Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent relationships using tables, flow diagrams, and equations.
• Find output values from input values, and vice versa.
• Draw simple graphs from tables or equations.
• Interpret relationships shown in graphs.
• Identify independent and dependent variables.

Lesson summary

This week, we delve deeper into functions, graphs, and relationships. Understanding these concepts is crucial, not just for succeeding in mathematics, but also for interpreting data and making informed decisions in everyday life. Imagine tracking the growth of a small business in your community, analyzing rainfall patterns affecting local agriculture, or even understanding cell phone data usage trends and costs. All of these rely on understanding relationships between variables and how they can be represented visually through graphs. This week builds upon the introduction from Week 1, focusing on identifying relationships, representing them in different ways, and interpreting basic graphs.

Lesson notes

2.1 What is a Relationship? A relationship exists when two or more things are connected or related to each other. 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 or vary. 2.2 Independent and Dependent Variables Independent Variable: This is the variable that we choose or that changes on its own. It's the input. We often represent it with the letter x.

Dependent Variable: This is the variable that depends on the independent variable. Its value is determined by the value of the independent variable. It's the output. We often represent it with the letter y.

Example: The amount of money you earn (dependent variable) depends on the number of hours you work (independent variable). 2.3 Representing Relationships We can represent relationships in several ways: Flow Diagrams: A flow diagram shows the process of transforming an input (independent variable) into an output (dependent variable) using a rule.

Tables: A table shows pairs of input and output values in an organized way.

Equations: An equation is a mathematical statement that shows the relationship between the input and output variables.

Graphs: A graph is a visual representation of the relationship between the input and output variables on a coordinate plane. 2.4 Flow Diagrams A flow diagram has an input, a rule (often described mathematically), and an output.

Example: Input --> (Multiply by 2, then add 1) --> Output If the input is 3: 3 --> (3 x 2 + 1) -->

7. The output is 7. 2.5 Tables A table organizes input and output values.

Example: | Input (x) | Output (y) | |---|---| | 1 | 3 | | 2 | 5 | | 3 | 7 | | 4 | 9 | From this table, we can see that when x = 1, y = 3; when x = 2, y = 5; and so on. 2.6 Equations An equation expresses the relationship between x and y.

Example: y = 2x + 1 This equation says that the output (y) is equal to twice the input (x) plus one. This equation describes the relationships shown in the flow diagram and table above. 2.7 Graphs A graph visually represents the relationship between x and y. Each pair of (x, y) values from the table becomes a point on the graph. Example (Using the equation y = 2x + 1): To draw the graph, we first create a table of values: | x | y = 2x + 1 | |---|---| | 0 | 2(0) + 1 = 1 | | 1 | 2(1) + 1 = 3 | | 2 | 2(2) + 1 = 5 | | 3 | 2(3) + 1 = 7 | Then we plot the points (0, 1), (1, 3), (2, 5), and (3, 7) on a coordinate plane and connect them with a straight line. This is a linear graph. Important

Note: Not all graphs are straight lines. 2.8 Finding the Output (y) Given the Input (x) If you know the equation and the input (x), you can find the output (y) by substituting the value of x into the equation.

Example: Equation: y = 5x - 2 Input: x = 4 Substitute x = 4 into the equation: y = 5(4) - 2 y = 20 - 2 y = 18 Therefore, the output (y) is 18 when the input (x) is 4. 2.9 Finding the Input (x) Given the Output (y) If you know the equation and the output (y), you can find the input (x) by solving the equation for x.

Example: Equation: y = 3x + 1 Output: y = 10 Substitute y = 10 into the equation: 10 = 3x + 1 Subtract 1 from both sides: 9 = 3x Divide both sides by 3: 3 = x Therefore, the input (x) is 3 when the output (y) is

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0. Guided Practice (With Solutions)

Question 1: Represent the relationship "The number of sweets, y, is twice the number of friends, x" using a flow diagram, a table (for x = 1, 2, 3, 4), and an equation.

Solution: Flow Diagram: Friends (x) --> (Multiply by 2) --> Sweets (y)

Table: | Friends (x) | Sweets (y) | |---|---| | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 | Equation: y = 2x

Commentary: This question introduces the basic representations of a relationship. Notice how the equation concisely captures the relationship expressed in words and visually in the flow diagram. The table provides specific examples.

Question 2: Given the equation y = -x + 5, find the output (y) when the input (x) is

2. Also, find the input (x) when the output (y) is

1. Solution: Finding y when x = 2: y = -x + 5 y = -(2) + 5 y = -2 + 5 y = 3 Finding x when y = 1: y = -x + 5 1 = -x + 5 Subtract 5 from both sides: -4 = -x Multiply both sides by -1: 4 = x

Commentary: This question practices the substitution of values into an equation to find the corresponding input or output. Pay attention to the negative sign in the equation.

Question 3: The table below shows the relationship between the number of hours a security guard works (x) and the amount he earns (y). | Hours (x) | Earnings (y) in Rands | |---|---| | 1 | 50 | | 2 | 100 | | 3 | 150 | a) What is the equation that represents this relationship? b) Draw a graph representing this relationship.

Solution: a) Observe that for every hour worked, the security guard earns R

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0. Therefore, the equation is: y = 50x b) To draw the graph, plot the points (1, 50), (2, 100), and (3, 150) on a coordinate plane.