017i - Composite Areas

Lesson 17 — Composite Shapes: Combining Triangle and Parallelogram Areas

Strand: Measurement | Descriptor: AC9M7M01 | Duration: 45 minutes

Learning Intentions

• To decompose a composite shape into rectangles, parallelograms and triangles.
• To calculate the area of a composite shape by addition or by subtraction.

Success Criteria

I can:

1. Split a composite shape into familiar parts and label each clearly.
2. Find missing side lengths from the given dimensions.
3. Calculate a composite area by adding parts, or by subtracting a hole from a whole.
4. Choose the more efficient of the two strategies and justify the choice.

Warmup

(6 minutes — retrieval relay, mini whiteboards)

1. Triangle, cm, cm.
2. Parallelogram, m, m.
3. Rectangle, cm by cm.
4. Triangle, cm, cm.

(Answers: , , , .)

Bridging question: If shapes 3 and 4 were joined — the triangle sitting on top of the rectangle — what would the total area be? (78 cm².) What if the triangle were instead cut out of the rectangle? (42 cm².)

Activities

Activity 1 — Explicit Instruction: the Addition Method (12 min)

The four-step protocol, modelled explicitly:

1. Split the shape with dashed lines into rectangles, parallelograms and triangles.
2. Label each part A, B, C, …
3. Find any missing lengths by subtracting known lengths.
4. Calculate each part, then total. Show a working column per part.

I do — an “L-shape”. An L-shaped room: the full width is m, the full height is m; a m by m rectangle has been removed from the top-right corner.

Split into rectangle A (, the bottom band) and rectangle B (, the upper left portion).

Finding missing lengths — model aloud. The upper band’s width is m; the bottom band’s height is m. Encourage students to write these onto the diagram before calculating.

We do — a “house” shape. A rectangle cm wide and cm tall, with a triangle of base cm and height cm on top.

You do: Four composite shapes from a worksheet, each requiring at least one missing length to be deduced.

Activity 2 — The Subtraction Method (8 min)

Some shapes are far quicker to treat as “whole minus hole”.

I do. A rectangular lawn m by m contains a triangular flowerbed with base m and height m. Find the grassed area.

Choosing the method — a rule of thumb:

If the shape looks like a full rectangle with pieces missing, subtract. If it looks like separate pieces joined, add.

Many shapes work either way. Have students solve one shape both ways and confirm the answers agree — an excellent self-check.

You do: A cm by cm rectangle with a cm by cm rectangle cut from one corner. Solve by subtraction, then verify by splitting and adding.

Activity 3 — Inquiry: the Arrow (10 min)

Pairs.

An arrow shape is made from a rectangle cm wide and cm long (the shaft), with a triangular head of base cm and height cm attached to one end of the shaft.

1. Sketch the shape and label all dimensions.
2. Find the total area.
3. A second arrow has the same total area but a shaft only cm wide. If its head is unchanged, how long must its shaft be?

Socratic scaffolding for Q3:

Prompt	Purpose
Understand: what is the unknown?	The new shaft length.
What is fixed, and what is changing?	The head and the total area are fixed; the shaft’s width and length change.
Find the total area first.	.
What area must the new shaft have?	Total minus head: .
Devise a plan	Shaft area width length, so length .
Carry it out	cm.
Looking back	Check: , plus the head , gives ✓
Notice	The shaft area is unchanged at — narrower means longer.

Answers: 2. ; ; total . 3. cm.

Checks for Understanding

(5 minutes — exit ticket)

1. A shape is made from a rectangle cm by cm with a triangle ( cm, cm) on top. Find its total area.
2. A m by m rectangle has a m by m rectangle removed from one corner. Find the remaining area.
3. A parallelogram ( cm, cm) sits beside a triangle ( cm, cm). Find the total area.
4. Reasoning. For a rectangular field with a square shed in the middle, which method — addition or subtraction — is more efficient? Explain.
5. A composite shape splits into three parts of , and . A student gives the answer as cm. What is wrong?

Answers: 1. ; 2. ; 3. ; 4. Subtraction — one large rectangle minus one square, rather than splitting the surround into four awkward pieces; 5. The units: area must be in , so the answer is .

Common Misconceptions

Misconception	How to pre-empt it
Double-counting the overlap region when splitting.	Require dashed split lines drawn on the diagram, and each region labelled exactly once.
Assuming a missing length equals a given one.	Insist that every deduced length is written on the diagram with its subtraction shown, e.g. "".
Adding when the problem calls for subtraction (or vice versa).	Ask before calculating: “Is something joined on, or cut out?”
Forgetting to halve for a triangular part.	Have students write the formula they are using on each line, not just the numbers.
Mixing units within one shape.	Convert all lengths as the first written step.
Giving the answer in linear units.	Model the units in every line of every worked example.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A square of side cm has a smaller square of side cm cut from one corner. What is the perimeter of the remaining shape?

Answer

The perimeter is unchanged at cm. Removing the corner square replaces two segments of cm each with two new segments of cm each — the removed and added lengths cancel exactly. (A useful contrast: the area does change, from to .)

E2 (AMC Junior style). A rectangle cm by cm has a triangle cut from it, with the triangle’s base being the full cm side and its apex on the opposite side. What fraction of the rectangle remains?

Answer

E3 (Challenge). A path m wide runs all the way around the outside of a rectangular garden measuring m by m. Find the area of the path.

Answer

Note the : the path adds m on each of two opposite sides.

E4 (Challenge). A shape is made of a cm by cm square with an equilateral-looking triangle of base cm and height cm attached to each of its four sides. Find the total area.

Answer

Homework

1. Find the total area of a rectangle cm by cm with a triangle ( cm, cm) on top.
2. A m by m rectangular yard has a m by m shed in one corner. Find the uncovered area.
3. A composite shape is made from a parallelogram ( cm, cm) and a triangle ( cm, cm). Find the total area.
4. An L-shaped room has overall dimensions m by m, with a m by m rectangle missing from one corner. Find the floor area. Carpet costs 45$ per square metre — find the total cost.
5. A rectangular sign cm by cm has a triangular corner ( cm, cm) cut off. Find the remaining area.
6. Reasoning. Explain why splitting the same composite shape in two different ways must give the same total area.
7. Challenge. A garden is a m by m rectangle. A triangular pond with base m occupies exactly one quarter of the garden’s area. Find the pond’s height.

Answers: Q1 — . Q2 — . Q3 — . Q4 — ; cost 22954000 - 300 = 3700\ \text{cm}^2240\ \text{m}^260\ \text{m}^2\tfrac12 \times 10 \times h = 60h = 12$ m.