each have area 60 m². 6. The records describe a 12 m² corner region; they do not prove its present contents, accessibility, or accuracy.
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