How many numbers locate a point?
Compare positions on a line and a grid to see why two directions are needed.
Begin with the activity ↓Is “two units across” enough to find a point?
Points can share a horizontal position but have different vertical positions. Give a reference and a position in both directions.
Imagine describing treasure on a grid to a friend. “Two units right” could point to several places at different heights. The horizontal position alone does not fix the vertical position.
First agree on the lower-left reference point. Choose rightward and upward directions, with one interval representing one unit. Without a shared reference, the same two numbers can lead to different places.
Find the point two units right and three units up from the reference. Count intervals between ticks, not the starting tick itself.
The points two across and one up, and two across and three up, share a horizontal position. Their heights differ, so they are different points. Agreement in one direction is not agreement in the whole position.
This activity locates intersections of grid lines, rather than numbered boxes. Keep intervals equal along each axis. Here one interval represents one unit in each direction.
Instead of describing both directions in a long sentence, we can write the two numbers in an agreed order. Next we put horizontal first and vertical second. You can begin on this grid before learning negative numbers.
Listen one paragraph at a time
Hold on to this distinction
Keep the horizontal position at two and change the vertical position. The different points show why a second number is needed.
Change it and check
Change each position separately. Hold one fixed and describe the direction in which the point moves.
We use integer grid intersections from 0 to 4. This does not mean every point in a plane has integer coordinates. The screen is not a ruler for measuring physical lengths.
Try with paper and objects
Draw a grid marked 0 through 4. Mark two points with the same horizontal position but different heights, and explain how to distinguish them.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Write an answer yourself
Return to the explanationCompare with a solution
(1, 3) · two numbers: horizontal, then vertical
Horizontal 1 comes first and vertical 3 second: (1, 3).
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Read points as ordered pairs and check what changes when the order is reversed.