Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Graphs
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Graphs
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Subject: General Mathematics
Class: Junior Secondary 2
Term: 2nd Term
Week: 3
Theme: Algebraic Processes
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Identify x–axis and y –axis Plot points on the Cartesian plane Prepare table of values Plot the graph of linear equations in two variables; In terpret the plotted graph Plot linear graphs from real life situations Solve quantitative aptitude problem.
information can be extracted from it.
Reading values: Given an x-value, find the corresponding y-value on the line (and vice versa).
Understanding trends: Observe if the line is going up (increasing relationship), down (decreasing relationship), or horizontal (constant relationship).
Finding intercepts: The point where the line crosses the x-axis (y=0) is the x-intercept. The point where it crosses the y-axis (x=0) is the y-intercept.
Worked Example 4: Interpreting a Graph Instruction: Using the graph of `y = 2x - 1` from Worked Example 3, find: a) The value of y when x = 0.5. b) The value of x when y =
2. Solution: a) To find y when x = 0.5: Locate 0.5 on the x-axis. Move vertically upwards from x = 0.5 until you reach the line. From that point on the line, move horizontally to the left until you reach the y-axis. Read the value on the y-axis. (It should be 0).
Calculation check: y = 2(0.5) - 1 = 1 - 1 =
0. So, when x = 0.5, y = 0. b) To find x when y = 2: Locate 2 on the y-axis. Move horizontally to the right from y = 2 until you reach the line. From that point on the line, move vertically downwards until you reach the x-axis. Read the value on the x-axis. (It should be 1.5).
Calculation check: 2 = 2x - 1 => 3 = 2x => x = 1.
5. So, when y = 2, x = 1.5. (Teacher should demonstrate tracing these points on the graph carefully.) G. Plotting Linear Graphs from Real-Life Situations Many real-life scenarios exhibit linear relationships. The key is to formulate an equation that represents the situation.
Steps:
1. Identify variables: Determine the two quantities that are related (e.g., number of items and cost, distance and time). Assign variables (e.g., x and y) to them.
2. Formulate an equation: Write a linear equation that expresses the relationship between the variables. This often involves identifying a constant rate of change and an initial value.
3. Prepare table of values: Choose relevant values for the independent variable and calculate the corresponding values for the dependent variable.
4. Plot the graph: Plot the points and draw the line, just as for abstract equations. Remember to label axes with the quantities they represent and include units (e.g., "Cost (Naira)", "Distance (km)").
5. Interpret: Use the graph to answer questions related to the real-life situation.
Worked Example 5: Real-Life Situation Graph Instruction: A vendor at a market sells yam tubers for N300 each. The cost of buying yam tubers from the vendor can be represented by a graph. a) Formulate an equation relating the number of yam tubers (x) to the total cost (y). b) Prepare a table of values for 0, 1, 2, 3, 4, 5 yam tubers. c) Plot the graph of the relationship. d) Use the graph to find the cost of 3.5 yam tubers. e) Use the graph to find how many yam tubers can be bought with N
1
2
0
0. Solution: a)
Formulate equation: Let x = number of yam tubers. Let y = total cost in Naira. Since each tuber costs N300, the total cost is 300 times the number of tubers.
Equation: `y = 300x` b)
Prepare table of values: | x (Number of Yam Tubers) | Calculation (300x) | y (Total Cost N) | (x, y) | | :----------------------- | :--------------------- | :--------------- | :--------------- | | 0 | 300 0 | 0 | (0, 0) | | 1 | 300 1 | 300 | (1, 300) | | 2 | 300 2 | 600 | (2, 600) | | 3 | 300 3 | 900 | (3, 900) | | 4 | 300 4 | 1200 | (4, 1200) | | 5 | 300 5 | 1500 | (5, 1500) | c)
Plot the graph: Draw axes. Label x-axis "Number of Yam Tubers (x)" and y-axis "Total Cost (N) (y)". * Choose a suitable scale: For x-axis, This section provides a detailed explanation of the core concepts required for teaching graphs. A. The Cartesian Plane (Coordinate Plane) The Cartesian plane is a two-dimensional surface formed by the intersection of two perpendicular number lines. It is used to plot points and graph equations. Origin (0,0): This is the central point where the x-axis and y-axis intersect. It represents the value zero for both axes. x-axis: This is the horizontal number line. Positive values are to the right of the origin, and negative values are to the left. y-axis: This is the vertical number line. Positive values are above the origin, and negative values are below.
Quadrants: The x-axis and y-axis divide the plane into four regions called quadrants, numbered counter-clockwise using Roman numerals (I, II, III, IV) starting from the top-right.
Quadrant I: (+x, +y)
Quadrant II: (-x, +y)
Quadrant III: (-x, -y)
Quadrant IV: (+x, -y) B. Coordinates and Plotting Points A point on the Cartesian plane is located using an ordered pair of numbers called coordinates, written as (x, y). x-coordinate (Abscissa): The first number in the ordered pair, indicating the horizontal distance from the origin along the x-axis. y-coordinate (Ordinate): The second number in the ordered pair, indicating the vertical distance from the origin along the y-axis. Steps for Plotting a Point (x, y):
1. Start at the origin (0,0).
2. Move horizontally along the x-axis according to the x-coordinate: Move right if x is positive. Move left if x is negative. Stay at the origin if x is zero.
3. From that horizontal position, move vertically along a line parallel to the y-axis according to the y-coordinate: Move up if y is positive. Move down if y is negative. Stay at the current position if y is zero.
4. Mark the final position with a dot and label it if necessary.
Worked Example 1: Identifying Axes and Plotting Points Instruction: Draw a Cartesian plane and identify the x-axis, y-axis, and origin. Then, plot the following points: A(3, 2), B(-2, 1), C(1, -3), D(-4, -2), E(0, 4), F(-3, 0).
Solution:
1. Draw two perpendicular lines intersecting at a point (the origin). Label the horizontal line 'x-axis' and the vertical line 'y-axis'. Mark the origin as '0'.
2. Scale the axes: Mark uniform intervals (e.g., 1 unit per square) along both axes, extending positive values to the right and up, and negative values to the left and down.
3. Plotting: A(3, 2): From origin, move 3 units right, then 2 units up. Mark the point. B(-2, 1): From origin, move 2 units left, then 1 unit up. Mark the point. C(1, -3): From origin, move 1 unit right, then 3 units down. Mark the point. D(-4, -2): From origin, move 4 units left, then 2 units down. Mark the point. E(0, 4): From origin, stay at 0 on x-axis, then move 4 units up. (Point is on the y-axis). F(-3, 0): From origin, move 3 units left, then stay at 0 on y-axis. (Point is on the x-axis). (Teacher should sketch this on the board, demonstrating each step visually) C. Linear Equations in Two Variables A linear equation in two variables (typically x and y) is an equation that can be written in the form `Ax + By = C` or `y = mx + c`, where A, B, C, m, and c are constants, and x and y are the variables. When graphed, a linear equation always forms a straight line. D. Preparing a Table of Values To graph a linear equation, we need at least two points. A table of values helps in systematically finding several (x, y) pairs that satisfy the equation. Steps for Preparing a Table of Values for an Equation (e.g., `y = 2x - 1`):
1. Choose a range of x-values. It's good practice to choose both negative, zero, and positive values (e.g., -2, -1, 0, 1, 2).
2. Substitute each chosen x-value into the equation to find the corresponding y-value.
3. Record the (x, y) pairs in a table. a Table of Values To graph a linear equation, we need at least two points. A table of values helps in systematically finding several (x, y) pairs that satisfy the equation. Steps for Preparing a Table of Values for an Equation (e.g., `y = 2x - 1`):
1. Choose a range of x-values. It's good practice to choose both negative, zero, and positive values (e.g., -2, -1, 0, 1, 2).
2. Substitute each chosen x-value into the equation to find the corresponding y-value.
3. Record the (x, y) pairs in a table.
Worked Example 2: Preparing a Table of Values Instruction: Prepare a table of values for the equation `y = 2x - 1` for x-values from -2 to
2. Solution: | x | Calculation (2x - 1) | y | (x, y) | | :---- | :------------------- | :-------- | :---------- | | -2 | 2(-2) - 1 = -4 - 1 | -5 | (-2, -5) | | -1 | 2(-1) - 1 = -2 - 1 | -3 | (-1, -3) | | 0 | 2(0) - 1 = 0 - 1 | -1 | (0, -1) | | 1 | 2(1) - 1 = 2 - 1 | 1 | (1, 1) | | 2 | 2(2) - 1 = 4 - 1 | 3 | (2, 3) | (Teacher should walk through each calculation step by step.)
E. Plotting the Graph of Linear Equations Steps for Plotting a Linear Graph:
1. Prepare a Table of Values: (As shown in Worked Example 2).
2. Draw Axes: Draw and label the x-axis and y-axis on graph paper, indicating the origin.
3. Choose a Scale: Select an appropriate scale for both axes based on the range of x and y values in the table. The scale should allow all points to fit comfortably on the graph paper and be easy to read (e.g., 1 cm represents 1 unit, or 1 cm represents 2 units). It's crucial for the scale to be uniform along each axis.
4. Plot Points: Plot all the (x, y) coordinate pairs from the table of values on the Cartesian plane.
5. Draw the Line: Use a ruler to draw a straight line connecting all the plotted points. Extend the line beyond the plotted points, and add arrows at both ends to indicate it continues indefinitely.
6. Label the Graph: Write the equation of the line on the graph (e.g., `y = 2x - 1`).
Worked Example 3: Plotting a Linear Graph Instruction: Plot the graph of the equation `y = 2x - 1` using the table of values from Worked Example
2. Solution:
1. Table of values: (From Worked Example 2) | x | y | | :---- | :---- | | -2 | -5 | | -1 | -3 | | 0 | -1 | | 1 | 1 | | 2 | 3 |
2. Draw Axes and Choose Scale: Draw x and y axes. Since x ranges from -2 to 2 and y from -5 to 3, a scale of 1 cm to 1 unit on both axes would be appropriate for standard graph paper.
3. Plot Points: Plot (-2, -5), (-1, -3), (0, -1), (1, 1), (2, 3).
4. Draw the Line: Connect all the plotted points with a straight line using a ruler. Extend the line slightly beyond the outermost points and add arrows.
5. Label: Write "y = 2x - 1" near the line. (Teacher should demonstrate this meticulously on the board or with an overhead projector/visual aid.) F. Interpreting the Plotted Graph Once a graph is plotted, information can be extracted from it.
Reading values: Given an x-value, find the corresponding y-value on the line (and vice versa).
Understanding trends: Observe if the line is going up (increasing relationship), down (decreasing relationship), or horizontal (constant relationship).
Finding intercepts: The point where the line crosses the x-axis (y=0) is the x-intercept. The point where it crosses the y-axis (x=0) is the y-intercept.
Worked Example 4: Interpreting a Graph* Instruction:* Using the graph of `y = 2x - 1` from Worked Example 3, find: a) The value of 0 | (0, 0) | | 1 | 300 1 | 300 | (1, 300) | | 2 | 300 2 | 600 | (2, 600) | | 3 | 300 3 | 900 | (3, 900) | | 4 | 300 4 | 1200 | (4, 1200) | | 5 | 300 5 | 1500 | (5, 1500) | c)
Plot the graph: Draw axes. Label x-axis "Number of Yam Tubers (x)" and y-axis "Total Cost (N) (y)".
Choose a suitable scale: For x-axis, 1 cm to 1 unit is good. For y-axis, since values go up to 1500, 1 cm to N300 or N150 would be appropriate (e.g., 2 cm to N300, or 1 cm to N150 if paper allows).
Plot the points: (0,0), (1,300), (2,600), (3,900), (4,1200), (5,1500). Draw a straight line through the points, starting from the origin. The line should be extended slightly. Label `y = 300x`. d) Cost of 3.5 yam tubers: Locate 3.5 on the x-axis. Move vertically to the line, then horizontally to the y-axis. Read the value on the y-axis. It should be N
1
0
5
0. Calculation check: y = 300 3.5 = 1050. e)
Yam tubers for N1200: Locate N1200 on the y-axis. Move horizontally to the line, then vertically to the x-axis. Read the value on the x-axis. It should be
4. Calculation check: 1200 = 300x => x = 4. ---
Market Price Trends in Nigeria: Application: Students can collect data on the weekly prices of staple foods (e.g., garri, rice, tomatoes) in their local market. They can then plot graphs of price (y-axis) against week number (x-axis) to observe trends like inflation, seasonal variations, or stable prices.
Integration: This connects to economics, consumer education, and data literacy. It helps students understand how graphs visually represent economic shifts and impacts their families' budgets. For instance, a steep upward slope in the price of tomatoes around planting season indicates scarcity and higher cost, which is a common experience in Nigeria. Distance-Time Graphs for Travel (e.g., Keke NAPEP/Okada): Application: Consider a Keke NAPEP (tricycle) that charges a base fare plus a certain amount per kilometer. Students can graph the total fare (y-axis) against the distance travelled (x-axis). This can be used to plan journeys or compare costs of different transport options.
Integration: This links to physics (motion), financial literacy, and practical decision-making. Students can analyze the 'steepness' of the graph to understand which Keke charges more per kilometer, helping them make informed choices about local transportation. They can also use it to estimate the cost of a trip to a nearby school or market. Population Growth in Local Government Areas (LGAs): Application: Obtain simplified population data for their LGA or state over a few decades. Students can plot the population (y-axis) against the year (x-axis). While actual population growth is often not perfectly linear, a linear approximation can be used to illustrate trends and make simple predictions for future population (e.g., if we assume a constant growth rate, what would the population be in 5 years?).
Integration: This connects to social studies, demography, and statistical analysis. It helps students visualize demographic changes and understand the implications for planning public services like schools, hospitals, and housing in their communities. ---