What makes a rectangle
A rectangle has four straight sides and four corners, and every corner is a square corner, like the corner of a sheet of paper. Its opposite sides are the same length: the top side matches the bottom side, and the left side matches the right side.
On a dot grid, a side that runs straight across a row of dots and a side that runs straight down a column of dots meet in a square corner. So a rectangle on the dots is made of sides that go straight across and straight down, and never on a slant.
Three corners find the fourth
Start at a dot and draw the top side 4 gaps across. From the left end, draw the left side 3 gaps straight down. That gives three corners of the rectangle.
The fourth corner lines up with two of the others. It is straight below the top-right corner, and it is in the same row as the bottom-left corner. There is only one dot that is both, so draw the right side 3 gaps down to it, and the bottom side 4 gaps across to meet it.
The top side is 4 gaps across and the left side is 3 gaps down: three corners are placed.
The fourth corner is below the top-right corner and level with the bottom-left corner. The rectangle is 4 gaps across and 3 gaps down.
A square has four equal sides
Now make the top side 3 gaps across instead of 4, and keep the left side 3 gaps down. Every side is 3 gaps long, so the rectangle is a square.
A square is a special rectangle. It has four square corners, like every rectangle, and all four of its sides are the same length. A rectangle that is 4 gaps across and 3 gaps down is not a square, because its sides are not all equal.
Every side is 3 gaps long: this rectangle is a square.
Two ways a drawing goes wrong
Four sides are not enough on their own. A shape whose sides lean over has four sides and four corners, but its corners are not square, so it is not a rectangle.
Counting the dots on a side, instead of the gaps, makes that side one gap too long, and the opposite sides stop matching. Count the gaps on the top side and the bottom side, and on the left side and the right side, and check that each pair is equal.
Worked example: The Squares Inside a Rectangle and Inside a Square on a Grid
Question Ali draws a rectangle on a square grid. It is 5 squares long and 3 squares wide. (a) How many grid squares are inside the rectangle? (b) Next Ali draws a square. It has 3 grid squares along each side. How many grid squares are inside the square?
1.Draw the rectangle. It has 3 rows, and each row has 5 squares.
The rectangle has 3 rows, and each row has 5 squares. 2.(a) Add the rows: 5 + 5 + 5 = 15. There are 15 grid squares inside the rectangle.
(a) 5 + 5 + 5 = 15. There are 15 grid squares inside the rectangle. 3.A square has the same number of grid squares along each side. Draw 3 rows with 3 squares in each row.
A square has the same number of grid squares along each side: 3 rows with 3 squares in each row. 4.(b) Add the rows: 3 + 3 + 3 = 9. There are 9 grid squares inside the square.
(b) 3 + 3 + 3 = 9. There are 9 grid squares inside the square.
Answer: (a) 15 squares; (b) 9 squares
Common mistakes
- Adding 5 + 3 = 8 for part (a). That adds one long side and one short side. It does not count the squares inside. Count row by row: 5 + 5 + 5 = 15.
- Counting only the squares round the edge. The squares in the middle are inside the shape too, so every row is counted in full.
Worked example: Telling a Square from a Rectangle by Its Sides
Question Three shapes are drawn on a square grid. Each shape has 4 sides and 4 square corners. Shape A is 4 squares long and 2 squares wide. Shape B is 3 squares long and 3 squares wide. Shape C is 5 squares long and 1 square wide. (a) Which shape is a square? (b) How many of the shapes are not squares?
1.A square has 4 equal sides. A shape that is longer than it is wide is not a square.
A square has 4 equal sides. A shape that is longer than it is wide is not a square. 2.Look at Shape A. It is 4 squares long and 2 squares wide. The sides are not equal, so Shape A is not a square.
Shape A is 4 squares long and 2 squares wide. The sides are not equal, so it is not a square. 3.(a) Look at Shape B. It is 3 squares long and 3 squares wide. All 4 sides are equal, so Shape B is a square.
(a) Shape B is 3 squares long and 3 squares wide. All 4 sides are equal, so Shape B is a square. 4.Look at Shape C. It is 5 squares long and 1 square wide. The sides are not equal, so Shape C is not a square.
Shape C is 5 squares long and 1 square wide. The sides are not equal, so it is not a square. 5.(b) Shape A and Shape C are not squares. That is 2 shapes.
(b) Shape A and Shape C are not squares. That is 2 shapes.
Answer: (a) Shape B; (b) 2 shapes
Common mistakes
- Saying that all three shapes are squares because each one has 4 sides and 4 square corners. A rectangle has those too. A square must also have 4 equal sides.
- Choosing Shape C because it is the longest shape. The longest shape is not what makes a square. Compare the length of a shape with the width of the same shape.