Area as a count of unit squares

0:00Here is a shape, and here are two completely different questions you could ask about it. How far is it round the edge? That is perimeter, and we have spent four videos on it. How much room is inside? That is area — a different measurement, with a different answer and a different unit. Perimeter you could walk. Area you would have to cover. So how do you measure a covering? You pick something to cover it with, and you count.

0:29And the thing we cover with is a square. One square of the grid — one square unit. Area is the number of those squares a shape encloses. It is a count, before it is ever a multiplication. Now, open the book at this page and you find something unusual. Two empty boxes. One for the area of a square. One for the area of a rectangle. Both printed blank.

0:58The teacher's note beside them says: hand out squared paper, and let them find the rules themselves. So the book is not withholding the rule to be awkward. It is refusing to hand you a sentence you could work out. And to be fair to it — four lines below those boxes, its own worked solution uses the rule anyway. So it is not a secret. It is an exercise. Let us do the exercise.

1:29Take a rectangle five units long and four units wide, and start covering it with unit squares. Lay the first row. One, two, three, four, five. Five squares in a row. Now the second row. Another five. Ten so far. Third row, fifteen. Fourth row, twenty. And the rectangle is full. Twenty unit squares — so the area is twenty square units. Twenty square metres, if those units are metres.

2:00Now look at what you actually did. Every row held five. There were four rows. Five, four times over. Which is five times four. That is where length times width comes from. It is not a definition. It is a fast way to count rows. Which also tells you when you may use it. Only when the shape really does break into equal rows. Rectangles do. Almost nothing else does.

2:31So. A floor five metres long and four metres wide. Twenty square metres. On it, a square carpet with sides of three metres. Three times three — nine square metres. How much floor is left uncovered? Twenty minus nine. Eleven square metres. And notice why that subtraction is allowed. It works because area is a count. Nine of the twenty squares are under the carpet. Eleven are not. You are simply counting what is left.

3:04Areas add and subtract for the same reason numbers of things do — because that is what they are. Now a piece of land twelve metres by ten. A hundred and twenty square metres. In each of its four corners, a square flower bed, four metres a side. Here is where almost everybody slips. One bed is four times four — sixteen square metres. Four beds is four times sixteen. Sixty-four.

3:36Not four times four. That number is one bed, and there are four of them. So the land left over is a hundred and twenty, minus sixty-four. Fifty-six square metres. Two steps, both easy, and the printed solution walks through both of them on purpose. Now the same idea running the other way — dividing, instead of subtracting. A coconut grove, a hundred metres by fifty. Five thousand square metres.

4:09Every tree needs twenty-five square metres to itself. What is the greatest number of trees? Five thousand divided by twenty-five. Two hundred trees. And that division is doing something very specific. It is asking how many twenty-five-square-metre patches fit inside five thousand. Which is the same question as before, except that the covering pieces are patches instead of unit squares. Every area problem in this section is a covering problem underneath.

4:46Now a shape that is not a rectangle at all. A stepped outline, like a staircase going down to the left. There is no formula for this. There is not going to be one. And you do not need one. Cut it. Any outline like this falls apart into rectangles, and you already know what to do with a rectangle. Cut it into flat strips. Four rectangles. Work each one out and add. Two, ten, seven, nine. Twenty-eight square metres.

5:18And the same move handles the second figure — an arch, five across and three down, with a rectangular bite out of the bottom. The whole block is fifteen. The bite is three by two, so six. Fifteen minus six is nine. Cut and add, or take the whole and subtract. Either way, you never needed a new rule. Now the book does something I was not expecting, and it is the best thing in the section.

5:49It puts the arithmetic away and hands you a tangram. One square, cut into seven pieces. Two big triangles, A and B. A medium triangle, F. Two small triangles, C and E. A square, D. And a parallelogram, G. No measurements are printed. No side lengths. No ruler is mentioned anywhere on the page. And then it asks you to compare their areas. Which sounds impossible — until you remember what area actually is.

6:24Because you do not need a number to compare two areas. You need to cover one with the other. Lay C on E. They match exactly, edge for edge. So C and E have the same area — and you measured nothing. Now take C and E together, and lay them on D, the little square. They cover it exactly. So D is two C's. D is C plus E, and those two are equal, so D is twice C.

6:52That is a proof, not an estimate, and there is not a number anywhere in it. And it kills a stubborn idea — that a square must somehow be bigger than a triangle. Shape and size are not the same thing. So keep going, with C as your unit, and cover everything with C. E is one C. D is two. Lay C's on F, the medium triangle — two again.

7:20G, the parallelogram, also takes two. So D and F and G are all exactly the same size, and they look nothing alike. The big triangles, A and B, take four C's each. Now total the whole set. Four and four is eight. Two and two and two is six. One and one is two. Eight, plus six, plus two. Sixteen. The whole square is sixteen times the smallest piece — and you found that by covering, with no ruler in the room.

7:55One more thing the tangram settles, quietly. Take the seven pieces and push them into a square. Sixteen units. Now break that up and lay the same seven pieces out anywhere — the book asks you to push them into a rectangle. Still sixteen, either way. Of course it is. Nothing was added, nothing taken away, nothing cut. Rearranging cannot change how much there is — and every splitting problem in this chapter leans on exactly that.

8:29When you cut a staircase into three rectangles and add them up, you are using this. It simply never announced itself. Last, a puzzle from the end of the chapter, and it is the sharpest test of everything so far. A rectangle, cut by one vertical line and one horizontal line into four parts. Top left, thirteen square centimetres. Top right, twenty-six. Bottom left, fifteen. Bottom right, blank. And here is the difficulty. The page gives you no lengths at all. Not one.

9:08So work with what you have. The two top parts share a height. One is twenty-six and the other thirteen, so one is exactly twice the other. Same height, twice the area — so the right column must be twice as wide as the left. Now the bottom row. Same two widths, and one shared height. So the bottom right is twice the bottom left. Twice fifteen. Thirty square centimetres.

9:37No ruler, no side length, no formula. Just the fact that area counts squares, and twice as wide holds twice as many. Next time, the question sitting underneath all of this. Why a square? Why not cover with circles, or triangles?