Part 1 — The Trapezoid Flower Bed

The city held a contest: design a small community park. Aisha's team needed exact areas — the judges would reject any plan with sloppy math. Their first feature was a flower bed shaped like a trapezoid, with parallel sides of 8 ft and 12 ft and a height of 5 ft.

Trapezoid area = ½ × (b₁ + b₂) × h = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50 square feet. The formula averages the two parallel sides, then multiplies by the height.

Analyze — Q1

Why does the trapezoid formula add the two parallel sides before multiplying by the height?

Solve It — #1

A second trapezoid bed has parallel sides of 6 ft and 10 ft and a height of 4 ft. Find its area.

Area (square feet):

Part 2 — The Composite Plaza

The central plaza combined a rectangle and a triangle (a stage area). "Decompose it," Aisha told her team. The rectangle measured 10 ft × 6 ft, and the triangle on top had a base of 10 ft and a height of 4 ft.

Rectangle: 10 × 6 = 60. Triangle: ½ × 10 × 4 = 20. Composite area = 60 + 20 = 80 square feet.

Solve It — #2

A different plaza is a 12 ft × 8 ft rectangle with a triangle on top (base 12 ft, height 5 ft). Find the total area.

Total area (square feet):

Part 3 — Comparing Two Designs

The team had two lawn designs. Design A covered 120 square feet; Design B covered 96 square feet. Sod (grass) costs $2 per square foot. "The judges care about budget," said Aisha. "Less area means less cost — but we must keep it usable."

Design A cost: 120 × $2 = $240. Design B cost: 96 × $2 = $192. Design B saves $48.

Solve It — #3

What is the sod cost for Design B (96 sq ft at $2 per sq ft)? Type the dollar amount.

Cost ($):

Analyze — Q2

Aisha says "less area means less cost — but we must keep it usable." What trade-off is she describing?

Part 4 — Working Backward

One rectangular path had to cover exactly 60 square feet, and the city required it to be 10 ft long. "What width keeps the area at 60?" asked a teammate. Aisha worked backward from the area formula.

Area = length × width, so width = area ÷ length = 60 ÷ 10 = 6 feet.

Solve It — #4

A parallelogram garden must have an area of 72 sq ft with a base of 9 ft. What height is needed? (Area = base × height.)

Height (feet):

Analyze — Q3

How does "working backward" from the area formula help the design team?

After You Read — Analytical Writing

Make a Mathematical Argument

Choose one prompt. Write a clear paragraph (5–7 sentences) using numbers from the story as evidence.

Prompt A — Defend a design. Recommend Design A or Design B to the judges. Use the area and sod cost as evidence, and explain the trade-off the team had to weigh.

Prompt B — Explain a strategy. Explain how decomposing a composite figure and "working backward" are both useful tools for a designer. Give a specific example with numbers from the story for each.

Optional academic frame: "The math shows ______; therefore I recommend ______. One trade-off is ______."

Write your argument here (included when you print or download):

Challenge Extension

Think Further

The team has a budget of $300 for sod at $2 per square foot. What is the largest lawn area they can afford? Explain how you found it. (Try it, then check with your teacher.)

How You Are Scored

Rubric

Category	4 — Advanced	3 — Proficient	2 — Developing
Comprehension & inference	All analysis questions correct; explains the reasoning	2 of 3 correct	1 correct
Multi-step math	All 4 Solve-It answers correct (trapezoid, composite, cost, work-backward)	3 correct	2 correct
Mathematical argument	Clear recommendation supported by specific numbers; names a trade-off	Claim with some evidence	States a claim with little evidence