Lesson 1: The Number Plane – Plotting Points and Simple Graphs

📘 Introduction

Have you ever played a game where you had to locate treasure using a map grid? That’s exactly what coordinates help us do in mathematics — find exact positions on a number plane.

In this lesson, we’ll explore how to:

• Understand the structure of the number plane
• Read and plot points using ordered pairs
• Identify quadrants
• Draw and interpret simple graphs from points

Let’s dive in and discover how to become a coordinate pro!

🧠 Key Concepts and Explanations

1. What is a Number Plane?

The number plane (or Cartesian plane) is a flat surface where we can locate points using two number lines:

• X-axis: Runs horizontally (left and right)
• Y-axis: Runs vertically (up and down)

The point where the x-axis and y-axis cross is called the origin, which is written as (0, 0).

2. Coordinates and Ordered Pairs

Each point on the plane is written as an ordered pair (x, y). The first value (x) tells you how far to move left or right, and the second value (y) tells you how far to move up or down.

3. Quadrants of the Number Plane

The plane is divided into four quadrants:

• Quadrant I: (+x, +y) – top right
• Quadrant II: (−x, +y) – top left
• Quadrant III: (−x, −y) – bottom left
• Quadrant IV: (+x, −y) – bottom right

Tip: Think of it like turning a clockwise wheel starting from the top right.

4. How to Plot Points (Step-by-Step)

To plot a point like (−3, 4):

1. Start at the origin (0, 0)
2. Move 3 units to the left (since x = −3)
3. Move 4 units up (since y = 4)
4. Mark the point

5. Drawing Simple Graphs

Once you plot several points, you can join them to form lines or shapes. Graphs help you visualize relationships between values.

Example 1: Line Graph

Plot the points A(0, 0), B(1, 2), C(2, 4), and D(3, 6). Connect them in order.

You’ll see a straight line going upward. This shows that as x increases, y also increases (a positive relationship).

Example 2: Closed Shape

Plot and connect points: A(1, 1), B(1, 4), C(4, 4), D(4, 1), and back to A.

You’ll get a rectangle!

🧮 More Sample Problem Walkthroughs

Example 3: What shape is formed by joining the points (−2, −1), (−2, 2), (1, 2), (1, −1)?

• Plot all four points
• Connect in order: You will form a rectangle in Quadrants II and I

Example 4: Draw a triangle with vertices at (0, 0), (3, 0), and (3, 4). Find the area of the triangle.

This triangle is right-angled. Use the formula for area of a triangle:

Base = 3 units, Height = 4 units. So:

✏️ Practice Exercises

Section A: Plotting and Identifying Points

1. Plot and label these points: P(2, 3), Q(−1, 4), R(−2, −3), S(4, −2)
2. State the quadrant for each point
3. Which point lies in Quadrant III?

Section B: Shape Construction

4. Plot the points A(0, 0), B(0, 3), C(4, 3), D(4, 0) and connect them. What shape is formed?
5. Now reflect point C(4, 3) across the x-axis. What are the coordinates of the new point C′?

Section C: Word Problem

6. A taxi driver starts at (0, 0), drives 5 km east to point A(5, 0), then 3 km north to point B(5, 3). Plot the path and find the total distance covered.

📝 Answers & Explanations

• Q1–Q3: P (Quadrant I), Q (Quadrant II), R (Quadrant III), S (Quadrant IV). R is in Quadrant III.
• Q4: A rectangle.
• Q5: Reflecting across x-axis changes y-sign: C′ = (4, −3)
• Q6: Distance = 5 + 3 = 8 km (movement is not diagonal)

🔁 Recap

• The number plane helps us describe positions using (x, y) coordinates
• It’s divided into four quadrants
• You can plot points, create shapes, and draw simple graphs
• Graphs help us understand relationships and solve problems visually

🪞Reflection Prompt

Why do you think we use coordinate planes in math, science, and even games? Can you think of an activity in your daily life that could use coordinates?