E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

The Cartesian Plane: All Four Quadrants

E1.3About 45 minutes

Learning goals

I can plot and read points with negative coordinates in all four quadrants.
I can name the quadrant a point is in by looking at the signs of its coordinates.
I can describe the translation that moves a point from one location to another, even across an axis.

Words to know

Cartesian plane
A flat grid made by a horizontal x-axis and a vertical y-axis that cross at the origin. It goes on forever in every direction.

quadrant
One of the four parts the axes divide the Cartesian plane into. They are numbered I, II, III and IV, going counterclockwise from the top right.

negative number
A number less than zero, such as the integer −3. On the x-axis, negatives are to the left of 0. On the y-axis, they are below 0.

origin
The point (0, 0), where the axes cross.

translation
A slide. Every point moves the same distance in the same direction.

Remember from Grade 5

In Grade 5 you plotted points like (15, 10) on grids with different scales. All your points were up and to the right of the origin, in one corner of the plane called the first quadrant. You also described translations, like "25 units right and 10 units down".

Let's start with a winter thermometer

On a cold January morning, the thermometer reads −8 °C. That's 8 degrees below zero. In the Number Sense unit you met integers like −8, which are less than zero.

A thermometer is like a number line standing up: positive numbers go up from 0 and negative numbers go down. Lay it on its side and the negatives are to the left of 0. Today you'll use negative numbers to find locations on a grid.

A number line from −6 to 6. The marked numbers are −4 (4 to the left of 0) and 3 (3 to the right of 0). They are 7 units apart: 4 units to get to 0, then 3 more.

Four quadrants

Now put two number lines together: the x-axis across, and the y-axis up and down. They cross at the origin (0, 0). The axes cut the plane into four quadrants.

To plot a point (x, y):

Start at the origin.
Look at x. Positive: go right. Negative: go left.
Then look at y. Positive: go up. Negative: go down.

Across first, then up or down, just like before!

A Cartesian plane from −5 to 5 on both axes, with one point in each quadrant: A (3, 4) in quadrant I, B (−4, 2) in quadrant II, C (−2, −3) in quadrant III and D (4, −4) in quadrant IV.

Worked example

Example 1: Read the points

What are the coordinates of B and C in the picture?

B: from the origin, go 4 left (x = −4), then 2 up (y = 2). B is at (−4, 2).
C: go 2 left (x = −2), then 3 down (y = −3). C is at (−2, −3).

B (−4, 2) and C (−2, −3).

Signs tell the quadrant

You can name the quadrant without plotting! Look at the signs:

(+, +) quadrant I
(−, +) quadrant II
(−, −) quadrant III
(+, −) quadrant IV

A point with a 0 in it, like (0, 4) or (−3, 0), sits on an axis and isn't in any quadrant.

Describing translations across the axes

To find how far a point slides, count the distance between the numbers on the number line. When you cross 0, count to 0 first, then past it. From −3 to 4 is 3 + 4 = 7.

Then pick the direction: x getting bigger means right, x getting smaller means left. y getting bigger means up, y getting smaller means down.

Point P (−3, 2) slides 7 units right and 3 units down to reach point Q (4, −1).

Worked example

Example 2: Describe the translation

Describe the translation that moves P (−3, 2) to Q (4, −1).

x: −3 to 4. Count to 0 (3 units), then to 4 (4 more): 7 units. x got bigger, so right.
y: 2 to −1. Count to 0 (2 units), then to −1 (1 more): 3 units. y got smaller, so down.

7 units right and 3 units down.

Worked example

Example 3: Where does it land?

Start at (2, −1). Translate the point 5 units left and 4 units up. Where does it land, and in which quadrant?

x: start at 2 and go 5 left: 2 → 1 → 0 → −1 → −2 → −3. x = −3.
y: start at −1 and go 4 up: −1 → 0 → 1 → 2 → 3. y = 3.
Signs (−, +) mean quadrant II.

(−3, 3), in quadrant II.

Just for fun

Why does the point (0, 0) never get lost?

It always knows where it came from: it's the origin!

Hands-on activity

Hidden treasure in four quadrants

You need: 2 sheets of grid paper, a ruler, 2 pencils and a partner. (Outside, you can draw a big grid with sidewalk chalk instead.)

Each player draws axes from −6 to 6 in the middle of a sheet. Number the lines carefully, with negatives to the left and down.
Secretly mark 3 "treasure" points, each in a different quadrant. Don't show your partner.
Take turns guessing a point by its coordinates, like "(−2, 5)". Your partner says "treasure!" or "miss". If it's a miss, they also say a translation that leads to their nearest treasure, like "3 units right and 1 unit down".
Mark every guess on your own sheet. First to find all 3 treasures wins.

Math talk: "How did you work out the distance when your translation crossed an axis?"

Your turn

Look at the first picture. What are the coordinates of A and D?

In which quadrant is (−6, 2)?

In which quadrant is (5, −7)?

Describe the translation from (−2, 3) to (4, 1).

Start at (1, −2). Translate the point 4 units left and 5 units up. Where does it land?

In which quadrant is (0, −4)?

More practice: warm-up

What are the signs of the coordinates in quadrant III?

What are the coordinates of the origin?

To plot a point with a negative x, do you go left or right from the origin?

Which is greater, −3 or −7?

Describe how to plot (−2, 0).

More practice: on your own

How far apart are −4 and 3 on a number line?

Describe the translation from (−5, −1) to (−1, 4).

Translate (3, 2) 6 units left and 5 units down. Where does it land, and in which quadrant?

Zoe plotted (−3, 4) by going 3 units left and 4 units down. What mistake did she make?

A point slides 8 units right and 2 units down and lands on (5, −6). Where did it start?

Challenge

Three corners of a square are at (−2, 3), (3, 3) and (3, −2). Find the fourth corner, the side length, the perimeter and the area.

A point in quadrant II is translated 4 units right and 3 units down. It lands on (1, −1). Where did it start? Describe the translation that would move it back.

A rectangle has corners at (−4, −1), (2, −1), (2, 3) and (−4, 3). Translate it 3 units left and 2 units down. Give the new corners and say which quadrants the new rectangle is in.

Check your understanding

Quiz

In which quadrant is (−4, 6)?

Which point is in quadrant IV?

Describe the translation from (−2, −3) to (3, 1).

Start at (−1, 4). Move 3 units right and 6 units down. Where do you land?

Which point is on the x-axis?

Both coordinates of a point are negative. Which quadrant is it in?

For parents and teachers

Big idea: The Ontario curriculum (E1.3) asks Grade 6 students to plot and read coordinates in all four quadrants of a Cartesian plane, and to describe the translations that move a point from one coordinate to another. The new part is negative coordinates.

Common difficulties:

Forgetting the sign. Have your child say "negative means left" for x and "negative means down" for y.
Counting across zero. From −3 to 4 is 7, not 1. Count to 0 first, then keep going.
Mixing up the order. Across (x) first, then up or down (y).
Quadrant numbers. They go counterclockwise from the top right: I, II, III, IV.

How to help: Use a thermometer to talk about negative temperatures. Play the treasure game on grid paper or with sidewalk chalk.