Pieces that fit with no gaps
Smaller shapes can fit together to make a bigger shape. The pieces must meet edge to edge, with no gap between them and no piece lying on top of another. Then together they fill the bigger shape exactly.
Start with one square. Cut it in a straight line from one corner to the corner across from it. The cut makes 2 triangles, and the two triangles match. Slide one triangle back against the other, along the cut, and they make the square again.
The gold triangle slides across from where it started, the dashed outline on the left. Along the cut it meets the white triangle, and the two fill the square with no gap.
One square, built three ways
The same square can be built from other pieces. Four smaller squares, in 2 rows of 2, fill it with nothing left over. Two rectangles, one above the other, fill it as well.
The pieces are different each time, but together they always cover the same square. Nothing sticks out past its edge, and no part of it is left uncovered.
Four small squares, 2 in each row and 2 rows, build the square.
One cut straight across the middle leaves two equal rectangles, and they build the square too.
Count the pieces, not the lines
To say how many pieces make a shape, count the pieces the lines leave, not the lines. One line across a square leaves 2 pieces. Two lines that cross leave 4.
Worked example: Triangles That Make Squares and Small Squares That Make Big Squares
Question Ben has paper triangles that are all the same. He puts 2 triangles together to make 1 square. (a) How many triangles does Ben need to make 3 squares? (b) Mei puts 4 small squares together to make 1 big square. How many small squares does Mei need to make 2 big squares?
1.Draw 1 square. Draw a line from one corner to the corner across from it. The line cuts the square into 2 triangles.
A line from one corner to the corner across from it cuts the square into 2 triangles. 2.Each square needs 2 triangles. Draw 3 squares and cut each one in the same way.
Each square needs 2 triangles. Draw 3 squares and cut each one in the same way. 3.(a) Count the triangles. 2 + 2 + 2 = 6. Ben needs 6 triangles.
(a) 2 + 2 + 2 = 6. Ben needs 6 triangles. 4.Draw 1 big square. Cut it into 2 rows with 2 small squares in each row. The big square has 4 small squares.
A big square has 2 rows with 2 small squares in each row. That is 4 small squares. 5.(b) Draw 2 big squares and count the small squares. 4 + 4 = 8. Mei needs 8 small squares.
(b) 4 + 4 = 8. Mei needs 8 small squares.
Answer: (a) 6 triangles; (b) 8 small squares
Common mistakes
- Writing 3 for part (a) because there are 3 squares. Each square is made of 2 triangles, so count 2 for every square: 2 + 2 + 2 = 6.
- Writing 6 for part (b) by adding 4 + 2. The 2 tells how many big squares there are. Each big square needs 4 small squares, so add 4 two times: 4 + 4 = 8.
Worked example: A House Made from a Square, a Triangle and Two Rectangles
Question Dad makes a picture of a house from paper shapes. The wall is 1 big square. The roof is 1 triangle on top of the wall. The door is 1 small rectangle. The window is 1 small rectangle too. (a) How many shapes in the picture have 4 sides? (b) How many shapes are in the picture in all?
1.Name each part of the house. The wall is a square. The roof is a triangle. The door and the window are rectangles.
The wall is a square. The roof is a triangle. The door and the window are rectangles. 2.Count the sides of each kind of shape. A square has 4 sides. A rectangle has 4 sides. A triangle has 3 sides.
A square has 4 sides. A rectangle has 4 sides. A triangle has 3 sides. 3.(a) The square and the 2 rectangles have 4 sides. 1 + 2 = 3. There are 3 shapes with 4 sides.
(a) The square and the 2 rectangles have 4 sides. 1 + 2 = 3 shapes. 4.(b) Count every shape once: 1 square, 1 triangle and 2 rectangles. 1 + 1 + 2 = 4. There are 4 shapes in all.
(b) Count every shape once: 1 + 1 + 2 = 4 shapes in all.
Answer: (a) 3 shapes; (b) 4 shapes
Common mistakes
- Writing 2 for part (a) because only the rectangles are counted. A square has 4 sides too, so the wall is counted with the door and the window.
- Counting the house as one more shape and writing 5 for part (b). The house is the whole picture. It is made of the 4 shapes, so it is not counted again.