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Geometry Critical Thinking Workshop

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Geometry Critical Thinking Workshop

A community team is planning a small outdoor space in Aotearoa New Zealand. Solve each design problem, show your reasoning, and include units where needed. Diagrams are not drawn to scale.

Angles and polygons

Use angle facts to check the design measurements.

1.Two straight paths cross. One of the angles is 68°. What is the size of the vertically opposite angle? Explain the angle fact you used.

2.A triangular garden bed has angles of 47° and 83°. Calculate the third angle and state the rule that makes your answer possible.

3.A regular hexagonal planter has six equal interior angles. Find the size of each interior angle. Show how you use the interior-angle sum.

Scale, area and reasoning

Use proportional reasoning and properties of shapes to test the proposed plans.

4.A similar triangular feature on the plan has a perimeter of 18 cm. The plan uses a scale of 1:50, meaning 1 cm on the plan represents 50 cm in the real space. What is the feature's real perimeter in metres?

5.A square paving panel has side length 4 m. A similar panel is made with side lengths multiplied by 1.5. What is the new panel's area? Explain why the area does not simply multiply by 1.5.

6.A student says, “Every rectangle is a square because both shapes have four right angles.” Is the claim correct? Give a counterexample and explain the difference between a rectangle and a square.

7.One interior angle of a parallelogram-shaped paved area is 112°. Find the other three interior angles. Explain how you know.

Explain and create

Communicate your geometric thinking clearly. More than one valid design may be possible.

8.A plan shows two similar triangles. One student says, “Their corresponding angles are equal, so their corresponding side lengths must also be equal.” Identify the error and give the correct relationship between the triangles' side lengths.

9.Write a short problem for another designer that uses at least two ideas from this workshop, such as angle rules, similarity, scale or area. Then solve it and justify your answer.