🟩 The counting game
The Square Counter
How do you measure the size of a shape that isn’t a rectangle? Most people count squares, but for lots of shapes there’s a quicker way.
Most people find the size of a shape by counting squares.
They count them one at a time.
That works, but it’s slow, and for lots of shapes there’s a quicker way.
The counting game
Count the squares that count
For an awkward shape, the usual method is to draw it on squared paper and count. A full square counts one, and a square that’s at least half covered also counts one. A square less than half covered counts nothing. The squares just over half and the ones just under roughly cancel out.
Your turn. Tap every square that counts. The blob’s edge cuts across the squares, so you’ll have to decide which ones are at least half covered.
Counting every square takes a long time.
With rectangles, you don’t have to.
The shortcut
A quicker way to count a rectangle
Counting works, but it’s slow. Drag the corner handle (or use the buttons) to stretch this rectangle as big as it goes, and the grid will count the squares for you as you go.
6 squares
3 across · 2 up
A rectangle is made of rows: 2 rows with 3 squares in each. So 3 × 2 should give you the same number without any counting. Decide whether you think it will, then check.
A rectangle is just rows stacked on top of each other.
Triangles look much harder, but there’s a quick way for them too.
The triangle trick
A right-angled triangle is half a rectangle
Triangles look hard to measure because of all the cut squares along the slope. But every right-angled triangle fits inside a rectangle, and it’s exactly half of it. Drag the corner to reshape the triangle, then complete the rectangle and watch how the two numbers change together.
Shapes with straight edges are easy to measure this way.
Curves are much harder.
The awkward one
Counting squares in a circle
Rectangles were easy, and triangles needed one extra step. Now try the circle. The half-rule is the same as before, but this time you choose how small the squares are. Smaller squares should give a better answer, so see how close you can get.
The estimate gets closer each time, but it’s never exact, because you can’t build a circle out of whole squares however small they are. To get the exact answer, you need a famous number that comes up in every circle. It’s written with a Greek letter, its digits never end, and you’ll meet it in secondary school.
People often mix up the next two measurements.
The fence around a field is one measurement, and the field inside it is another.
The same length of fence can hold different amounts of space.
The fence
The same fence can hold different amounts of carpet
Lots of people mix up area and perimeter, but they measure different things. Perimeter is the walk around the edge, which is a length measured in cm. Area is the carpet inside, which is a number of squares measured in cm². This pen uses exactly 12 cm of fence, and the dot is walking round it. Reshape the pen and watch which number changes.
You can count squares, or multiply rows to save time.
The fence is the walk around the edge, and the carpet covers the inside.
They’re two different measurements, and you can now tell them apart.
One more thing
You’ll come across both of these all the time, and mixing them up can cost real money. Skirting board goes round the edge of a room, so you buy it in metres. Carpet covers the floor, so you buy it in square metres. A picture frame goes round the edge, and the poster inside it is an area. How much paint you need depends on how many square metres of wall you’re covering.
If someone works out the area of a rectangle by adding up the sides, they’ve found the perimeter instead. You can show them how to multiply the sides to get the area.