Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent linear relationships using tables, flow diagrams, and equations.
• Draw graphs from tables of values and equations.
• Interpret information from graphs like intercepts and steepness.
• Identify direct proportion relationships.
• Solve simple real-world problems using graphs and equations.

Lesson summary

Functions, graphs, and relationships are fundamental to understanding how quantities change and interact. This week, we will focus on representing relationships between two variables graphically and algebraically. Understanding these concepts is crucial, not just for further studies in mathematics, but also for interpreting data and making informed decisions in everyday life. For example, understanding the relationship between cell phone data usage and cost, or the trend of rainfall patterns in your region, all rely on the principles we will cover this week.

Lesson notes

2.1 Representing Relationships: A relationship describes how two or more things are connected. In mathematics, we often deal with relationships between two variables. A variable is a symbol (usually a letter, like x or y) that represents a quantity that can change. We call these variables the input (or independent variable) and the output (or dependent variable). The output depends on the input. 2.1.1 Tables: A table of values shows pairs of input and output values that satisfy a particular relationship.

Example: A taxi charges a fixed call-out fee of R10 plus R8 per kilometer travelled. Let x be the number of kilometers travelled, and y be the total cost. We can create a table to represent this relationship: | Kilometers (x) | Total Cost (y) | |-------------------|-------------------| | 0 | 10 | | 1 | 18 | | 2 | 26 | | 3 | 34 | | 4 | 42 | 2.1.2 Flow Diagrams: A flow diagram visually shows the steps involved in converting an input value to an output value.

Example (Using the taxi example above): Input (x: Kilometers) → Multiply by 8 → Add 10 → Output (y: Total Cost) 2.1.3 Algebraic Equations: An algebraic equation is a mathematical statement that expresses the relationship between variables using symbols and operations. In this context, we're focusing on linear equations.

Example (Taxi example): The equation representing the taxi fare is: y = 8x + 10 2.2 Graphing Relationships: A graph is a visual representation of the relationship between two variables. We plot the input values on the horizontal axis (x-axis) and the output values on the vertical axis (y-axis). Each pair of values from the table becomes a coordinate point (x, y) on the graph.

Example (Taxi example): Using the table above, we would plot the points (0, 10), (1, 18), (2, 26), (3, 34), and (4, 42). Connecting these points gives us a straight line. The line represents all possible combinations of kilometers travelled and the total cost.

Important: Always label the axes of your graph and include a title. For example, "Taxi Fare vs. Kilometers Travelled." 2.3 Interpreting Graphs: Intercepts: The y-intercept is the point where the graph crosses the y-axis (where x = 0). In the taxi example, the y-intercept is 10, representing the call-out fee. The x-intercept is the point where the graph crosses the x-axis (where y = 0). This represents the number of kilometers travelled when the fare is zero (which isn't practical in this example, but important conceptually).

Slope (Qualitative): The slope describes the steepness of the line. A steeper line indicates a larger change in y for a given change in x. In our taxi example, the slope reflects the cost per kilometer; a steeper line would mean a higher cost per kilometer. We won't be calculating the slope numerically this week, but understanding the qualitative meaning is essential. 2.4 Direct Proportion: Direct proportion is a special type of linear relationship where y is directly proportional to x. This means that as x increases, y increases at a constant rate, and the graph passes through the origin (0, 0). The equation for direct proportion is y = kx, where k is a constant.

Example: The cost of electricity is directly proportional to the number of units consumed. If 1 unit costs R2, then 2 units cost R4, 3 units cost R6, and so on. The equation is y = 2x, where x is the number of units and y is the total cost. The graph of this relationship would be a straight line passing through the origin. Guided Practice (With Solutions)

Question 1: A street vendor sells vetkoek for R5 each. Create a table, flow diagram, and equation to represent the relationship between the number of vetkoek sold (x) and the total revenue (y). Then, draw a graph of this relationship for x values from 0 to

5. Solution: Table: | Vetkoek (x) | Revenue (y) | |---------------|---------------| | 0 | 0 | | 1 | 5 | | 2 | 10 | | 3 | 15 | | 4 | 20 | | 5 | 25 | Flow Diagram: Input (x: Vetkoek) → Multiply by 5 → Output (y: Revenue)

Equation: y = 5x Graph: (You would draw a graph with x on the horizontal axis and y on the vertical axis. Plot the points from the table and connect them with a straight line. The line should pass through the origin (0,0)).

Commentary: This is a direct proportion relationship because the line passes through the origin, and the revenue increases proportionally with the number of vetkoek sold.

Question 2: A domestic worker earns a fixed salary of R2000 per month plus R50 for each additional hour of overtime worked. Let x be the number of overtime hours worked and y be the total monthly earnings. Write an equation for this relationship, then create a table for x = 0, 5, 10, and

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5. Solution: Equation: y = 50x + 2000 Table: | Overtime Hours (x) | Total Earnings (y) | |----------------------|----------------------| | 0 | 2000 | | 5 | 2250 | | 10 | 2500 | | 15 | 2750 |

Commentary: The R2000 represents the y-intercept.