A four-quadrant coordinate grid with a bright plotted point.
Unit 3 · Negative Numbers & the Coordinate Plane · 6.NS

Lesson 3.3 — The Four-Quadrant Coordinate Plane

Two number lines cross to make a map of the whole plane.

🎙️ Narration script

Alright, let's build a map for numbers. It's called the coordinate plane.

Here's the idea. Take one number line lying flat, going left and right. That's the x-axis. Now take a second number line standing straight up, going up and down. That's the y-axis. Cross them right at their zeros, and the spot where they meet is called the origin, which is the point zero, zero.

Those two crossing lines split the whole plane into four regions, and we call each region a quadrant. We number them with Roman numerals, starting in the top right and going counterclockwise: Quadrant one, two, three, and four.

To name any point, we use an ordered pair, written as x first, then y, inside parentheses. The order really matters. Here's a little phrase to remember it: walk before you climb. You move sideways first, then up or down.

Let's plot the point three, two. Start at the origin. The first number is three, so move three to the right. The second number is two, so move two up. Mark your point. That lands you up in Quadrant one, where both numbers are positive.

The signs actually tell you the quadrant. Quadrant one is positive, positive. Quadrant two is negative, positive. Quadrant three is negative, negative. And Quadrant four is positive, negative. So a point like negative two, negative three goes down and to the left, into Quadrant three.

Quick recap. The coordinate plane is the x-axis and y-axis crossing at the origin. A point is an ordered pair, across first, then up or down. And the quadrants run one, two, three, four, counterclockwise. Great job today!

1 Core idea

The coordinate plane is two number lines crossing at the origin (0, 0): the horizontal x-axis and the vertical y-axis. They split the plane into four quadrants. Any point is an ordered pair (x, y): move x left/right first, then y up/down.

🧩 Think of it like… giving directions from the center of a city: "from the town square, go 3 blocks east, then 2 blocks north." Every spot has one exact address, and the order is fixed — east-amount first, north-amount second — just like (x, y).
Where it breaks: a real city only uses positive directions ("3 blocks east") and stops at the city limits, but the coordinate plane also goes west and south with negative numbers and stretches forever in all four directions — that's what the lower-numbered quadrants capture.

2 Key terms

x-axis / y-axis
The horizontal and vertical number lines.
Origin
Where the axes cross: (0, 0).
Ordered pair (x, y)
Coordinates of a point — x first, y second.
Quadrant
One of four regions, numbered I–IV counterclockwise from the top right.

3 Real-life examples

  • Maps & games: a treasure at "3 right, 2 up" is the point (3, 2).
  • Reflections: (3, 2) and (−3, 2) are mirror images across the y-axis.
  • Signs locate the quadrant: (−2, −3) is down-and-left → Quadrant III.
🤔 Pause & think: Is the point (3, 2) the same spot as (2, 3)? Why does the order inside the parentheses matter so much?
Reveal the thinking
No — they're different points. (3, 2) means right 3, up 2; (2, 3) means right 2, up 3. They both land in Quadrant I, but at different places. The order matters because the first number always controls sideways and the second always controls up/down — swap them and you've given a different address. "Walk before you climb."

4 Common doubts

Which number is x and which is y?

(x, y): x comes first (sideways), y second (up/down). "Walk before you climb."

What signs go with each quadrant?

I (+, +), II (−, +), III (−, −), IV (+, −) — counterclockwise from top-right.

How do reflections work?

Flip the sign of x → reflect across the y-axis. Flip the sign of y → reflect across the x-axis.

5 Step-by-step (plot a point)

  1. Start at the origin (0, 0).
  2. Move x: right if positive, left if negative.
  3. Move y: up if positive, down if negative.
  4. Mark the point and name its quadrant.

📊 See it · plotting (3, 2)

x y I (+,+)II (−,+) III (−,−)IV (+,−) (3, 2)

From the origin: right 3, then up 2 → the point (3, 2) lands in Quadrant I.

✅ Check yourself
  1. In which quadrant is the point (−4, 5)?
    answer Quadrant II. x is negative (left) and y is positive (up) → (−, +) is II.
  2. Reflect (3, 2) across the x-axis — what are the new coordinates?
    answer (3, −2). Reflecting across the x-axis flips the sign of y only, so the point drops into Quadrant IV.
⚡ Quick recap. The plane = x-axis & y-axis crossing at the origin. A point is (x, y) — across first, then up/down. Quadrants run I (+,+), II (−,+), III (−,−), IV (+,−) counterclockwise.

Grounded in CA CCSS-M, Grade 6 · 6.NS.6 & 6.NS.8 (coordinate plane, ordered pairs), California Department of Education. Hero image generated with Gemini Nano Banana Pro.