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Where is it? Above, below, beside, between, left and right

Geometry · level 1, lesson 3 of 5

A question to start

Stand face to face with a friend. You both hold up your right hand. Look at the two hands. Are they on the same side of the room?

No! Your right hand is near one wall, and your friend's right hand is near the other wall. From where you stand, your friend's right hand looks like it is on your left. Nobody is wrong. "Left" and "right" depend on which way you are facing. That is one of the surprising things about position words, the words we use to say where things are. In this lesson you will learn them, and you will learn how to give directions so clear that a robot could follow them.

The idea

Geometry is also about where

Geometry is not only about the shapes of things. It is also about where things are, and how to get from one place to another. Every time you say "the cat is under the table", you are doing a little bit of geometry. Builders, pilots, and the people who drive rovers on Mars all need very clear words for where things are.

Above and below

Above means higher up. Below means lower down.

  • The lamp is above the table.
  • The rug is below the table.

On top of means above and touching. Under (or underneath) means below, often with something covering. A book can be on top of the table, and a cat can be under it.

Look at this shelf.

top middle bottom
  • The ball is on the top shelf. It is above the cube.
  • The cylinder and the box are on the bottom shelf. They are below the cube.
  • The cube is on the middle shelf.

Notice something important. The ball is above the cube. So the cube is below the ball. The same fact can be said in two ways, by swapping the words around. "Above" and "below" are a pair of opposites.

Beside, next to, and between

Beside (or next to) means right at the side, close by, at the same height.

Between means in the middle of two things, with one thing on each side. "Between" always needs two other things. You cannot be between just one thing!

On the middle shelf, the cube is between the two cones. Each cone is beside the cube.

Left and right

Now look at this row of shapes.

1 2 3 4 5 left right

When we read a row, we start at the left and go to the right, just like reading words in a book.

  • The circle (1) is at the left end of the row.
  • The green triangle (5) is at the right end.
  • The square (3) is in the middle, with two shapes on its left and two shapes on its right.
  • The blue triangle (2) is to the left of the square. It is also next to the square.
  • The rectangle (4) is to the right of the square, and next to it.
  • The square (3) is between the blue triangle (2) and the rectangle (4).

Here is a trick to remember which is left. Hold both hands out in front of you, palms away, fingers up. Stick your thumbs out sideways. The hand where your thumb and first finger make the letter L is your left hand.

Why left and right can be tricky

Above and below are easy, because "up" is the same for everybody: it is away from the ground. But left and right depend on which way you are facing.

That is what happened in the question at the start. When you face your friend, everything is swapped: your left is on their right side, and your right is on their left side. When you stand side by side, facing the same way, your lefts and rights match again.

So when you give directions to someone, it helps to say which way they are facing first, or to use a thing you can both see: "the door side of the room", "the window side".

Giving directions on a grid

A grid is a set of squares in rows and columns, like a chess board or a sheet of squared paper. Grids make directions very exact. Here is a robot called R.

R

The robot can only move one square at a time: up, down, left or right. It cannot move corner-to-corner.

To send the robot to the triangle, you could say: "Go 2 squares right, then 2 squares up." Try it with your finger. Start at R. Right 1, right 2. Up 1, up 2. You land on the triangle!

There is often more than one way to get to the same place. "Up 2, then right 2" also works. Both paths take 4 steps.

Worked examples

Example 1. Use the shelf. What is on the shelf below the cone on the left?

  1. Find the cone on the left. It is on the middle shelf.
  2. "Below" means lower down. The shelf below the middle shelf is the bottom shelf.
  3. On the bottom shelf, right under the left cone, is the cylinder.

Example 2. Between in a row. In the row of five shapes, what is between the circle and the square?

  1. Find the circle: it is shape 1, at the left end.
  2. Find the square: it is shape 3.
  3. What is in the middle of 1 and 3? Shape 2, the blue triangle.

Now a harder one: what is between shape 1 and shape 5? Everything in the middle: shapes 2, 3 and 4. "Between" can mean more than one thing when the two ends are far apart.

Example 3. Directions to the star. Tell the robot how to get to the star in as few steps as possible.

  1. The star is at the top right. The robot is at the bottom left.
  2. Count how far right: from the robot's column to the star's column is 4 squares.
  3. Count how far up: from the bottom row to the top row is 3 squares.
  4. So: "Go 4 squares right, then 3 squares up." That is 4 + 3 = 7 steps.
  5. Could there be a shorter way? No. The robot must go right 4 times and up 3 times somehow, whatever order it does them in. Each step only moves one square. So 7 steps is the fewest possible.

That is a real reason, not a guess. Even if the robot zig-zags, "right, up, right, up, ...", it still uses 4 rights and 3 ups.

Try it

Use the pictures above.

  1. Which shape is on the top shelf?
  2. On the bottom shelf, is the box to the left or to the right of the cylinder?
  3. In the row, which shape is at the right end?
  4. In the row, which two shapes are next to the rectangle?
  5. In the grid, tell the robot how to get to the green square.
  6. In the grid, how many steps does the robot take to go to the triangle and then on to the star? (Triangle first!)
  7. Your friend sits facing you and says "the cup is on my right". Is the cup on your left or your right?
  8. Make up a riddle for a grown-up: "I am above the ... and between the ... and the ...".
Answers
  1. The ball (a sphere). It is above everything else.
  2. To the right. The cylinder is on the left side of the bottom shelf and the box is on the right side.
  3. The green triangle, shape 5.
  4. The square (shape 3, on its left) and the green triangle (shape 5, on its right).
  5. "Go 3 squares right." The square is in the same row as the robot, so it does not need to go up or down at all.
  6. To the triangle: 2 right and 2 up, that is 4 steps. Then from the triangle to the star: the star is 2 columns further right and 1 row higher, so 2 right and 1 up, that is 3 steps. Altogether 4 + 3 = 7 steps. Interesting: that is the same as going straight to the star, because the triangle is "on the way" (the robot never had to go left or down).
  7. On your left. When two people face each other, left and right are swapped.
  8. Any riddle that is true! For example: "I am above the cube, and I am on the top shelf all by myself. What am I?" (The ball.) Or: "I am between two cones. What am I?" (The cube.)

Common mistake

Mixing up left and right when facing someone. Your friend's right hand is on your left side when you face each other. If you want to point the same way as your friend, turn around so you face the same way, or use the L trick on your own hands.

Using "between" with one thing. "The cube is between the cone" does not make sense. "Between" needs a thing on each side: "the cube is between the two cones."

Where it leads