Chapter 2 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

M3.9.2 Chapter 2

All four quarters of the grid

What if the place I want is to the left of the start, or below it?

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 points with negative coordinates, like (−2, 3)
  • Find the fourth corner of a square from three corners

In the last chapter the grid had only numbers from 0 up. But a number line goes below 0 too. On a grid, the x axis goes left of the origin. The y axis goes down below it.

Left of 0 the numbers are −1, −2, −3 and so on. Below 0 the y axis goes −1, −2, −3 in the same way. So both axes have numbers on both sides of the origin.

The two axes cut the grid into four quarters. A negative across number sends you left. A negative up number sends you down.

The first number of a point is its x coordinate. The second number is its y coordinate.

Draw it first

Plot the point (−2, 3)

  1. A grid with the x axis across and the y axis up. Step 1: an arrow along the x axis from the origin to the left, marked 2 left. Step 2: an arrow going up from there, marked 3 up. Step 3: the dot A at (−2, 3), in the top left quarter.2 left−3−2−1123−2−112340xy

    1 The first number is −2. A negative number means left. Go 2 left from the origin.

  2. A grid with the x axis across and the y axis up. Step 1: an arrow along the x axis from the origin to the left, marked 2 left. Step 2: an arrow going up from there, marked 3 up. Step 3: the dot A at (−2, 3), in the top left quarter.2 left3 up−3−2−1123−2−112340xy

    2 The second number is 3. A positive number means up. Go 3 up from there.

  3. A grid with the x axis across and the y axis up. Step 1: an arrow along the x axis from the origin to the left, marked 2 left. Step 2: an arrow going up from there, marked 3 up. Step 3: the dot A at (−2, 3), in the top left quarter.A2 left3 up−3−2−1123−2−112340xy

    3 Mark a dot and write A. The point (−2, 3) is in the top left quarter.

A grid with the x axis across and the y axis up. Four dots, one in each quarter. A (3, 2) is in the top right. B (−3, 2) is in the top left. C (−3, −2) is in the bottom left. D (3, −2) is in the bottom right.A (3, 2)B (−3, 2)C (−3, −2)D (3, −2)−5−4−3−2−112345−5−4−3−2−1123450xy
Top right: both numbers positive. Top left: across negative. Bottom left: both negative. Bottom right: up negative.

Worked example

Plot the point B (−4, −3).

  1. The first number is −4. Go 4 left from the origin.
  2. The second number is −3. Go 3 down from there.
  3. Mark the dot. It is in the bottom left quarter.

Answer: 4 left and 3 down

A grid with the x axis across and the y axis up. The dot B (−4, −3). A dashed line goes from B up to the x axis at −4. A dashed line goes from B across to the y axis at −3.B−5−4−3−2−112345−5−4−3−2−1123450xy
B is 4 left of the y axis and 3 below the x axis.

In this chapter the sides of the square run along the grid lines.

Worked example

Three corners of a square are A (−2, −1), B (2, −1) and C (2, 3). Find the fourth corner D.

  1. From A to B the square goes 4 across, from −2 to 2. So each side is 4 long.
  2. From B to C it goes 4 up, from −1 to 3.
  3. D is above A. It has the same across number as A, which is −2.
  4. D has the same up number as C, which is 3. So D is (−2, 3).

Answer: (−2, 3)

A grid with the x axis across and the y axis up. A square with corners A (−2, −1), B (2, −1) and C (2, 3) shown as dots, and an open dot D at the fourth corner. The sides are drawn as a dashed square.ABCD−5−4−3−2−112345−5−4−3−2−1123450xy
D is the open dot. Its across number matches A and its up number matches C.

See it three ways

With things

Stand on a spot. Take 2 steps to the left. Then take 3 steps forward. Mark where you stop.

As a picture

A grid with the x axis across and the y axis up. A dot A (−2, 3). A dashed line goes from A down to the x axis at −2. A dashed line goes from A across to the y axis at 3.A−5−4−3−2−112345−5−4−3−2−1123450xy

In symbols

The place is (−2, 3). The first number is −2, so go left. The second number is 3, so go up.

Try it

A floor grid

You need: chalk, or sticks and stones, a flat floor or yard, a friend.

  1. Draw a long cross on the ground. The two lines are the axes. Mark the middle with a stone. This is the origin.
  2. Mark every step along each line with a stone or a chalk mark. Count left of the middle as negative.
  3. Your friend stands on the origin. Say a point, for example (−2, 3).
  4. Your friend walks 2 steps left, then 3 steps forward. Check where they stop.

What you will see: A negative across number sends your friend left. A negative up number sends them back, towards the bottom of the picture. Forward is up.

If you cannot try it, this is what happens.
A grid with the x axis across and the y axis up. A dot S at the origin where your friend starts, and a dot F at (−2, 3) where they stop after 2 steps left and 3 steps forward.SF−5−4−3−2−112345−5−4−3−2−1123450xy

A wrong idea many people have

The point (−3, −2) is in the top right quarter, because two minus signs make a plus.

Why it feels right: Two minus signs do make a plus when you multiply negative numbers. So it seems both can cancel.

The test that breaks it: Plot (−3, −2). Go 3 left, then 2 down. You are in the bottom left quarter, not the top right.

What is true: The two numbers do not multiply. Each one moves the point by itself. A minus on the across number is left. A minus on the up number is down.

Key words

quarter
one of the four parts the two axes cut the grid into.
x coordinate
the first number of a point. It tells how far across, left or right.
y coordinate
the second number of a point. It tells how far up or down.

Back to your guess

The answer is B: The point (−2, 3) is 2 to the left of the origin and 3 up.

A negative first number means left. A negative second number means down. So (−2, 3) is 2 left and 3 up. It is not the same place as (2, 3), which is 2 right and 3 up.

Check yourself

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

  1. Which point is in the bottom left quarter?

    A grid with the x axis across and the y axis up. Four dots. A (3, 2), B (−3, 2), C (−3, −2) and D (3, −2).A (3, 2)B (−3, 2)C (−3, −2)D (3, −2)−5−4−3−2−112345−5−4−3−2−1123450xy
    Tap your answer.

    1. Not this one. A is 3 right and 2 up. That is the top right quarter. Go to M3.9.2. Try another.

    2. Not this one. B is left, which fits the bottom left. But it is 2 up. The bottom left needs a down number. Go to M3.9.2. Try another.

    3. Yes, that is right. The bottom left quarter has both numbers negative. C is (−3, −2).

    4. Not this one. D is 2 down, which fits. But it is 3 right. The bottom left needs a left number. Go to M3.9.2. Try another.

    See the answer

    Point C

    The bottom left quarter has both numbers negative. C is (−3, −2).

    • If you chose A: A is 3 right and 2 up. That is the top right quarter. Go to M3.9.2.
    • If you chose B: B is left, which fits the bottom left. But it is 2 up. The bottom left needs a down number. Go to M3.9.2.
    • If you chose D: D is 2 down, which fits. But it is 3 right. The bottom left needs a left number. Go to M3.9.2.

  2. Where is the point (−4, 2)?

    Tap your answer.

    1. Not this one. That is the point (4, −2). The signs are the other way round. Go to M3.9.2. Try another.

    2. Yes, that is right. The first number, −4, is 4 left. The second number, 2, is 2 up.

    3. Not this one. The second number, 2, has no minus sign. So it goes up. Go to M3.9.2. Try another.

    4. Not this one. You swapped the numbers. The first number, −4, is across. Go to M3.9.1. Try another.

    See the answer

    4 left and 2 up

    The first number, −4, is 4 left. The second number, 2, is 2 up.

    • If you chose A: That is the point (4, −2). The signs are the other way round. Go to M3.9.2.
    • If you chose C: The second number, 2, has no minus sign. So it goes up. Go to M3.9.2.
    • If you chose D: You swapped the numbers. The first number, −4, is across. Go to M3.9.1.

  3. What are the coordinates of the point M?

    A grid with the x axis across and the y axis up. A dot M. Dashed lines go from M up to the x axis and across to the y axis.M−5−4−3−2−112345−5−4−3−2−1123450xy
    Tap your answer.

    1. Not this one. M is left of the y axis and below the x axis. Both numbers are negative. Go to M3.9.2. Try another.

    2. Not this one. You swapped the numbers. The line up from M meets the x axis at −3. Across comes first. Go to M3.9.1. Try another.

    3. Not this one. M is below the x axis, so the second number is negative. Go to M3.9.2. Try another.

    4. Yes, that is right. The dashed line up meets the x axis at −3. The dashed line across meets the y axis at −4. So M is (−3, −4).

    See the answer

    (−3, −4)

    The dashed line up meets the x axis at −3. The dashed line across meets the y axis at −4. So M is (−3, −4).

    • If you chose A: M is left of the y axis and below the x axis. Both numbers are negative. Go to M3.9.2.
    • If you chose B: You swapped the numbers. The line up from M meets the x axis at −3. Across comes first. Go to M3.9.1.
    • If you chose C: M is below the x axis, so the second number is negative. Go to M3.9.2.

  4. Which point is on the y axis?

    Tap your answer.

    1. Not this one. This point has 0 up, so it sits on the x axis. Go to M3.9.2. Try another.

    2. Not this one. This point is 3 left and 3 down. It is in the bottom left quarter, not on an axis. Go to M3.9.2. Try another.

    3. Yes, that is right. A point on the y axis is 0 across. It has no left or right move. The point (0, −3) is 3 down from the origin.

    See the answer

    (0, −3)

    A point on the y axis is 0 across. It has no left or right move. The point (0, −3) is 3 down from the origin.

    • If you chose A: This point has 0 up, so it sits on the x axis. Go to M3.9.2.
    • If you chose B: This point is 3 left and 3 down. It is in the bottom left quarter, not on an axis. Go to M3.9.2.

  5. Three corners of a square are A (−3, −2), B (1, −2) and C (1, 2). What is the fourth corner D?

    A grid with the x axis across and the y axis up. Three dots, A (−3, −2), B (1, −2) and C (1, 2), at corners of a square. The fourth corner is not drawn.ABC−5−4−3−2−112345−5−4−3−2−1123450xy
    Tap your answer.

    1. Yes, that is right. D is above A, so its across number is −3. It is level with C, so its up number is 2.

    2. Not this one. You swapped the numbers. D is above A, so it has A’s across number, −3. Go to M3.9.2. Try another.

    3. Not this one. You went 4 right of C. D is on the other side of the square, level with C. Go to M3.9.2. Try another.

    4. Not this one. You went down from A. D is 4 above A, not below it. Go to M3.9.2. Try another.

    See the answer

    (−3, 2)

    D is above A, so its across number is −3. It is level with C, so its up number is 2.

    • If you chose B: You swapped the numbers. D is above A, so it has A’s across number, −3. Go to M3.9.2.
    • If you chose C: You went 4 right of C. D is on the other side of the square, level with C. Go to M3.9.2.
    • If you chose D: You went down from A. D is 4 above A, not below it. Go to M3.9.2.

  6. Three corners of a square are (−2, −2), (3, −2) and (3, 3). What is the fourth corner?

    See the answer

    (−2, 3)

    The fourth corner is above the first, so it has across number −2. It is level with the third, so it has up number 3.

  7. How many units to the left of the y axis is the point (−4, 3)?

    See the answer

    4

    The first number, −4, says 4 left. The point is 4 units left of the y axis.

  8. From an earlier chapter

    A point is 7 across and 2 up from the origin. What are its coordinates?

    See the answer

    (7, 2)

    Across comes first, so the first number is 7. The second number is 2.

Keep it

Copy this into your exercise book.

Left of the origin is negative across. Below the origin is negative up. The axes make four quarters. Each number moves the point by itself.

Draw: Draw a grid from −5 to 5. Plot (−2, 3), (3, 2), (−3, −2) and (3, −2). Write each quarter next to its dot.

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.