Year 8 · Unit 3 Integers · Lesson 6 of 6

Across first, then up or down

Two number lines at right angles give every point a name. Negative numbers open up three more quadrants, and the distance ideas from earlier in the unit still work.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. How far apart are −5 and 3 on a number line?
    Answer8.
  2. Work out 2 − (−4).
    Answer6.
  3. Which is greater, −6 or −1?
    Answer−1: it is further right.
Words used in this lesson
axis (plural axes)
one of the two number lines on the grid. For example: the x-axis goes across.
origin
where the axes cross, (0, 0). For example: start every point here.
ordered pair
two numbers in a fixed order, (x, y). For example: (−4, 3).
coordinate
one of the two numbers in an ordered pair. For example: the x-coordinate of (5, −2) is 5.
quadrant
one of the four quarters of the grid. For example: (−3, −5) is in Quadrant III.
reflect
flip over a mirror line. For example: (3, −2) reflects to (3, 2) in the x-axis.

Two number lines make a map

A number line gives a position on a line. Put a second one across it, at right angles, and you can give a position anywhere on a flat surface. The across line is the x-axis, the up-and-down line is the y-axis, and they cross at the origin, (0, 0).

A point is named by an ordered pair (x, y): how far across first, then how far up or down. With negative numbers the axes extend left and down, and the plane splits into four quadrants, numbered I to IV anticlockwise from the top right. Maps of the whole Earth do the same: latitude is positive north of the Equator and negative south of it.

Predict, then check: where will P be for (−3, −5)? Which quadrant?

−4
3
(−4, 3)P
Quadrant IIwhere P is
6across from P to Q
4up or down from P to Q

P is 4 left and 3 up from the origin: (−4, 3), in Quadrant II. To reach Q at (2, −1) you go 6 across and 4 down.

Show me the steps
  1. P = (−4, 3): from the origin go 4 left, then 3 up. Quadrant II, where x is negative and y positive.
  2. P = (−3, −5): 3 left, 5 down. Quadrant III, where both are negative.
  3. P = (5, −2): 5 right, 2 down. Quadrant IV.
  4. P = (0, 4) is on the y-axis, in no quadrant.
  5. From P = (−4, 3) to Q = (2, −1): 6 across and 4 down.

(x, y): across first, then up or down. Negative x is left of the origin; negative y is below it.

Quadrant I (+, +), II (−, +), III (−, −), IV (+, −). A point on an axis is in no quadrant.

The order matters. (−3, 5) and (5, −3) are different points in different quadrants. If you say “along the corridor, then up the stairs”, you will read x first.

Worked examples

Plot A(3, −2) and name its quadrant.

  1. From the origin, go 3 right (x = 3).
  2. Then 2 down (y = −2). Positive x, negative y: Quadrant IV.

How far is it from A(−5, 2) to B(3, 2)?

  1. The y-coordinates are the same, so the path is straight across.
  2. It is the distance between −5 and 3 on a number line: 3 − (−5) = 8.

Your go. The first step is given. A rectangle has corners (−3, 2), (4, 2), (4, −1) and (−3, −1). Find its area.

  1. Width: from x = −3 to x = 4 is 4 − (−3) = 7.
  2. Find the height, then the area, and check.
Check the answer

Height: from y = −1 to y = 2 is 2 − (−1) = 3. Area = 7 × 3 = 21 square units.

Check before you go on

1. Which point is in Quadrant III?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Point P is 4 units left of the origin and 2 units down. What is its x-coordinate?

3. Which quadrant is (5, −3) in?

Level 2 · Building

4. The point (3, −2) is reflected in the x-axis. What is the y-coordinate of its image?

5. How far is it from (−6, −1) to (4, −1)?

6. A(−2, −3) moves 5 units right and 7 units up. What is the new y-coordinate?

Level 3 · Stretch

7. A rectangle has corners (−4, 3), (2, 3), (2, −2) and (−4, −2). What is its area, in square units?

8. Three corners of a square are (−1, 4), (3, 4) and (3, 0). The fourth corner is (−1, y). What is y?

Something to think about

Which points are the same distance from (0, 0) and from (4, 0), when you may only travel along grid lines?

One way to think about it

Every point with x = 2, such as (2, 5) or (2, −3). Each is 2 across from both points, and the same distance up or down, so the two journeys are equally long. Any point with a different x is nearer one of them.

Exit ticket

9. Leo plots (−3, 5) by going 5 left and 3 up from the origin. What went wrong?