024i - Problem Solving with Composite Shapes

Lesson 24 — Problem Solving with Irregular and Composite Shapes

Strand: Measurement | Descriptor: AC9M8M01 | Duration: 45 minutes

Learning Intentions

• To solve multi-step practical problems involving the area and perimeter of irregular and composite shapes.
• To work backwards from a given area or perimeter to find an unknown dimension.

Success Criteria

I can:

1. Solve multi-step problems combining area (materials) and perimeter (boundary) for the same composite shape.
2. Work backwards from a known area or perimeter to find a missing dimension.
3. Justify a rounding decision appropriate to the real context.
4. Communicate a full solution clearly, with a diagram, working and a concluding sentence.

Warmup

(6 minutes — mixed retrieval, mini whiteboards)

1. A rectangle has area and one side cm. Find the other side.
2. An L-shape’s bounding rectangle is m by m, and the L-shape’s own area is . Find the area of the notch removed.
3. A composite shape (single corner notch) has perimeter m. One pair of bounding-rectangle sides is m each. Find the other pair.
4. True or false: “If you know a composite shape’s total area, you can always find its perimeter.” Explain.

Answers: 1. cm; 2. ; 3. , so m; 4. False — many different shapes can share the same area but have very different perimeters, and vice versa.

Activities

Activity 1 — Explicit Instruction: Combining Area and Perimeter in One Problem (12 min)

Real renovation and landscaping problems usually need both — materials by area, boundary trims by perimeter.

I do — a backyard renovation. The yard is an L-shape: overall m by m, with a m by m notch removed from one corner for a shed pad. Turf costs 12\text{m}^2$7$ per metre.

We do — a pool deck. Overall m by m, with a m by m notch removed for a pool house. Paving costs 54\text{m}^2$15$ per metre.

You do: A courtyard renovation with new dimensions and rates — find the area, perimeter, and total cost, presenting full working and a concluding sentence.

Activity 2 — Explicit Instruction: Working backwards (12 min)

Sometimes the area or perimeter is known, and a dimension must be found.

I do — backwards from area. An L-shaped room has area . Its overall shape is metres by m, with a m by m notch removed from one corner. Find .

We do — backwards from perimeter. An L-shaped garden (single corner notch) has perimeter m. One side of the bounding rectangle is m. Find the other.

You do: Three “working backwards” problems — two from area, one from perimeter — using new dimensions.

Activity 3 — Applied Problem: the Courtyard Renovation (10 min)

Pairs. A genuine multi-step problem requiring the full toolkit.

A homeowner’s courtyard is an L-shape: the overall block is m by m, with a rectangular garden bed removed from one corner for planting (left unpaved). Concrete paving must cover exactly — the supplier’s minimum delivery.

1. Find the area of the garden bed needed to achieve this.
2. If the garden bed’s width is m, find its other side length.
3. Find the perimeter of the paved area, and the cost of edging at 16.50$ per metre.

Socratic scaffolding:

Prompt	Purpose
Understand: what is fixed, and what is unknown?	The paved area () is fixed; the garden bed’s dimensions are unknown.
What do you know about the overall block?	total.
Devise a plan for the notch’s area	Paved area whole notch, so notch whole paved.
Carry out	.
Devise a plan for the missing side	Notch area width other side.
Carry out	m.
Now the perimeter — what shortcut applies?	Corner notch perimeter equals the bounding rectangle’s perimeter (Lesson 21).
Carry out	m; cost 759$.
Look back	Does a m by m garden bed fit inside a m by m block? Yes, comfortably within both sides.

Answers: 1. . 2. m. 3. m; 759$.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A composite shape has area , formed from a rectangle cm by cm minus a cm by cm notch. Find .
2. An L-shaped patio (single corner notch) has perimeter m. One side of the bounding rectangle is m. Find the other side.
3. A rectangular lawn m by m has a garden bed removed from a corner, leaving of lawn. Find the area of the garden bed.
4. Reasoning. Two different composite shapes both have an area of . Must they have the same perimeter? Explain.
5. A paving company charges 58\text{m}^2$1445\ \text{m}^232$ m. Find the total cost.

Answers: 1. cm; 2. m; 3. ; 4. No — shape and proportions affect perimeter independently of area; two shapes of equal area can have very different boundaries; 5. 3058$.

Common Misconceptions

Misconception	How to pre-empt it
Multiplying perimeter by an area rate, or area by a per-metre rate.	Underline each rate’s units before multiplying, and check they match the quantity being costed.
Assuming equal areas imply equal perimeters (or vice versa).	Activity 2 and the CFU reasoning question confront this directly with a counterexample.
Working backwards with the wrong operation — dividing when multiplying was needed, or vice versa.	Write the original equation first (e.g. ""), then solve it as an equation, rather than guessing an operation.
Forgetting to add back (or subtract) the notch when reversing a composite-area calculation.	Always re-state the original forward relationship in words before working backwards.
Not checking whether a deduced dimension is realistic (e.g., larger than the whole shape, or negative).	Make “looking back” a required, written step, not an afterthought.
Rounding a costing answer inconsistently (nearest dollar vs nearest cent).	Agree a rounding convention (usually to the cent) at the start of every costing problem.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A square garden of side has a square pond of side m removed from a corner, leaving of garden. Find .

Answer

E2 (AMC Junior style). A rectangular block of land has whole-number side lengths and area . A shed occupies one corner. The perimeter of the remaining usable land is m. Find the block’s dimensions.

Answer

Since the shed is a corner notch, the usable land’s perimeter equals the block’s own perimeter: . Also . These give , so:

The block is m by m.

E3 (Challenge). A pentagon is a cm by cm rectangle with a triangular corner removed, where one leg of the removed triangle is cm. The pentagon’s area is . Find the other leg.

Answer

E4 (Challenge). A rectangular garden m by m has a square pond removed from the middle of one side (not a corner), with side length . The perimeter of the remaining garden edge is m. Find .

Answer

Using the Lesson 21 result (a middle-of-side notch increases the perimeter by twice its depth):

Homework

1. A composite shape has area , formed from a rectangle cm by cm minus a cm by cm notch. Find .
2. An L-shaped courtyard (single corner notch) has perimeter m. One side of the bounding rectangle is m. Find the other side.
3. A rectangular oval m by m has an equipment shed removed from a corner, leaving of usable oval. Find the area of the shed.
4. A paving company charges 62\text{m}^2$1775\ \text{m}^238$ m. Find the total cost.
5. A square courtyard of side has a m by m garden bed removed from a corner, leaving an area of . Find .
6. Reasoning. A student claims: “If two composite shapes have the same perimeter, they must have the same area.” Give a specific counterexample using two rectangles.
7. Challenge. A rectangular garden m by m has a square pond removed from the middle of one side (not a corner), with side length . The perimeter of the remaining garden edge is m. Find .

Answers: Q1 — cm. Q2 — m. Q3 — . Q4 — 5296s^2 - 4 = 77 \Rightarrow s^2 = 81 \Rightarrow s = 910 \times 224208 \times 424322(22+16)=7676 + 2s = 84 \Rightarrow s = 4$ m.