E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Coordinate Grids with Different Scales

E1.4About 40 minutes

Learning goals

I can plot and read points on a coordinate grid when the lines count by 2s, 5s, 10s or more.
I can choose a good scale for a grid.
I can describe the translation (slide) that moves a point from one spot to another.

Words to know

coordinate grid
A grid with a horizontal x-axis and a vertical y-axis, used to give exact locations. (Its full name is the first quadrant of a Cartesian plane.)

ordered pair
Two numbers in brackets, like (15, 10), that give a point's location. The first number goes across (x) and the second goes up (y).

origin
The point (0, 0), where the two axes meet.

scale
How much each space between grid lines is worth, like 2, 5 or 10.

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

Remember from Grade 4

In Grade 4 you plotted points like (3, 5) on a coordinate grid. You started at the origin (0, 0), went across the x-axis first, then up the y-axis. Each grid line counted by 1. You also described how a point slides from one spot to another.

Let's start with a winter trail map

The park ranger is making a map of the ski trails. The park is 40 m wide on her map grid. If every grid line counted by 1, she would need 40 squares across, which is far too many to draw neatly! Instead, she makes each grid space worth 5. Now 8 spaces is enough: 5, 10, 15, 20, 25, 30, 35, 40.

The amount each space is worth is called the scale.

A trail map on a grid with a scale of 5. A (the skating rink) is at (10, 15), B (the warming hut) is at (35, 5) and C (the lookout) is at (15, 10).

Reading a grid with a scale

Read the numbers on the axes first. Find out what each space is worth.
Start at the origin. Go across to the point. Read the x-number.
Then go up to the point. Read the y-number.
Write the ordered pair (x, y).

If a point is halfway between two lines, its number is halfway between the two numbers. On a grid that counts by 2s, halfway between 6 and 8 is 7.

Worked example

Example 1: Read the trail map

What are the coordinates of the skating rink (A) and the warming hut (B)?

The axes count by 5s. Each space is worth 5.
A: go across 2 spaces: 5, 10. Then up 3 spaces: 5, 10, 15. A is at (10, 15).
B: go across 7 spaces: 35. Then up 1 space: 5. B is at (35, 5).

A (10, 15) and B (35, 5).

A grid with a scale of 2. P is at (4, 6), Q is at (10, 2), and R is halfway between grid lines at (7, 3).

Worked example

Example 2: A point between lines

On the grid that counts by 2s, what are the coordinates of point R?

Across: R is halfway between the lines for 6 and 8. Halfway is 7.
Up: R is halfway between the lines for 2 and 4. Halfway is 3.

R (7, 3)

Describing a translation

To describe how a point slides from one spot to another, compare the numbers:

Compare the x-numbers. If x gets bigger, the point moves right. If x gets smaller, it moves left. The difference tells you how far.
Compare the y-numbers. If y gets bigger, the point moves up. If y gets smaller, it moves down.

Say the across part first, then the up-or-down part.

On a grid with a scale of 10, point A (10, 40) slides 40 units right and 30 units down to reach point B (50, 10).

Worked example

Example 3: How did the point move?

Describe the translation from A (10, 40) to B (50, 10).

x-numbers: 10 → 50. It got bigger by 50 − 10 = 40. So 40 units right.
y-numbers: 40 → 10. It got smaller by 40 − 10 = 30. So 30 units down.
Check on the grid: on this grid, 40 units right is 4 spaces, and 30 units down is 3 spaces.

A moves 40 units right and 30 units down to B.

Worked example

Example 4: Choose a scale

You want to plot points with x-numbers up to 90 and y-numbers up to 60. Your paper has room for about 10 squares across and 7 squares up. What scale should you use?

With a scale of 1, you'd need 90 squares across. Too many!
With a scale of 10, you'd need 90 ÷ 10 = 9 squares across and 60 ÷ 10 = 6 squares up. That fits.

Use a scale of 10 (count by 10s on both axes).

Read the scale first

Always look at the axis numbers before you read or plot. A very common mistake is to count squares instead of reading the scale. "3 spaces" on a grid that counts by 5s is 15, not 3. When you catch yourself making this mistake, you are learning!

Just for fun

Can't remember which number comes first? Think of a building: you have to walk across the lobby before you can ride the elevator up. Across first (x), then up (y)!

Hands-on activity

Treasure map

You need: grid paper (or draw a grid with a ruler), a pencil, coloured pencils and a partner.

Draw a coordinate grid with 10 spaces across and 10 up. Number each axis by 10s (0, 10, 20 ... 100).
Draw a map of your home, your yard or your street on the grid. Mark 5 places, like the front door, a tree or the mailbox, with a dot on a grid point or halfway between grid lines.
Write the coordinates of each place in a list.
Hide a "treasure" at one point. Give your partner clues using translations: "Start at the front door. Move 30 units right and 20 units up."
Your partner follows the clues and says the coordinates where they land.

Math talk: "How did you choose the scale for your grid?"

Your turn

On the trail map, what are the coordinates of the lookout, C?

On the trail map, describe the translation from the skating rink A (10, 15) to the warming hut B (35, 5).

On a grid that counts by 2s, what number is halfway between the lines for 6 and 8?

A point moves from (12, 30) to (40, 30). Describe the translation.

Start at (40, 50). Move 30 units left and 15 units up. Where do you end?

You need to plot points with numbers up to about 1000. Would a scale of 1, 10 or 100 work best on a normal sheet of grid paper?

More practice: warm-up

In an ordered pair (x, y), which number tells you how far across to go?

What are the coordinates of the origin?

On a grid that counts by 10s, what number is the 3rd line after 0?

On which axis is the point (0, 20)?

On a grid that counts by 5s, how much are 4 spaces worth?

More practice: on your own

On the grid that counts by 2s, what are the coordinates of P and Q?

Describe the translation from P (4, 6) to Q (10, 2).

A graph shows heights from 0 to 50 cm. You have room for 5 squares up. What scale should you use for the y-axis?

Kai says (5, 15) and (15, 5) are the same point. Is he right?

A point is at (25, 40). It slides 15 units left and 20 units down. Where is it now?

Challenge

Point A is at (15, 40). It moves 25 units right and 20 units down, then 10 units left and 35 units up. Where does it end? Describe one single translation from the start to the end.

Mia was asked to plot (20, 5) on a grid that counts by 5s. She went up 4 spaces and right 1 space. What point did she plot? What should she have done?

Three corners of a rectangle are at (10, 10), (50, 10) and (50, 30). Where is the fourth corner? How long are the sides?

Check your understanding

Quiz

On a grid that counts by 5s, a point is 3 spaces across and 2 spaces up. What are its coordinates?

On a grid that counts by 2s, a point is halfway between the 4 line and the 6 line across. What is its x-number?

A point moves from (20, 10) to (20, 45). How did it move?

Which is the translation from (60, 30) to (20, 50)?

You need to plot numbers up to 200 with only 10 spaces. What scale should you use?

Start at (35, 15). Move 10 units left and 25 units up. Where do you end?

For parents and teachers

Big idea: The Ontario curriculum (E1.4) asks Grade 5 students to plot and read coordinates in the first quadrant of a Cartesian plane using various scales, and to describe the translations that move a point from one coordinate to another. The new idea this year is the scale: grid lines can count by 2s, 5s, 10s or more.

Common difficulties:

Counting squares instead of reading the scale. Have your child say the axis numbers out loud before reading a point.
Mixing up the order. Across first, then up. A memory trick: "You walk across the room before you climb the stairs."
Points between lines. Practise finding halfway values.

How to help: Make treasure maps together, or look at a paper map or a city map app that uses a grid. Ask "How far right and how far up is it from here?"