Introduction to Graphs and the Cartesian Plane for JSS 2

Mathematics Lesson Plan

Subject: Mathematics
Class: JSS 2
Term: Second Term
Week: Week 10
Topic: Graphs

• Cartesian Plane
• Constructing Cartesian Plane
• Coordinate/Ordered Pair
• Choosing Scales
• Plotting Points on a Cartesian Plane
• Graphs of Linear Equations
• Plotting Graphs from a Table of Values

Previous Lesson:

Topic: Geometry – Solid Shapes (Cubes, Cuboids, Cylinders, Cones, Capacities)

Behavioral Objectives:

By the end of the lesson, students will be able to:

1. Understand the Cartesian Plane and its components.
2. Plot points accurately on the Cartesian Plane.
3. Understand and graph linear equations.
4. Plot graphs from a table of values.

Materials Needed:

• Whiteboard or chalkboard
• Markers or chalk
• Printed Cartesian Plane worksheets
• Rulers
• Graph paper or Graph Book
• Pencils
• Online Materials (for digital platforms)

Content Breakdown:

Introduction to the Cartesian Plane:

• The Cartesian Plane is a graph consisting of two perpendicular number lines that intersect at a right angle.
  • The x-axis is the horizontal axis.
  • The y-axis is the vertical axis.
  • The intersection point, where the x-axis and y-axis meet, is called the origin (0,0).

Coordinates and Ordered Pairs:

• Each point on the Cartesian Plane is represented by an ordered pair (x, y).
  • The first number (x) shows how far to move along the x-axis.
  • The second number (y) shows how far to move along the y-axis.

Example:

• Point (3, 5): Move 3 units to the right on the x-axis and 5 units up on the y-axis.

Constructing the Cartesian Plane:

1. Draw two perpendicular lines.
2. Label the x-axis and y-axis.
3. Mark the origin (0,0) where the lines intersect.

Choosing Scales:

• Scales are chosen for the axes to make the graph easy to interpret.
  • Example: For plant heights, 1 unit on the graph = 10 cm.

Plotting Points on the Cartesian Plane:

1. To plot a point (x, y):
   • Start at the origin (0,0).
   • Move x units along the x-axis (right if positive, left if negative).
   • Move y units along the y-axis (up if positive, down if negative).

Example:

• Plot (2, 3): Move 2 units to the right and 3 units up.

More Examples:

• (-2, 4) → 2 units left, 4 units up.
• (0, -3) → 3 units down (on y-axis).
• (5, 0) → 5 units to the right (on x-axis).

Graphs of Linear Equations:

• Linear equations are written in the form y = mx + b, where:
  • m is the slope (steepness of the line).
  • b is the y-intercept (where the line crosses the y-axis).

Example:

• Equation: y = 2x + 3.
  • Start at (0, 3) (y-intercept).
  • Use the slope (2) to go up 2 units and 1 unit to the right to find the next point (1, 5).

Plotting from a Table of Values:

• Choose values for x and use the equation to find y.
• Plot these points on the Cartesian Plane and draw a line through them.

Example:
For y = -3x + 4, if x = -2, -1, 0, 1, 2, calculate y-values and plot points.

Evaluation Questions:

1. Which of the following best describes the Cartesian Plane?
   a) a shape
   b) a type of graph
   c) a type of measurement
   d) a type of calculation
   Answer: b) a type of graph

2. What is the origin of the Cartesian Plane?
   a) the first point you plot on the graph
   b) the point where the x-axis and y-axis intersect
   c) the point where the x-axis and y-axis are perpendicular
   d) the highest point on the graph
   Answer: b) the point where the x-axis and y-axis intersect

3. Which of the following is an example of an ordered pair?
   a) (2, 3)
   b) 2 + 3
   c) 2 x 3
   d) 2 – 3
   Answer: a) (2, 3)

4. How do you plot a point on a Cartesian Plane?
   a) by choosing a random location
   b) by using the x and y coordinates
   c) by counting how many lines intersect at the point
   d) by guessing the location
   Answer: b) by using the x and y coordinates

5. What is the slope of a linear equation?
   a) the y-intercept
   b) the point where the line crosses the x-axis
   c) the steepness of the line
   d) the point where the line crosses the y-axis
   Answer: c) the steepness of the line

Lesson Presentation:

Introduction (5 minutes):

1. Ask students if they have heard of the Cartesian Plane or graphing before.
2. Explain the objective: understanding the Cartesian Plane, linear equations, and graphing.

Part 1: The Cartesian Plane (10 minutes):

1. Explain the Cartesian Plane’s components.
2. Draw and label the axes, origin, and coordinates on the board.
3. Distribute printed worksheets for students to practice plotting points.

Part 2: Linear Equations (15 minutes):

1. Define linear equations and slope-intercept form.
2. Explain slope and y-intercept with examples.
3. Students practice graphing equations on graph paper.

Part 3: Plotting Graphs from a Table of Values (10 minutes):

1. Demonstrate how to use a table of values to plot graphs.
2. Students work on exercises, plotting points from a table of values.

Assessment (10-20 minutes):

1. Have students complete a worksheet that includes plotting points, graphing linear equations, and plotting graphs from tables of values.
2. Review worksheets for understanding.

Conclusion (5 minutes):

1. Review the lesson’s objectives and ask if students feel they have understood the concepts.
2. Answer any remaining questions on Cartesian Plane, linear equations, or graphing.
3. Assign homework as necessary.

Fill-in-the-Blank Questions with Options (a, b, c, d) for the Topic “Graphs”

1. The Cartesian plane is made up of two perpendicular lines, called the __________ axis and __________ axis.
   a) X, Y
   b) Y, Z
   c) X, Z
   d) Y, W

2. In a Cartesian plane, the horizontal axis is called the __________ axis.
   a) Z
   b) Y
   c) X
   d) W

3. The point where the X and Y axes meet is called the __________.
   a) Origin
   b) Vertex
   c) Intersection
   d) Point of Interest

4. A coordinate pair is written in the form __________.
   a) (X, Y)
   b) [X, Y]
   c) {X, Y}
   d) (Y, X)

5. The coordinates of the origin are __________.
   a) (0, 1)
   b) (1, 0)
   c) (0, 0)
   d) (1, 1)

6. The vertical axis in the Cartesian plane is known as the __________ axis.
   a) X
   b) Z
   c) Y
   d) W

7. The coordinates of a point on the Cartesian plane are represented as (3, 4), where 3 is the __________ coordinate.
   a) Y
   b) X
   c) Z
   d) W

8. To plot the point (2, -3) on the Cartesian plane, we move __________ units along the X-axis and __________ units down along the Y-axis.
   a) 2, 3
   b) 2, -3
   c) -2, 3
   d) -2, -3

9. The line formed by plotting all points where y = 2x is called a __________.
   a) Curved graph
   b) Linear graph
   c) Parabolic graph
   d) Horizontal line

10. When plotting a graph from a table of values, the first column usually represents the __________ values.
    a) Y
    b) X
    c) Z
    d) W

11. A table of values is helpful when __________.
    a) Plotting a point
    b) Creating a coordinate system
    c) Graphing linear equations
    d) All of the above

12. When graphing a linear equation, the slope determines the __________ of the line.
    a) Shape
    b) Direction
    c) Steepness
    d) Length

13. The ordered pair (4, 2) represents the point that is __________ units to the right and __________ units up from the origin.
    a) 4, 2
    b) 2, 4
    c) -4, -2
    d) -2, -4

14. In the equation y = 3x + 1, the number 3 represents the __________.
    a) Y-intercept
    b) Slope
    c) X-coordinate
    d) Y-coordinate

15. A scale is chosen when constructing a graph to represent __________.
    a) Only the Y values
    b) Only the X values
    c) Both X and Y values proportionally
    d) Random values

Frequently Asked Questions (FAQs) for the Topic “Graphs”

1. What is a Cartesian plane?
   The Cartesian plane is a two-dimensional grid formed by two perpendicular number lines: the X-axis (horizontal) and the Y-axis (vertical), used to plot points.

2. How do I plot a point on a Cartesian plane?
   To plot a point, locate the x-coordinate on the X-axis, then the y-coordinate on the Y-axis, and draw a point where these two meet.

3. What does an ordered pair represent?
   An ordered pair represents the coordinates of a point on the Cartesian plane, written as (x, y).

4. What is the origin in a Cartesian plane?
   The origin is the point where the X-axis and Y-axis intersect, with coordinates (0, 0).

5. How do you read the coordinates (3, -4)?
   The first number (3) is the X-coordinate, indicating 3 units to the right, and the second number (-4) is the Y-coordinate, indicating 4 units down from the origin.

6. What is the difference between the X-axis and the Y-axis?
   The X-axis runs horizontally, while the Y-axis runs vertically in a Cartesian plane.

7. How do I choose a scale when drawing a graph?
   Choose a scale based on the range of values you want to represent, ensuring that both axes can accommodate the data without overcrowding.

8. What is the slope of a line?
   The slope is the rate of change between the Y and X values, and it describes how steep the line is.

9. What are linear equations and how are they graphed?
   Linear equations are equations of the form y = mx + b, where m is the slope and b is the y-intercept. These equations are graphed as straight lines.

10. What does the y-intercept represent in a linear equation?
    The y-intercept is the point where the graph of the equation crosses the Y-axis, with the x-coordinate equal to 0.

11. How do I plot a graph from a table of values?
    For each value in the table, plot the corresponding x and y coordinates as points on the Cartesian plane, then connect the points to form a graph.

12. What is the difference between a graph of a linear equation and a curve?
    A linear equation produces a straight line, while a curve results from non-linear equations where the rate of change between x and y is not constant.

13. Why is choosing the right scale important when graphing?
    Choosing the right scale ensures that your graph is clear, readable, and accurately represents the data points without overcrowding or misrepresentation.

14. How do I plot a point when the coordinates are negative?
    If the coordinates are negative, move left for negative x-values and down for negative y-values from the origin.

15. What is the relationship between x and y in a linear equation?
    In a linear equation, x and y have a constant rate of change, represented by the slope of the line.

10 Evaluation Questions for the Topic “Graphs”

1. What does the ordered pair (4, -5) represent on the Cartesian plane?

2. Plot the point (3, 2) on the Cartesian plane. What are the X and Y coordinates of the point?

3. In the equation y = -2x + 3, what is the slope of the line?

4. Explain how you would plot the point (0, 6) on the Cartesian plane.

5. What does the y-intercept of a line tell you about the graph?

6. What is the difference between plotting a point on the X-axis and the Y-axis?

7. How do you find the slope of a line from a table of values?

8. Why is it important to choose a suitable scale when constructing a graph?

9. Describe how to plot a graph from a table of values. What steps do you follow?

10. Given the equation y = x + 4, how would you plot the graph of this linear equation?