Primary & Middle School Mathematics · Grades 1–9

5Shapes and Space

Windows are rectangles, wheels are circles, roofs hide triangles: shapes are everywhere. This chapter learns to recognize them by counting sides and corners, and to say precisely where things are: left, right, above, below, between.

5.1 The four basic shapes

Definition 5.1 (Square, rectangle, triangle, circle)

the triangle has $3$ sides and $3$ corners;
the square has $4$ equal sides and $4$ corners;
the rectangle has $4$ sides and $4$ corners, with opposite sides equal (a stretched square);
the circle is perfectly round: no sides, no corners.

The four basic shapes. To name a shape, do not trust the first glance: count its sides and corners. — The four basic shapes. To name a shape, do not trust the first glance: *count* its sides and corners.

Example 5.2 (Tilted shapes are still shapes)

A square standing on its corner is still a square — four equal sides, four corners — even if it looks like a kite. Turning a shape does not change what it is.

Example 5.3 (Shape hunt)

In the classroom: the board is a rectangle, the clock a circle, a folded napkin can be a triangle, some windows are squares. Real objects are not perfect shapes, but their faces are close to them (solids and their faces return in Chapter 11).

5.2 Where is it?

Definition 5.4 (Position words)

To say where something is: left / right, above / below, on / under, in front of / behind, between, inside / outside.

Position words in action. Careful: your left and the left of someone facing you are opposite! — Position words in action. Careful: *your* left and the left of someone facing you are opposite!

5.3 Paths on a grid

Method 5.5 (Describing a path)

On grid paper, a path is a list of steps: $\to$ (one square right), $\leftarrow$ (left), $\uparrow$ (up), $\downarrow$ (down).

To follow a path: start on the marked square and obey the arrows one by one;
to describe a path: walk it in your head and write the arrows in order;
two different paths can join the same two squares — some shorter, some longer.

$A path from start to goal, one arrow per step: \ \ \ \ \ \ — seven steps. Can you find another path with the same number of steps?$

A path from start to goal, one arrow per step:

\to\ \to\ \uparrow\ \uparrow\ \to\ \to\ \to

— seven steps. Can you find another path with the same number of steps?

5.4 Exercises

Exercise 5.1 ★

For each shape, count the sides and corners and name it: a shape with $3$ corners; a round shape with no corner; a shape with $4$ equal sides.

Solution

Solution of Exercise 5.1.

$3$ corners: a triangle. Round, no corner: a circle. $4$ equal sides: a square.

Exercise 5.2 ★

Find at home: two rectangles, one circle, one triangle, one square. (Doors? plates? road signs?)

Solution

Solution of Exercise 5.2.

Answers vary: a door and a book are rectangles; a plate or a clock is a circle; a road-warning sign is a triangle; a tile can be a square.

Exercise 5.3 ★

True or false? “A square turned on its corner stops being a square.” Explain with Example 5.2.

Solution

Solution of Exercise 5.3.

False: a square turned on its corner still has four equal sides and four corners — turning does not change the shape.

Exercise 5.4 ★

Draw on grid paper: a square of $3$ squares by $3$ ; a rectangle of $5$ by $2$ ; a triangle using a diagonal of the grid.

Solution

Solution of Exercise 5.4.

Grid drawings: the square is $3 \times 3$ squares, the rectangle $5 \times 2$ , and the triangle uses a slanted grid line as one side.

Exercise 5.5 ★

Place three toys in a row. Describe the row with position words: which is between the others? Which is on the left?

Solution

Solution of Exercise 5.5.

Answers vary. Example: the ball is on the left, the car is between the ball and the doll, the doll is on the right.

Exercise 5.6 ★

Draw a circle. Draw a cross inside it, a star outside it, and a dot on the circle itself.

Solution

Solution of Exercise 5.6.

The cross is drawn inside the circle, the star outside, and the dot right on the circle’s line.

Exercise 5.7 ★

On grid paper, mark a start square. Follow: $\to\ \uparrow\ \to\ \uparrow\ \to$ . Mark the arrival. Then write the path that goes straight back to the start.

Solution

Solution of Exercise 5.7.

Following $\to\ \uparrow\ \to\ \uparrow\ \to$ moves $3$ squares right and $2$ up. The way back is $\downarrow\ \downarrow$ then $\leftarrow\ \leftarrow\ \leftarrow$ (or any path returning to the start).

Exercise 5.8 ★

How many corners in total: two triangles and one square? One circle and two rectangles?

Solution

Solution of Exercise 5.8.

Two triangles and one square: $3 + 3 + 4 = 10$ corners. One circle and two rectangles: $0 + 4 + 4 = 8$ corners.

Exercise 5.9 ★★

Take $12$ matches. Can you outline a square using all of them, with whole matches and the same number on each side? How many matches per side? Could you do it with $10$ matches?

Solution

Solution of Exercise 5.9.

$12$ matches make a square with $3$ matches on each side ( $3 + 3 + 3 + 3 = 12$ ). With $10$ matches it cannot be done: $10$ does not split into $4$ equal whole parts (each side would need $2$ matches and a half).