Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent relationships using tables.
• Describe relationships with a verbal rule.
• Draw graphs from tables or rules.
• Identify independent and dependent variables.
• Find input and output values from tables, rules, or graphs.

Lesson summary

Functions, graphs, and relationships are fundamental building blocks in mathematics. They allow us to model real-world situations, predict outcomes, and understand how different quantities are related. For example, we can use graphs to represent the relationship between the amount of rainfall and crop yield in a farm or to model the cost of airtime as a function of minutes purchased. Understanding these concepts empowers you to analyse trends, solve problems, and make informed decisions in various aspects of life, from managing your finances to understanding scientific phenomena. This week, we will focus on representing relationships between variables in tables, rules and graphs.

Lesson notes

2.1 Relationships between Variables: In mathematics, a variable is a symbol (usually a letter like x or y) that represents a quantity that can change or vary. A relationship between two variables describes how changes in one variable affect the other. 2.2 Independent and Dependent Variables: The independent variable is the variable that we change or control. It's often called the input. The dependent variable is the variable that changes in response to the independent variable. It's often called the output. Its value depends on the value of the independent variable. For example, if we're looking at the relationship between the number of hours worked and the amount earned, the number of hours worked is the independent variable (we choose how many hours to work), and the amount earned is the dependent variable (it depends on how many hours we worked). 2.3 Representing Relationships: We can represent relationships between variables in several ways: Tables of Values: A table shows pairs of corresponding values for the independent and dependent variables. The independent variable is usually listed on the left (or top), and the dependent variable on the right (or bottom).

Verbal Rule: A verbal rule describes the relationship in words.

For instance: "The cost is equal to R5 times the number of items." Graphs: A graph is a visual representation of the relationship on a coordinate plane. The independent variable is usually plotted on the horizontal axis (x-axis), and the dependent variable on the vertical axis (y-axis). 2.4 Understanding Functions A function is a special type of relationship where each input value (independent variable) has exactly one output value (dependent variable). In simpler terms, you put something in, and you get only one specific thing out.

Example 1: Table of Values Consider a tuck shop selling sweets for R2 each. | Number of Sweets (x) | Cost (y) | |---|---| | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 | Independent variable (x): Number of sweets Dependent variable (y): Cost Verbal Rule: The cost is equal to R2 multiplied by the number of sweets.

Example 2: Verbal Rule Let's say a taxi charges a flat rate of R15 plus R8 per kilometre travelled.

Independent variable (x): Number of kilometres travelled Dependent variable (y): Total cost Table of Values: | Kilometres (x) | Cost (y) = 15 + 8x | |---|---| | 0 | 15 | | 1 | 23 | | 2 | 31 | | 3 | 39 | | 4 | 47 | Example 3: Graphing a Relationship Let's graph the relationship from Example 1 (sweets costing R2 each). We can plot the points from the table (1, 2), (2, 4), (3, 6), and (4, 8) on a coordinate plane. Join the points with a straight line since you can buy fractions of sweets, in some instances, as part of bigger packages. The x-axis is "Number of Sweets" and the y-axis is "Cost (R)".

Understanding the Graph: The graph shows that as the number of sweets increases, the cost also increases at a constant rate.

Example 4: Determining Input and Output from Graphs Imagine the following context: Thando is walking to school. The graph below represents her distance from home (in metres) over time (in minutes). X axis = Time (minutes) Y axis = Distance (meters)

The graph shows the following points: (0,0), (5,250), (10,250), (15,500), (20,750) How far has Thando walked after 10 minutes?

Answer: 250 meters After 5 minutes, Thando is walking at constant pace, what happens from 5 - 10 minutes?

Answer: She maintains the same distance (250m) indicating that she has stopped walking. If the school is 750 meters away, how long does Thando take to reach school?

Answer: 20 minutes Guided Practice (With Solutions)

Question 1: Complete the following table for the relationship: y = 3x + 2 | x | y | |---|---| | 0 | | | 1 | | | 2 | | | 3 | | Solution: When x = 0, y = 3(0) + 2 = 2 When x = 1, y = 3(1) + 2 = 5 When x = 2, y = 3(2) + 2 = 8 When x = 3, y = 3(3) + 2 = 11 Completed Table: | x | y | |---|---| | 0 | 2 | | 1 | 5 | | 2 | 8 | | 3 | 11 | Question 2: A cellphone company charges R1.50 per minute for calls. Write a verbal rule, create a table of values for the first 5 minutes, and identify the independent and dependent variables.

Solution: Verbal Rule: The total cost of the call is R1.50 multiplied by the number of minutes.

Table of Values: | Minutes (x) | Cost (y) = 1.50x | |---|---| | 0 | 0 | | 1 | 1.50 | | 2 | 3.00 | | 3 | 4.50 | | 4 | 6.00 | | 5 | 7.50 | Independent variable: Number of minutes (x)

Dependent variable: Cost (y)

Question 3: The graph below represents the number of litres of water in a tank decreasing each hour due to a small leak. (Assume the graph is a straight line connecting the points (0, 20) and (4, 0)) What was the initial amount of water in the tank? After how many hours will the tank be empty? How many litres of water will be left after 2 hours?

Solution: Initial amount of water: From the graph, when time (x) = 0, the amount of water (y) = 20 litres.

Tank empty: From the graph, when the amount of water (y) = 0, time (x) = 4 hours.