Advanced Geometry in the Real World

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Part 1: Calculate and Choose Carefully

Show your working, include units, and check whether your answer is reasonable.

1. A rectangular garden is 6 m long and 4 m wide.

a) Calculate the distance around the garden.

b) Calculate the amount of ground inside the garden.

c) Which answer would be needed to buy fencing? Which answer would be needed to buy soil or grass? Explain why.

2. A worker measures each of the four sides of the garden 1 cm too long.

a) How much too large will the total boundary measurement be?

b) Explain why a small measurement error can become more important when it is repeated around a boundary.

3. An 8 m by 5 m dog exercise area needs both fencing and a new grass surface.

a) Calculate the distance around the area.

b) Calculate the area inside the area.

c) State which calculation is useful for the fencing decision and which is useful for the grass decision.

4. Two students calculate the dog exercise area.

Student A: 8 × 5 = 40, so 40 m is needed for the fence.

Student B: 8 + 5 + 8 + 5 = 26, so the area is 26 m².

Critique both methods. Identify the error in each statement and give the correct result with its unit.

Part 2: Reason, Communicate and Design

5. A student writes for the 6 m by 4 m garden: “Perimeter = 6 × 4 = 24 m.” Correct the calculation and explain what quantity the student has actually found.

6. A supplier offers 40 m² of grass for the 8 m by 5 m dog exercise area. Is this amount enough to cover the area exactly? Use a calculation and a reasonableness check to justify your answer.

7. Draw and label a rectangle representing the garden. Show the length and width, and write one statement explaining whether your diagram helps communicate the perimeter or the area problem.

8. Design one similar real-world problem using a rectangle. It must include two side measurements and ask for both a boundary decision and an inside-area decision. Solve your problem, include units, and explain how you checked that your answers are reasonable.

Answer Key and Assessment Rubric

Answer Key

1. a) 6 + 4 + 6 + 4 = 20 m. b) 6 × 4 = 24 m². c) Fencing requires the perimeter because fencing follows the boundary. Soil or grass requires the area because it covers the inside surface.

2. a) The error is 1 cm on each of 4 sides: 1 + 1 + 1 + 1 = 4 cm too large. b) The error is repeated once for every side, so the total boundary error is greater than the error on one measurement.

3. a) 8 + 5 + 8 + 5 = 26 m. b) 8 × 5 = 40 m². c) The perimeter, 26 m, is useful for fencing. The area, 40 m², is useful for grass or another surface covering.

4. Student A has calculated the area, not the boundary distance. The correct area is 40 m², so it cannot be written as 40 m for fencing. Student B has calculated the perimeter, not the area. The correct perimeter is 26 m, so it cannot be written as 26 m² for the surface.

5. The correct perimeter is 6 + 4 + 6 + 4 = 20 m. The calculation 6 × 4 = 24 finds the area, which is 24 m², not the perimeter.

6. Yes. The area is 8 × 5 = 40 m², so 40 m² covers it exactly. A reasonable check is that 8 groups of 5 square metres, or 5 groups of 8 square metres, make 40 square metres.

7. The diagram should be a rectangle labelled 6 m and 4 m. It can communicate either calculation, but labels and the stated decision make it clear whether the focus is the boundary or the inside surface.

8. Answers will vary. A complete answer must use a rectangle, provide two side measurements, calculate a perimeter and an area, match each result to a real-world decision, use correct units, and include a reasonable check.

Rubric

Method: 3 marks for selecting and carrying out the correct perimeter or area calculation; 2 marks for mostly correct working with a minor error; 1 mark for an attempted method; 0 marks for no relevant method.

Units: 2 marks for consistently using metres for boundary distance and square metres for area; 1 mark for mostly correct units; 0 marks for missing or unsuitable units.

Explanation: 3 marks for clearly linking the selected quantity to fencing, covering the ground, or the stated decision; 2 marks for a partly developed explanation; 1 mark for a simple statement; 0 marks for no explanation.

Checking: 2 marks for a sensible reasonableness check that supports the answer; 1 mark for an attempted check; 0 marks for no check.