A lake or a leaf has no formula. Lay a grid over it: count the full squares, add half the part squares, and multiply by the area of one square. A finer grid gives a better estimate; straight pieces round the edge estimate the perimeter.
How big is a lake? Councils, farmers and park rangers estimate areas like this from maps all the time, with a grid and some careful counting.
An irregular shape has curved or uneven edges, so no formula fits. Estimate instead:
Area: put a grid over the shape. Count the full squares, and count each part square as half (some are mostly in, some mostly out, so on average half). Multiply by the area of one square. A finer grid gives a closer estimate.
Perimeter: join points on the edge with straight pieces and add their lengths. More, shorter pieces follow the curve more closely.
Always say how precise your estimate is ("about 25 km², to the nearest whole km²").
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.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Only the full squares
Look at the squares the edge passes through.
TryTry it on the board and compare the result with your prediction.
Count and refine
Count the squares colours full squares green and part squares orange.
Try it yourselfYour edits are saved in this browser.
Count the squares, then make the grid finer. Was your prediction right? How much did the estimate change?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Why half?
Read Sam's working. Is every orange square all lake?
Sam estimates his lake's area. Click the line where his method goes wrong.
ModifyChange the problem on purpose and see what happens.
Closer estimates
Need a hint?
Watch the perimeter estimate as n grows.
Need a hint?
Press Finer grid twice.
Another hint
Then Count the squares.
Try it yourselfYour edits are saved in this browser.
Use 32 straight pieces. Does the perimeter estimate get bigger or smaller?
Use the board. This ticks when you get there.
Use the finest grid (0.25 km squares) and count. How close is the estimate to the measured area?
Use the board. This ticks when you get there.
Estimate the area of a leaf on a 0.5 cm grid.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Estimating
A shape covers 30 full squares and part of 12 more. Each square is 1 m². Estimate its area in m².
On a grid of 0.5 m squares, a shape covers 40 full squares and part of 20 more. Estimate its area in m².
Review: a rectangle 8 m by 5 m has a 2 m by 2 m square cut from one corner. What is its area in m²?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
An island
Then write your estimate with its precision, and estimate the coastline with 32 pieces.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Count on each grid, coarsest first.
Another hint
The measured area appears on the finest grid.
Estimate the island's area to within 2 km² of the measured area.
Use the board. This ticks when you get there.
For teachers
Students estimate the area of irregular shapes by counting full and part squares on finer and finer grids, and the perimeter with straight pieces round the edge (AC9M8M01, node 2). About 35 minutes.
I can estimate an irregular area by counting full squares and half the part squares.
I can explain why a finer grid gives a better estimate.
I can estimate a curved perimeter and say how precise my answers are.
Before class
Do the Run board yourself first. Students need composite areas (previous lesson).
The one idea: count full squares, add half the part squares, multiply by one square's area; finer grids and more straight pieces get closer. The common mistake is counting every square the shape touches as whole (Sam's mistake).
The board's measured area (shown on the finest grid) comes from the exact outline; the estimates are within about 1 km² there. Straight pieces always fall a little short of a curve.
Wider: estimate a real area from a map or an aerial photo (a school oval, a local lake) with a printed grid.
Running the lesson
Predict: ask what the orange squares are worth.
Run: ask how much the estimate changed with the finer grid.
Investigate: ask Sam to look at one orange square.
Modify: ask when the extra effort of a finer grid is worth it.
Make: ask how they will say how precise their estimate is.
The hard part (on purpose)
Deciding what a part square is worth. Ask students to look at several orange squares and say how much of each is inside.
When students get stuck
A student counts every square the shape touches as whole. Ask: Is this orange square all inside? Fix: Part squares count as half.
A student forgets the size of a square on a finer grid. Ask: What is the area of one 0.5 by 0.5 square? Fix: Multiply the count by one square's area: 0.25.
A student reports many decimal places. Ask: How precise can a 1 km grid be? Fix: Round to what the grid can tell you, and say so.
Stretch and support
Stretch: Trace a leaf from outside onto 1 cm grid paper. Estimate its area on 1 cm squares and then 0.5 cm squares. How much did the estimate change?
Support: Give a printed grid and two coloured pencils: one for full squares, one for part squares.
Where it leads
This completes AC9M8M01.
Where it leads: estimating areas under graphs (Years 11 and 12), and scale drawings and maps.
What each step is for
only the full squares: Committing to what counting only full squares gives shows whether they see that the edge squares hold area too.
count and refine: Counting on a coarse grid then a finer one shows the estimate settling as the squares shrink.
why half?: Part squares count as half because edges cut them at random (Sam counts them as whole); straight pieces always fall a little short of a curve.
closer estimates: More pieces and finer grids close in on the true values; a different shape tests the method; the answer states its precision.
estimating: The count rule with different square sizes, mixed; counting part squares as whole is the planted slip; the composite review keeps exact areas fresh.
an island: Estimating to a stated precision brings the counting method and the finer grid together.
Now you can
Estimate an irregular area by counting full squares and half the part squares.
Explain why a finer grid gives a better estimate.
Estimate a curved perimeter and say how precise my answers are.