Across Then Up
An ordered pair is a set of directions from the origin: go across , then up . The order matters — and are different places.
Two number lines crossed at right angles let you name any point with two numbers: how far right (the -coordinate), then how far up (the -coordinate). The order matters — and are different points. Reading a graph is just reading these two numbers off.
From the origin, move right and then up.
Point is units to the right of the origin and units up. Its ordered pair is : across first, up second.
Reading the pair in the wrong order. "Along the hall, then up the stairs": first, second.
Plot and . Which one is on the -axis?
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is across and up, so it sits on the -axis. is across and up, on the -axis. Swapping the numbers moved the point from one axis to the other.
In Plain Terms & More Examples
An ordered pair means go across first, then up . Swapping the numbers lands you somewhere else.
From the origin, move right and up.
Move across (stay on the vertical axis) and up.
Starting the count at . The origin is , so the first grid line to the right is . Counting the origin itself as "" puts every point one square too far.
Start at . Move units right and units down. Where are you?
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Right adds to : . Down subtracts from : . You are at .
Distances and Shapes
Two points with the same lie on a horizontal line; the distance between them is the difference of the -coordinates. Same : vertical, subtract the 's.
Corners : width , height . It is a square of area .
Both points have , so they lie on one vertical line. The distance is the difference of the -coordinates: units.
Counting grid lines instead of spaces. From to you cross grid lines, but the distance is units. Count the jumps between the lines, not the lines.
Three corners of a rectangle are and . Find the fourth corner and the perimeter.
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Fourth corner . Width , height , perimeter .
Word Problems
On a coordinate map, east and west change the -coordinate (east adds, west subtracts); north and south change the -coordinate (north adds, south subtracts).
Keeping the two directions apart is the whole trick. A move east never changes how far north you are, so only one number changes at a time, and the moves can be done in either order.
Start at , go east and north: you land at .
Mia saves each week. After weeks she has . As ordered pairs (week, dollars): . Plotted, the points line up: each step right goes up.
Mixing up the directions. " east, then north from " is . Adding the to and the to gives , a different spot on the map.
From school at , Ravi walks west and then south. Where does he end up, and what point is that?
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, the origin.
Patterns and Graphs
Two number patterns can be paired up term by term to make ordered pairs. Plotting the pairs shows how the patterns are related.
If one pattern adds each time and the other adds , the second is always twice the first: after steps they are at and . The plotted points fall on a straight line through the origin, and the picture makes "twice as much" visible.
Pattern A starts at and adds : . Pattern B starts at and adds : . The pairs are . Each is double its : , , .
A plant's height is plotted each week: . It starts at cm and grows cm each week. Following the pattern, at week it would be cm.
Plotting by accident. The pair means across and up. Plotting it across and up puts it on a different line, one where each is half its .
Pattern starts at and adds ; pattern starts at and adds . Write the first four ordered pairs. How does compare with ?
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: . : . Pairs: . Each is times its : , , .
Why It Works
The coordinate plane turns geometry into arithmetic. A horizontal move from to is units, because only the -coordinate changed. A vertical move is a difference of -coordinates. Every fact about distances, shapes and lines on the plane can be computed from coordinates — which is what makes graphing, mapping, and every computer screen work.
Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.