Area problems
Pick the base and the height at a right angle to it, put both lengths in the same unit, and split a shape into rectangles and triangles. Area is in square units: m², cm².
How much paint for the end wall of a house, with its triangle of roof? Real shapes mix rectangles and triangles, and real measurements mix metres and centimetres.
Area problems come with extra information. Three moves sort them out:
- Choose the base and its height. The height is at a right angle to the base; a slanted side is not needed.
- Use one unit. Change every length to the same unit (100 cm = 1 m) before you multiply. Area comes out in square units: m² or cm².
- Split the shape. A shape made of rectangles and triangles has area equal to the sum of the parts (or a big shape minus a missing part).
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
Metres and centimetres
Write both lengths in metres first.
A 4 m by 2.5 m room
Each grid square is 1 m by 1 m: one square metre.
Make a rectangle with base 4 m and height 2.5 m (250 cm). What is its area? Was your prediction right?
Use the board. This ticks when you get there.
Choose, convert, split
Here are the three moves.
- Choose: the base and the height at a right angle to it. Ignore the slanted 5 cm side.
- Convert: both lengths in metres, then multiply.
- Split: a rectangle (6 × 4) plus a triangle (½ × 6 × 3).
- Or subtract: a rectangle with a corner missing is the big rectangle minus the corner.
Your working can be short notes. Then choose the reason for the key step.
Read Ivy's working. Are both lengths in the same unit?
The end wall of a house is a rectangle 6 m wide and 4 m high, with a triangle of roof on top: base 6 m, height 3 m. Find the area of the whole wall. Write your steps.
Your working
A model solution
- Rectangle: m² a rectangle's area is length × width
- Triangle: m² a triangle is half of base × height
- Wall: m² the whole is the sum of its parts
Check your own working
Ivy finds the area of a rug. Click the line where her working goes wrong.
One change at a time
Each task changes one thing.
Need a hint?
The sail's base is along the boom (3 m) and its height up the mast (5 m).
A sail is a right-angled triangle: 3 m along the bottom and 5 m up the mast. Make it and read its area.
Use the board. This ticks when you get there.
A rectangular garden 8 m by 5 m has a 3 m by 2 m shed in one corner. What area of garden is left, in m²?
How many square centimetres are in 1 square metre?
A wall 5 m wide and 2.4 m high is painted. One litre of paint covers 4 m². How many litres are needed?
Area problems
A flag is a 10 cm by 4 cm rectangle with a triangle on one end: base 4 cm, height 3 cm. Total area, in cm²?
A path is 300 cm long and 2 m wide. Its area, in m²?
A right-angled triangle has base 12 cm, height 5 cm and sloping side 13 cm. Area, in cm²?
Review: find 25% of 48.
A tall sail
Then work out the cost of the sailcloth at $15 per m².
Need a hint?
½ × b × h = 12, so b × h = 24.
Another hint
Pick a pair where the height is bigger than the base.
Design a right-angled sail with area 12 m² that is taller than it is wide.
Use the board. This ticks when you get there.
For teachers
Students solve area problems by choosing the base and perpendicular height, converting to one unit, and splitting composite shapes into rectangles and triangles (adding or subtracting parts), with reasoning they own (AC9M7M01, node 2). About 45 minutes.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- I can pick the right base and height when a shape has extra lengths.
- I can change lengths to the same unit before finding an area.
- I can find the area of a shape made of rectangles and triangles.
Before class
Do the Run board yourself first. Students need the parallelogram and triangle formulas (previous lessons) and converting cm and m.
The one idea: choose, convert, split. The common mistake is multiplying lengths in different units (Ivy's mistake); extra information (slanted sides) is planted in several tasks. 1 m² = 10 000 cm² surprises most students.
The house-wall task is a reasoning check: the working is never marked, only the final area and the reason for the roof step.
Running the lesson
- Predict: students who chose 1000 m² did not convert.
- Run: ask how the half-metre row shows on the grid.
- Investigate: ask two students to share different splits of the house wall.
- Modify: for 1 m², draw a 1 m square and ask how many 1 cm squares fit along one side.
- Make: compare designs, then cost them.
The hard part (on purpose)
Planning a composite area before calculating. Ask students to sketch the parts and label each with its own lengths first; resist giving the split.
When students get stuck
- A student multiplies cm by m (Ivy's mistake). Ask: What unit is each length in? Fix: Change both to metres (or both to cm) first.
- A student uses every number given. Ask: Which lengths meet at a right angle? Fix: Use the base and its height; ignore slanted sides.
- A student cannot start a composite shape. Ask: Can you see a rectangle and a triangle in it? Fix: Sketch the parts, find each area, then add (or subtract a missing part).
Stretch and support
Stretch: Find the area of the letter L, T or H drawn on squared paper in two different ways (adding parts, and subtracting from a rectangle).
Support: Use squared paper; convert all lengths to the smaller unit and write them on the sketch.
Where it leads
Next: Volume (AC9M7M02): the area of the base times the height.
Where it leads: area of trapeziums, circles and composite shapes with circles (Year 8), and surface area (Year 9).
What each step is for
- metres and centimetres: Committing to an area for a 4 m by 250 cm room shows whether they convert to one unit before multiplying.
- a 4 m by 2.5 m room: Building the room in metres on the board shows the area in m², with a half-metre row of squares.
- choose, convert, split: Area problems need the right lengths, one unit and a split into parts; Ivy multiplies centimetres by metres. Students own the working for the house shape and pick the reason at the key step.
- one change at a time: One change at a time: a sail on the board, a shape with a missing corner (subtract), square metres to square centimetres, a two-step paint problem, then choosing a sensible unit.
- area problems: A split shape, a unit change and extra information, mixed; mixed units are the planted slip; the review keeps percentages fresh.
- a tall sail: Designing a sail with a set area and a shape condition means searching base and height pairs, not just using a formula forwards.
- Pick the right base and height when a shape has extra lengths.
- Change lengths to the same unit before finding an area.
- Find the area of a shape made of rectangles and triangles.