Chapter 1 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.1 Chapter 1

Coordinates

How can I tell a friend exactly where a spot is, using only numbers?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Plot a point like (3, 5), going across first and then up
  • Read the coordinates of a point on a grid

On a town map, a letter and a number can name each square. A grid in mathematics does the same with two numbers. The two numbers name one exact place.

A grid has two number lines. The one that goes across is the x axis. The one that goes up is the y axis. They cross at the origin, where both numbers are 0.

The two numbers for a place are its coordinates. We write them in brackets, with a comma: (3, 5). The first number tells how far across. The second tells how far up.

To remember the order, think of a corridor and a staircase. You walk along the corridor first. Then you climb the stairs. Across first, then up.

Draw it first

Plot the point (3, 5)

  1. Step 1: a grid with the x axis across and the y axis up, both numbered from 0 to 6. Step 2: an arrow along the x axis from the origin, marked 3 across. Step 3: an arrow going up from there, marked 5 up, ending at the dot A (3, 5).3 across1234561234560xy

    1 Start at the origin, where the two axes cross. Its coordinates are (0, 0). Go 3 across along the x axis.

  2. Step 1: a grid with the x axis across and the y axis up, both numbered from 0 to 6. Step 2: an arrow along the x axis from the origin, marked 3 across. Step 3: an arrow going up from there, marked 5 up, ending at the dot A (3, 5).3 across5 up1234561234560xy

    2 Stay where you are across. Now go 5 up.

  3. Step 1: a grid with the x axis across and the y axis up, both numbered from 0 to 6. Step 2: an arrow along the x axis from the origin, marked 3 across. Step 3: an arrow going up from there, marked 5 up, ending at the dot A (3, 5).A (3, 5)3 across5 up1234561234560xy

    3 Mark a dot. Write the letter A beside it. The point is (3, 5).

A grid. The origin, 0, is where the x axis and the y axis cross. A point X (4, 0) is on the x axis, 4 across. A point Y (0, 3) is on the y axis, 3 up.X (4, 0)Y (0, 3)1234561234560xy
A point on the x axis has 0 up. A point on the y axis has 0 across.

Worked example

Plot the point A (3, 5) on a grid.

  1. The first number is 3. Start at the origin and go 3 across.
  2. The second number is 5. Go 5 up from there.
  3. Mark the dot. It is 3 across and 5 up.

Answer: 3 across and 5 up

A grid with a dot P. Dashed lines go from P down to the x axis and across to the y axis. The one going down meets the x axis at 5. The one going across meets the y axis at 3.P1234561234560xy
Look down to the x axis for the first number. Look across to the y axis for the second.

Worked example

Read the coordinates of the point P on the grid above.

  1. Look straight down from P to the x axis. It meets the axis at 5.
  2. Look straight across from P to the y axis. It meets the axis at 3.
  3. Write across first: (5, 3).

Answer: (5, 3)

See it three ways

With things

Put a finger on the bottom left corner of squared paper. Move it 3 squares along. Then move it 5 squares up. Make a dot.

As a picture

A grid with a dot A (3, 5). A dashed line goes down from A to the x axis at 3. A dashed line goes across from A to the y axis at 5.A (3, 5)1234561234560xy

In symbols

The place is written (3, 5). The first number is across and the second is up.

Try it

Hide a treasure on squared paper

You need: squared paper, a pencil, a friend.

  1. Draw a line along the bottom of the paper and a line up the left side. Number each line 0, 1, 2, 3 and so on.
  2. Mark a treasure on a crossing. Do not show your friend.
  3. Tell your friend its coordinates, for example (4, 3).
  4. Your friend marks the place. Then you compare the two papers.

What you will see: If your friend goes across first and then up, the two marks sit in the same place. If your friend goes up first, the marks are in different places. That is why everyone uses the same order.

If you cannot try it, this is what happens.
A grid with the x axis and the y axis. A dot T (4, 3) is the treasure and an open dot U (3, 4) where a friend who went up first would mark it.T (4, 3)U (3, 4)1234561234560xy

A wrong idea many people have

The order of the two numbers does not matter.

Why it feels right: Both numbers are in the brackets, so it seems they name the same place whichever comes first.

The test that breaks it: Plot (2, 5) and then (5, 2). Count across first, then up, each time. The dots are in different places.

What is true: The order matters. The first number is across and the second is up. Only a point with two equal numbers, like (4, 4), is the same both ways.

Key words

axis
a number line on a grid. The x axis goes across and the y axis goes up.
origin
the place where the two axes cross. Its coordinates are (0, 0).
coordinates
two numbers that name a place on a grid. Across comes first, then up.
plot
to mark a point on a grid.

Back to your guess

The answer is B: Go 3 across, then 5 up. The first number is how far across.

The first number tells how far across and the second tells how far up. So (3, 5) is 3 across and 5 up. The place (5, 3) is a different place: 5 across and 3 up.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which dot is the point (4, 2)?

    A grid with the x axis and the y axis, and four dots. A is at (2, 4). B is at (4, 2). C is at (4, 4). D is at (2, 2).ABCD1234561234560xy
    Tap your answer.

    1. Not this one. Dot A is 2 across and 4 up. You went up first. The first number is across. Go to M3.9.1. Try another.

    2. Not this one. Dot C is 4 across but also 4 up. The second number of (4, 2) is 2. Go to M3.9.1. Try another.

    3. Yes, that is right. The point (4, 2) is 4 across and 2 up. Dot B is the one there.

    4. Not this one. Dot D is 2 across and 2 up. The first number of (4, 2) is 4. Go to M3.9.1. Try another.

    See the answer

    Dot B

    The point (4, 2) is 4 across and 2 up. Dot B is the one there.

    • If you chose A: Dot A is 2 across and 4 up. You went up first. The first number is across. Go to M3.9.1.
    • If you chose B: Dot C is 4 across but also 4 up. The second number of (4, 2) is 2. Go to M3.9.1.
    • If you chose D: Dot D is 2 across and 2 up. The first number of (4, 2) is 4. Go to M3.9.1.

  2. What are the coordinates of the point P?

    A grid with a dot P. Dashed lines go from P down to the x axis and across to the y axis.P1234561234560xy
    Tap your answer.

    1. Not this one. You gave the up number first. Write across first: the line down meets the x axis at 5. Go to M3.9.1. Try another.

    2. Not this one. The line across from P meets the y axis at 3, not 5. Go to M3.9.1. Try another.

    3. Not this one. The line down from P meets the x axis at 5, not 3. Go to M3.9.1. Try another.

    4. Yes, that is right. P is 5 across and 3 up. So its coordinates are (5, 3).

    See the answer

    (5, 3)

    P is 5 across and 3 up. So its coordinates are (5, 3).

    • If you chose A: You gave the up number first. Write across first: the line down meets the x axis at 5. Go to M3.9.1.
    • If you chose B: The line across from P meets the y axis at 3, not 5. Go to M3.9.1.
    • If you chose C: The line down from P meets the x axis at 5, not 3. Go to M3.9.1.

  3. Which sentence tells you how to plot the point (6, 1)?

    Tap your answer.

    1. Not this one. That plots (1, 6). The first number is across. Go to M3.9.1. Try another.

    2. Yes, that is right. Coordinates count from the origin. The first number, 6, is across. The second number, 1, is up.

    3. Not this one. Coordinates always count from the origin. Starting anywhere else gives another place. Go to M3.9.1. Try another.

    See the answer

    Start at the origin. Go 6 across, then 1 up.

    Coordinates count from the origin. The first number, 6, is across. The second number, 1, is up.

    • If you chose A: That plots (1, 6). The first number is across. Go to M3.9.1.
    • If you chose C: Coordinates always count from the origin. Starting anywhere else gives another place. Go to M3.9.1.

  4. The point (0, 4) is plotted. Where is it?

    Tap your answer.

    1. Not this one. The across number is 0, so it does not move right. It is on the y axis. Go to M3.9.1. Try another.

    2. Not this one. The origin is (0, 0). This point goes 4 up. Go to M3.9.1. Try another.

    3. Yes, that is right. It is 0 across, so it stays on the y axis. It is 4 up.

    See the answer

    On the y axis, 4 up from the origin.

    It is 0 across, so it stays on the y axis. It is 4 up.

    • If you chose A: The across number is 0, so it does not move right. It is on the y axis. Go to M3.9.1.
    • If you chose B: The origin is (0, 0). This point goes 4 up. Go to M3.9.1.

  5. What are the coordinates of the origin?

    Tap your answer.

    1. Yes, that is right. The origin is where the axes cross. It is 0 across and 0 up.

    2. Not this one. The axes start counting at 0, not at 1. Go to M3.9.1. Try another.

    3. Not this one. That is 1 up from the origin. The origin is 0 across and 0 up. Go to M3.9.1. Try another.

    See the answer

    (0, 0)

    The origin is where the axes cross. It is 0 across and 0 up.

    • If you chose B: The axes start counting at 0, not at 1. Go to M3.9.1.
    • If you chose C: That is 1 up from the origin. The origin is 0 across and 0 up. Go to M3.9.1.

  6. Mei walks 6 squares across and 2 squares up from the origin. What are her coordinates?

    See the answer

    (6, 2)

    The first number is across, 6. The second number is up, 2. So she is at (6, 2).

  7. A point has coordinates (2, 6). How many units up is it from the x axis?

    See the answer

    6

    The second number is the number up. The point (2, 6) is 6 up.

Keep it

Copy this into your exercise book.

Coordinates are two numbers in brackets. The first is across and the second is up. Start at the origin, (0, 0). Across first, then up.

Draw: Draw a grid from 0 to 6. Plot (3, 5). Draw a dashed line down to the x axis and across to the y axis.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (761 KB). It opens on any phone.