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Lesson 1 ~25 min Unit 2 · Linear +85 XP

The Cartesian Plane

Understand how two perpendicular number lines create a system for locating any point in 2D space using ordered pairs $(x, y)$.

Today's hook: Captain Grid has buried treasure. Her map uses a code: each location is written as a pair of numbers. The first number tells you steps east (or west if negative) from the old oak tree, the second tells you steps north (or south if negative). The treasure is at $(4, 3)$. Which quadrant is that?
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Printable Worksheets

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Think First
warm-up

Before we dive in — Captain Grid's map says the treasure is at $(4, 3)$, a clue is at $(-2, 5)$, and a trap is at $(3, -4)$. Can you figure out which quadrant each location lies in? What do you notice about the sign of the numbers in each quadrant?

Record your answer in your workbook.
1
The Big Idea
+5 XP

In the 17th century, René Descartes (1596–1650) had a brilliant insight: any position on a flat surface could be described using just two numbers. By crossing a horizontal number line with a vertical one, he created a system that turned geometry into algebra.

The Cartesian plane is a flat surface where we can locate any point using two numbers. Imagine two number lines. Stand one up vertically so they cross at their zero points, forming a perfect "+" shape. Every point on the plane now has a unique "address" called an ordered pair $(x, y)$. This system connects algebra with geometry — and powers GPS, computer graphics, and video games.

x y O 1 2 3 4 -1 -2 -3 -4 1 2 3 4 -1 -2 -3 -4 I II III IV A(3,2) B(-3,3) C(-2,-2) D(4,-3)
Every point has a unique address $(x, y)$
Horizontal first
The $x$-coordinate is always first. Move left/right along the corridor.
Vertical second
The $y$-coordinate is second. Move up/down the stairs.
Descartes' insight
Algebra + geometry = coordinate geometry. This powers GPS and gaming.
2
What You'll Master
objectives

Know

  • The Cartesian plane consists of two perpendicular number lines
  • The horizontal axis is the x-axis; the vertical axis is the y-axis
  • The point where axes intersect is the origin $(0, 0)$
  • Coordinates are written as ordered pairs $(x, y)$

Understand

  • How the Cartesian plane creates a system for locating any point in 2D space
  • The sign patterns of coordinates in each of the four quadrants
  • Why the order of coordinates matters ($x$ before $y$)

Can Do

  • Identify and label the x-axis, y-axis, origin, and quadrants
  • Read coordinates of points on the Cartesian plane
  • Determine the quadrant of a point from its coordinates
3
Words You Need
vocabulary
Cartesian PlaneA 2D coordinate system formed by two perpendicular number lines (axes) that intersect at the origin.
x-axisThe horizontal number line. The first number in an ordered pair tells you the position along this axis.
y-axisThe vertical number line. The second number in an ordered pair tells you the position along this axis.
OriginThe point $(0, 0)$ where the x-axis and y-axis intersect. The reference point for all coordinates.
CoordinatesAn ordered pair $(x, y)$ that describes the position of a point on the Cartesian plane.
QuadrantOne of the four regions created by the axes, numbered I, II, III, IV anticlockwise from the top right.
4
Spot the Trap
heads-up

Wrong: "The first number in $(3, 2)$ tells me to move 3 units up." No — the first number is always horizontal ($x$).

Right: "$x$ first (along the corridor), $y$ second (up the stairs)." The first coordinate is always horizontal.

Wrong: "Quadrants go clockwise from the top right." They actually go anticlockwise: I (top right) → II (top left) → III (bottom left) → IV (bottom right).

Right: Points on the axes do NOT belong to any quadrant. $(3, 0)$ is on the x-axis; $(0, 4)$ is on the y-axis.

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The Axes and the Origin
+5 XP

The Cartesian plane has two axes: the x-axis (horizontal, left to right) and the y-axis (vertical, bottom to top). Where they cross is the origin at $(0, 0)$ — the starting point for all coordinates.

The x-axis runs horizontally. Positive $x$ values are to the right of the origin; negative $x$ values are to the left. The y-axis runs vertically. Positive $y$ values are above the origin; negative $y$ values are below. The origin O(0, 0) is the intersection point — the "address zero" of the whole plane.

x-axis y-axis Origin O(0,0)
x-axis (horizontal) ⊥ y-axis (vertical)
Memory tip
The letter y has a vertical line going down — just like the y-axis!
Positive direction
Positive $x$ = right; positive $y$ = up. Both increase away from origin.
Origin = zero
The origin $(0, 0)$ is on both axes and belongs to no quadrant.
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The Four Quadrants
+5 XP

The two axes divide the plane into four quadrants, numbered I, II, III, and IV going anticlockwise from the top right. The sign of the coordinates tells you the quadrant.

Quadrant x y Sign
I (top right)positivepositive$(+,+)$
II (top left)negativepositive$(-,+)$
III (bottom left)negativenegative$(-,-)$
IV (bottom right)positivenegative$(+,-)$
Numbered anticlockwise from top right
Anticlockwise
Think backwards "C": I → II → III → IV, not like a clock.
Check both signs
Quadrant III has TWO negatives $(-, -)$. Always check both coordinates.
On axes = no quadrant
Points on the axes are boundaries between quadrants — not inside any.

Watch Me Solve It · 3 examples

Watch Me Solve It · Read coordinates
+15 XP per step
Q1
PROBLEM
Find the coordinates of point $P$ which is 2 units to the right of the origin and 2 units above it.
  1. 1
    Start at the origin
    $O(0, 0)$
    The origin is always our starting reference point.
  2. 2
    Move horizontally (x-coordinate)
    2 units to the right → $x = +2$
  3. 3
    Move vertically (y-coordinate)
    2 units up → $y = +2$
    Both coordinates are positive, so $P$ is in Quadrant I.
Answer$P(2, 2)$ — Quadrant I
Watch Me Solve It · Find the quadrant
+15 XP per step
Q2
PROBLEM
In which quadrant does each point lie? (a) $(-4, 7)$   (b) $(5, -2)$   (c) $(-1, -6)$
  1. 1
    Part (a): $(-4, 7)$
    $x = -4$ (negative), $y = 7$ (positive) → pattern $(-, +)$
    $(-, +)$ corresponds to Quadrant II (top left).
  2. 2
    Part (b): $(5, -2)$
    $x = 5$ (positive), $y = -2$ (negative) → pattern $(+, -)$
    $(+, -)$ corresponds to Quadrant IV (bottom right).
  3. 3
    Part (c): $(-1, -6)$
    $x = -1$ (negative), $y = -6$ (negative) → pattern $(-, -)$
    $(-, -)$ corresponds to Quadrant III (bottom left). Both negative!
Answer(a) Quadrant II   (b) Quadrant IV   (c) Quadrant III
Watch Me Solve It · Points on the axes
+15 XP per step
Q3
PROBLEM
The point $(4, 0)$ does not lie in any quadrant. Explain why, and describe its location.
  1. 1
    Identify the y-coordinate
    $y = 0$ → zero vertical distance from the x-axis
    When $y = 0$, the point sits directly on the x-axis.
  2. 2
    Why not in any quadrant?
    Quadrants are regions BETWEEN the axes, not on them
  3. 3
    Describe the location
    $(4, 0)$ lies on the x-axis, 4 units to the right of the origin
    Only points with BOTH $x \neq 0$ and $y \neq 0$ can lie inside a quadrant.
AnswerOn the x-axis, 4 units right of origin — not in any quadrant

Common Pitfalls

8
Common Pitfalls
heads-up
Mixing up x and y
In $(3, 2)$, students sometimes think "3 up, 2 right." But coordinates are always $(x, y)$ — $x$ is horizontal first, then $y$ vertical.
Fix: Say "along the corridor, up the stairs" — $x$ across, then $y$ up/down.
Wrong quadrant numbering direction
Quadrants are numbered anticlockwise from the top right. Many students assume they go clockwise like a clock — they do not!
Fix: Think of a backwards "C": I (top right) → II (top left) → III (bottom left) → IV (bottom right).
Forgetting that axes are not in any quadrant
Points ON the x-axis have $y = 0$, and points ON the y-axis have $x = 0$. These are the boundaries between quadrants, not inside them.
Fix: A point is in a quadrant ONLY if both $x \neq 0$ and $y \neq 0$.
Copy Into Your Books

The Cartesian Plane

  • Two perpendicular number lines (axes)
  • x-axis = horizontal; y-axis = vertical
  • Origin O(0, 0) where axes intersect

Coordinates

  • Written as ordered pair $(x, y)$
  • $x$ first (horizontal), $y$ second (vertical)
  • "Along the corridor, up the stairs"

Four Quadrants (anticlockwise)

  • I (top right): $(+, +)$
  • II (top left): $(-, +)$
  • III (bottom left): $(-, -)$
  • IV (bottom right): $(+, -)$

Points on Axes

  • On x-axis: $y = 0$, e.g. $(3, 0)$
  • On y-axis: $x = 0$, e.g. $(0, -2)$
  • NOT in any quadrant

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Cartesian Plane
4 problems

Four drill problems to sharpen your coordinate skills. Work each, then reveal the answer.

  1. 1 Which quadrant is $(5, 2)$ in?

    Both coordinates are positive: $(+, +)$.Quadrant I

  2. 2 Which quadrant is $(-2, -6)$ in?

    Both coordinates are negative: $(-, -)$.Quadrant III

  3. 3 The point $(0, 5)$ lies on which axis?

    When $x = 0$, the point is on the vertical axis.y-axis

  4. 4 Does the origin belong to any quadrant?

    It sits on both axes, so it cannot be inside any quadrant.No — it lies on both axes
Complete in your workbook.

Quick Check · 5 questions

1
Which quadrant is the point $(-3, 5)$ located in?
+10 XP
2
The point $(0, -4)$ lies on which axis?
+10 XP
3
What are the coordinates of the origin?
+10 XP
4
A point has coordinates $(a, b)$ where $a > 0$ and $b < 0$. Which quadrant?
+10 XP
5
The point $(-2, -5)$ has which sign pattern?
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 3 MARKS

Q6. Plot the following points on a Cartesian plane and state which quadrant each lies in (or which axis): $A(3, 2)$, $B(-1, 4)$, $C(-2, -3)$, $D(5, -1)$, $E(0, 3)$.

Answer in your workbook.
Understand Easy 2 MARKS

Q7. A point $P$ is located 4 units to the left of the origin and 6 units up. Write the coordinates of $P$ and state which quadrant it lies in.

Answer in your workbook.
Reason Hard 4 MARKS

Q8. Three points $A(2, 3)$, $B(-2, 3)$, and $C(2, -3)$ are plotted. A fourth point $D$ is added so that $ABCD$ forms a rectangle. Find the coordinates of $D$ and explain your reasoning.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B — Quadrant II. $x = -3$ (negative), $y = 5$ (positive) → pattern $(-, +)$.

2. B — y-axis. When $x = 0$, the point is on the vertical y-axis.

3. C — $(0, 0)$. The origin is where both axes intersect.

4. D — Quadrant IV. $a > 0$ (positive $x$), $b < 0$ (negative $y$) → pattern $(+, -)$.

5. C — $(-, -)$. Both $x = -2$ and $y = -5$ are negative.

Show Your Working Model Answers

Q6 (3 marks): A(3,2) → Quadrant I [1]; B(-1,4) → Quadrant II; C(-2,-3) → Quadrant III; D(5,-1) → Quadrant IV; E(0,3) → y-axis [1 each, accept any 3 correct].

Q7 (2 marks): 4 units left → $x = -4$ [1]. 6 units up → $y = 6$ [1]. $P = (-4, 6)$, Quadrant II. Pattern $(-, +)$.

Q8 (4 marks): $A(2,3)$ and $B(-2,3)$ share $y = 3$ (horizontal side). $A(2,3)$ and $C(2,-3)$ share $x = 2$ (vertical side). For rectangle, $D$ must match $B$'s $x$ and $C$'s $y$ [2]. So $D = (-2, -3)$ [1]. Check: $BD$ is vertical (same $x$), $CD$ is horizontal (same $y$) [1].

Stretch Challenge · +25 XP, +10 coins

The Rectangle Puzzle

A point $P(a, b)$ lies in Quadrant II. If $a = -7$ and the point is exactly 25 units from the origin, find the value of $b$. (Hint: use Pythagoras — the horizontal and vertical distances form a right triangle with the origin.)

Reveal solution

Since $P$ is in Quadrant II, $b > 0$. By Pythagoras: $7^2 + b^2 = 25^2$, so $49 + b^2 = 625$, so $b^2 = 576$, and $b = 24$. Point $P = (-7, 24)$.

R
Quick Review

The Plane

Two perpendicular number lines crossing at the origin

Coordinates

$(x, y)$ — $x$ horizontal first, $y$ vertical second

Four Quadrants

I $(+,+)$, II $(-,+)$, III $(-,-)$, IV $(+,-)$ — anticlockwise

Origin

$(0, 0)$ — on both axes, not in any quadrant

Axes = no quadrant

$y = 0$ → x-axis; $x = 0$ → y-axis. Not inside any quadrant

Memory trick

"Along the corridor, up the stairs" = $(x, y)$

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