Cross two number lines and every point on the page gets its own address (x, y).
At the movies you find your seat by a row and a number; "two blocks east, three north" finds a street corner. A flat surface is two-dimensional, so it takes exactly two numbers to pin down any spot on it. Cross two number lines at right angles, agree to read across first, then up, and every point on the page earns one unique address — an ordered pair (x, y). This lesson builds that map, names its four fields, and shows how the very same two numbers already measure how far a point sits from each axis and from the center.
This is the foundation the whole Functions strand stands on. Once a point is a pair of numbers, the geometry moves you already know turn into pure arithmetic, and the graphs of functions have somewhere to live. Everything ahead — lines, parabolas, the idea of a function itself — is drawn on the plane we set up here.
To say where something is, you need a frame of reference and the right number of readings. On a straight road there is only one freedom — forward or back — so a single number does the job: "mile marker 47." On a line, one number locates a point.
A flat surface is different. Knowing only "row C" in a theatre still leaves a whole row of seats; knowing only "seat 8" leaves a whole column. You need both: row and seat. A delivery driver hears "two blocks east, then three blocks north" — again two readings, each a direction with a distance. A flat surface has two independent freedoms, so it takes two numbers to pin a point down.
A point on a line takes one number to locate. A point on a flat surface takes two. Mathematics turns "row and seat" into a precise, universal version of the same idea.
You have already met the number line: one line, a zero, positive to the right and negative to the left. That single line gives us one number to work with. To map a whole surface, we use two of them.
Lay one number line down flat — call it the x-axis. Stand a second number line straight up, crossing the first exactly at its zero — that is the y-axis. The two lines meet at right angles at a single point we call the origin, written O = (0, 0). On the x-axis, right is positive and left is negative; on the y-axis, up is positive and down is negative. Together these two crossed, scaled lines make the coordinate plane — also called the Cartesian plane, after René Descartes, who first joined algebra to geometry this way in the 1600s.
One line, one number. Two lines crossed at right angles, two numbers — and a flat page becomes a map where every point has an address.
Here is how to read a point P on the plane. Drop a straight line from P down to the x-axis; the number you land on is P's x-coordinate (its old name is the abscissa) — how far across it sits. Drop a straight line across to the y-axis; that reading is the y-coordinate (the ordinate) — how high it sits. Then write the two numbers down in order, x first, inside one set of parentheses:
P = (x, y) — x across, then y up.
Order is the whole point. The address (3, 2) means "3 across, 2 up"; the address (2, 3) means "2 across, 3 up" — a different place entirely. So (3, 2) ≠ (2, 3), and that is exactly why we call (x, y) an ordered pair: the order carries information.
The reading works both directions. Given a point, you read off its address. Given an address, you find the point: start at the origin, walk x along the x-axis, then turn and walk y parallel to the y-axis. Where you stop is the one and only point with that address.
Never swap the pair. The first number is always the across-reading (x), the second is always the up-reading (y). Writing (y, x) by accident sends you to the wrong place.
The two axes cut the plane into four regions called quadrants. We number them Ⅰ, Ⅱ, Ⅲ, Ⅳ going counterclockwise, starting from the top-right. The lovely part: you never have to look at the picture to know which quadrant a point is in — the signs of its coordinates tell you instantly.
What about a point sitting right on an axis? It belongs to no quadrant at all — it is on the border between two of them. A point on the x-axis has height zero, so its y = 0; a point on the y-axis has no across-reach, so its x = 0; and the origin, where both axes meet, is (0, 0).
| Quadrant | x | y | Example |
|---|---|---|---|
| Ⅰ | + | + | (3, 2) |
| Ⅱ | − | + | (−3, 2) |
| Ⅲ | − | − | (−3, −2) |
| Ⅳ | + | − | (3, −2) |
| on x-axis | any | 0 | (4, 0) |
| on y-axis | 0 | any | (0, −5) |
The Locate & address widget above already reports the quadrant live — step a coordinate down to 0 and watch it announce "on the axis" instead. The signs do all the work.
Here is a quiet bonus: the two numbers that name a point also measure it. Distance is never negative, so we read the size of a coordinate with the absolute value.
That last one is the Pythagorean theorem in disguise. Drop a guide from P to the x-axis: you get a right triangle whose legs are the across-distance |x| and the up-distance |y|, with the straight reach OP as its hypotenuse. So
OP = √(|x|² + |y|²) = √(x² + y²).
Work an example. Take P = (3, 4). It is 4 units from the x-axis (height 4), 3 units from the y-axis (across 3), and from the origin it is √(3² + 4²) = √(9 + 16) = √25 = 5. A clean 3–4–5 right triangle, read straight off the coordinates.
Where is P = (−3, 4) relative to everything? Its quadrant: signs (−, +) → Quadrant Ⅱ. From the x-axis: |4| = 4. From the y-axis: |−3| = 3. From the origin: √((−3)² + 4²) = √(9 + 16) = √25 = 5. The signs flipped the position, but the distances care only about size.
Cross two number lines at right angles through a shared zero and you build the coordinate plane: the horizontal x-axis, the vertical y-axis, and the origin O = (0, 0) where they meet.
Next, in 20.2 Symmetry & Translation, we watch the geometry moves — reflections and slides — become tidy arithmetic on these very coordinates.
Plot (−2, 5) in your head and name its quadrant.
Signs are (−, +) — across is negative, up is positive — so the point is top-left: Quadrant Ⅱ.
Write the coordinates of the point that sits 4 right and 3 down from the origin.
Right is positive x, down is negative y, so the address is (4, −3) — that lands in Quadrant Ⅳ.
Is (3, 2) the same point as (2, 3)? Explain.
No. The pair is ordered: (3, 2) is 3 across and 2 up, while (2, 3) is 2 across and 3 up — two different spots. Order carries meaning, so (3, 2) ≠ (2, 3).
A point lies on the y-axis. Which one of its coordinates must be 0, and why?
Its x-coordinate is 0. The y-axis is the line of "no across-reach," so the across-reading must be zero; the point looks like (0, y).
Find the distance of (−3, 4) from the origin.
Use √(x² + y²): √((−3)² + 4²) = √(9 + 16) = √25 = 5. (The negative sign disappears when you square.)
A point is 6 units from the y-axis, 0 units from the x-axis, and lies to the left. Name it.
Distance 0 from the x-axis means y = 0 — it sits on the x-axis. Distance 6 from the y-axis means |x| = 6, and "to the left" makes x negative. The point is (−6, 0).
Six questions to lock it in. Tap the answer you think is right.
This lesson opens the Functions strand by giving algebra a place to live. The single big idea is that a flat surface is two-dimensional, so an ordered pair — two numbers in a fixed order — pins down any point. Everything that follows (transformations as arithmetic, then functions and their graphs) depends on students being fluent and unhurried here: read across first, then up; let the signs name the quadrant; and treat the coordinates as both an address and a ruler.
Three misconceptions are worth watching for. First, swapping the pair — plotting (2, 3) when the point is (3, 2). Drilling "across, then up" out loud, and contrasting the two locations on the plane, fixes it fast. Second, believing a point on an axis lives "in" a quadrant; emphasize that axis points have a zero coordinate and belong to no quadrant. Third, reading a distance from an axis as the signed coordinate rather than its absolute value: the distance from the x-axis is |y|, never −y, so a point at (3, −4) is still 4 units away.
Alignment with the US Common Core: 5.G.A.1–2 (defining the coordinate plane; plotting and interpreting ordered pairs), 6.NS.C.6 (signs of coordinates, locating points by quadrant, and reflections across the axes), and 8.G.B.8 / G-GPE.B.7 (finding the distance between two points using the Pythagorean theorem — here, the distance from a point to the origin). The interactive widgets let students test their own readings against the live coordinate, which is exactly the self-correction these standards reward.