Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent relationships using tables, ordered pairs, and equations.
• Identify independent and dependent variables.
• Draw and interpret simple linear graphs.
• Find output values from input values (and vice versa).
• Describe the relationship shown in a graph.

Lesson summary

This week, we delve deeper into the fascinating world of functions, graphs, and relationships. Understanding how things relate to each other mathematically is crucial, not just in the classroom, but in everyday life. From predicting cellphone data usage to budgeting your pocket money, the principles we learn this week are applicable everywhere. In South Africa, understanding data and relationships is especially important, as we navigate issues like resource allocation, economic trends, and social statistics. Imagine being able to predict the demand for electricity in your area based on temperature or understanding how the price of bread is related to inflation!

Lesson notes

2.1 Relationships Between Variables: A relationship exists when two or more quantities are linked. In mathematics, we often focus on relationships between two variables.

Variable: A variable is a symbol (usually a letter like x or y) that represents a quantity that can change or vary.

Independent Variable: The independent variable is the variable that you control or choose. Its value determines the value of the other variable. We often represent it on the x-axis. Think of it as the "cause." Dependent Variable: The dependent variable is the variable whose value depends on the value of the independent variable. We often represent it on the y-axis. Think of it as the "effect."

Example: Consider the relationship between the number of hours you work (x) and the amount of money you earn (y). The number of hours you work is the independent variable (you decide how many hours to work), and the amount of money you earn is the dependent variable (it depends on how many hours you worked). 2.2 Representing Relationships: Relationships can be represented in several ways: Tables: A table shows pairs of values for the independent and dependent variables. | Hours Worked (x) | Money Earned (y) (Rands) | | :----------------: | :-----------------------: | | 1 | 50 | | 2 | 100 | | 3 | 150 | | 4 | 200 | Ordered Pairs: Ordered pairs are written in the form (x, y), where x is the independent variable and y is the dependent variable. From the table above, we can create ordered pairs: (1, 50), (2, 100), (3, 150), (4, 200).

Equations: An equation expresses the relationship between the variables mathematically. In this example, the equation could be y = 50x (where y is the money earned and x is the hours worked). 2.3 Functions: A function is a special type of relationship where each input (x-value) has exactly one output (y-value). In simpler terms, for every 'x' you put in, you get only one 'y' out. 2.4 Linear Functions and Graphs: A linear function is a function that can be represented by a straight line on a graph. Its equation typically takes the form y = mx + c, where: 'm' is the gradient (slope) of the line (how steep the line is). 'c' is the y-intercept (the point where the line crosses the y-axis).

Steps to draw a linear graph: Create a Table of Values: Choose a few x-values (at least three is recommended). Substitute these x-values into the equation to find the corresponding y-values.

Plot the Ordered Pairs: Plot the ordered pairs (x, y) on the Cartesian plane. Remember, the x-axis is horizontal, and the y-axis is vertical.

Draw a Straight Line: Use a ruler to draw a straight line that passes through all the plotted points.

Example 1: Earning Money Let's say Zola earns R20 for every hour he helps his grandfather in the garden.

Equation: y = 20x (where y is the amount earned and x is the number of hours worked).

Table of Values: | Hours Worked (x) | Money Earned (y) | | :----------------: | :----------------: | | 0 | 0 | | 1 | 20 | | 2 | 40 | | 3 | 60 | Ordered Pairs: (0, 0), (1, 20), (2, 40), (3, 60)

Graph: Plot these points on a graph with 'Hours Worked' on the x-axis and 'Money Earned' on the y-axis, and draw a straight line through them.

Example 2: Cellphone Data Usage Suppose your cellphone uses 50MB of data per hour of streaming videos.

Equation: y = 50x (where y is the data used in MB and x is the hours of streaming).

Table of Values: | Hours of Streaming (x) | Data Used (y) (MB) | | :----------------------: | :-----------------: | | 0 | 0 | | 1 | 50 | | 2 | 100 | | 3 | 150 | Ordered Pairs: (0, 0), (1, 50), (2, 100), (3, 150)

Graph: Plot these points on a graph with 'Hours of Streaming' on the x-axis and 'Data Used' on the y-axis, and draw a straight line through them.

Example 3: Converting Celsius to Fahrenheit (Relevant as SA weather can be quite varied) The formula to convert Celsius (°C) to Fahrenheit (°F) is F = (9/5)C +

3

2. Equation: F = (9/5)C + 32 Table of Values: | Celsius (°C) (C) | Fahrenheit (°F) (F) | | :---------------: | :-----------------: | | 0 | 32 | | 10 | 50 | | 20 | 68 | | 30 | 86 | Ordered Pairs: (0, 32), (10, 50), (20, 68), (30, 86)

Graph: Plot these points on a graph with 'Celsius' on the x-axis and 'Fahrenheit' on the y-axis, and draw a straight line through them. This graph can be useful for understanding temperature differences! Guided Practice (With Solutions)

Question 1: Complete the table of values for the equation y = 3x + 1. | x | y | |---|---| | 0 | | | 1 | | | 2 | | Solution: When x = 0, y = (3 0) + 1 = 1 When x = 1, y = (3 1) + 1 = 4 When x = 2, y = (3 2) + 1 = 7 Completed Table: | x | y | |---|---| | 0 | 1 | | 1 | 4 | | 2 | 7 |

Commentary: We substitute each x-value into the equation to calculate the corresponding y-value. Remember to follow the order of operations (multiplication before addition).

Question 2: Identify the independent and dependent variables in the following scenario: The number of bags of oranges bought and the total cost.