025i - Consolidation Area and Perimeter

Lesson 25 — Consolidation and Check: Area and Perimeter

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

Learning Intentions

• To consolidate all strategies for finding the area and perimeter of irregular and composite shapes.
• To apply appropriate units and solve a substantial multi-step applied problem confidently and accurately.

Success Criteria

I can:

1. Select and apply the most efficient method — addition, subtraction, or both — for a given composite shape.
2. Convert and apply units correctly throughout a multi-step problem.
3. Solve a substantial real-world problem involving both area and perimeter, showing full working.
4. Check and justify the reasonableness of a final answer.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example.

1. Cutting a rectangular notch from a corner changes the perimeter of a shape.
2. Cutting a triangular (diagonal) corner off a rectangle decreases its perimeter.
3. Splitting a composite shape a different way changes its total area.
4. A composite shape’s area can be found without knowing every one of its side lengths.

Answers: 1. Never — the two edges removed are exactly replaced. 2. Always — the hypotenuse is always shorter than the sum of the two legs it replaces (triangle inequality). 3. Never — the region covered is identical regardless of how it is split. 4. Sometimes — subtraction can need only the outer dimensions and the missing piece’s dimensions, not every internal length.

Activities

Activity 1 — Diagnostic Review Circuit (10 min)

Four short, independent problems — one from each strand of the week. Work through them briskly, then check as a class.

A. Area (addition/subtraction). An L-shape has overall dimensions m by m, with a m by m notch removed from a corner. Find its area.

B. Perimeter. Find the perimeter of the same L-shape.

C. Units. A shape’s overall dimensions are m by cm, with a cm by m notch removed from a corner. Convert to a single unit and find the area in .

D. Chamfered corner. A rectangle cm by cm has a diagonal corner cut removing a triangle with legs cm and cm. Find the remaining area.

Answers: A — . B — corner notch, m. C — . D — .

Activity 2 — Guided Practice: Multi-step Review (10 min)

Problem 1. A courtyard’s overall dimensions are m by m, with a rectangular notch removed from a corner. The paved area is . Find the notch’s area, and — if one side of the notch is m — find its other side.

Problem 2. An L-shaped garden bed (single corner notch) has perimeter m. Edging costs 9.20$ per metre. Find the total edging cost.

Answers: Problem 1 — notch ; other side m. Problem 2 — 331.20$.

Activity 3 — The Check Problem: the New Deck (14 min)

Pairs, then whole-class share. This problem draws on every skill from this unit.

A rectangular backyard deck is planned with overall dimensions m by cm. To fit around an existing tree, the back-right corner of the deck is cut off along a diagonal, removing a right triangle with legs m and m.

Decking boards cost 84\text{m}^2$22$ per metre, and is needed around the entire perimeter of the deck, including the diagonal edge.

1. Convert all measurements to a single consistent unit.
2. Find the area of the deck.
3. Find the perimeter of the deck, including the diagonal edge.
4. Find the total cost of decking boards and edge trim.
5. The homeowner has budgeted 6000$. Is this enough? If so, by how much is the budget under- or over-spent?

Socratic scaffolding:

Prompt	Purpose
Understand: what is the shape, and what is unusual about it?	A rectangle with one corner cut off diagonally — a pentagon.
What is fixed, and what must be found?	Dimensions are fixed once converted; area, perimeter, cost and a budget comparison are all required.
Devise a plan	Convert units, then: area (rectangle minus triangle), perimeter (using the diagonal, not the two legs), cost of each part, then compare to budget.
Why is the diagonal edge a whole number?	Legs and form a –– Pythagorean triple — a rule you’ll meet properly later this year.
Carry out (area)	.
Carry out (perimeter)	m.
Carry out (cost)	4469.12$.
Look back	Is 4469.12$6000$1530.88$ to spare.
Reflect	Which costs more — the boards or the trim? What does that suggest about where cost-cutting has the biggest effect?

Answers: 2. . 3. m. 4. 4469.12$1530.88$ under budget.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Find the area of an L-shape: overall m by m, notch m by m removed from a corner.
2. Find the perimeter of the same L-shape.
3. A rectangle cm by cm has a diagonal corner cut removing a triangle with legs cm and cm. Find the remaining area.
4. Convert (a) to (b) to .
5. Reasoning. A composite shape’s perimeter can increase, decrease, or stay unchanged depending on how material is added or removed, while its area only tracks how much material changes. Using an example, explain why area and perimeter don’t always change together.

Answers: 1. ; 2. Corner notch, m; 3. ; 4. (a) (b) ; 5. E.g. cutting a corner notch decreases area but leaves perimeter unchanged, while cutting a middle-of-side notch decreases area and increases perimeter — the two quantities respond independently to the same kind of change.

Common Misconceptions

A consolidated checklist of the week’s key errors.

Misconception	How to pre-empt it
Perimeter is the sum of only the labelled numbers on the diagram.	Trace the full outline every time, deducing and labelling every missing segment first (Lesson 21).
Subtracting a rectangle instead of a triangle for a chamfered (diagonal) corner.	Always identify and label the two legs of the missing right triangle before calculating (Lesson 22).
Using the linear conversion factor instead of the squared factor for an area conversion.	Show the squaring step explicitly every time: “the linear factor is , so the area factor is ” (Lesson 23).
Assuming equal areas imply equal perimeters, or vice versa.	Keep a running counterexample on display, e.g. two rectangles with equal perimeter but different area (Lesson 24).
Reversing the operation when working backwards from a known area or perimeter.	Write the forward equation first, then solve it, rather than guessing which operation to apply (Lesson 24).
Not checking whether a diagonal length is a recognisable whole-number triple.	Keep a short list of common triples (3–4–5, 6–8–10, 5–12–13) visible as a checking tool.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A plus-shaped (cross) tile pattern is made from five congruent squares, each of side mm. Find the perimeter of the whole cross, in centimetres.

Answer

cm. Using the Lesson 21 result ( side for a plus-pentomino):

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

Corner notch perimeter is unaffected: . Also , giving :

The block is m by m.

E3 (Challenge). A rectangular deck m by cm has a diagonal corner cut removing a right triangle with legs cm and cm (hypotenuse cm — a scaled –– triple). Find (a) the deck’s area in (b) its perimeter in metres.

Answer

Convert: cm.

E4 (Challenge — capstone). A garden plot is a rectangle m by m. A m by m rectangular notch is removed from one corner (for a shed), and the opposite corner is chamfered diagonally, removing a right triangle with legs m and m (a –– triple). Find (a) the remaining area (b) the remaining perimeter.

Answer

The rectangular notch leaves the perimeter unchanged; the diagonal chamfer changes it by (hypotenuse legs) :

Homework

1. Find the area and perimeter of an L-shape: overall m by m, notch m by m removed from a corner.
2. A rectangle cm by cm has a diagonal corner cut removing a triangle with legs cm and cm. Find (a) its area (b) its perimeter.
3. Convert: (a) to (b) to (c) to .
4. A composite paved courtyard has area and perimeter m. Paving costs 59\text{m}^2$16$ per metre. Find the total cost.
5. An L-shaped room has area . Its overall shape is metres by m, with a m by m notch removed. Find .
6. Reasoning. Explain, with reference to this unit’s lessons, why “same perimeter” does not guarantee “same area” — use a specific example.
7. Challenge. A rectangular deck m by cm has its back corner chamfered, removing a right triangle with legs m and m. Find (a) the deck’s area in (b) its perimeter in metres.

Answers: Q1 — area ; perimeter (corner notch) m. Q2 — (a) (b) cm. Q3 — (a) (b) (c) . Q4 — 432010w-12=72 \Rightarrow w=8.4650\ \text{cm}=6.59\times6.5-6=52.5\ \text{m}^22(9+6.5)-7+5=29$ m.