Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Represent relationships using tables, descriptions, flow diagrams, and equations.
• Identify independent and dependent variables.
• Draw and interpret linear graphs.
• Find input and output values using graphs, tables or equations.
• Recognise and extend patterns.

Lesson summary

This week, we delve into the fascinating world of functions, graphs, and the relationships they represent. Understanding these concepts is not just about scoring well on tests; it's about developing critical thinking skills applicable to various aspects of life, from budgeting your pocket money to understanding population growth in South Africa. Graphs help us visualise data, identify trends, and make informed decisions. In the South African context, this knowledge is crucial for understanding economic indicators, environmental changes, and social trends.

Lesson notes

2.1 What is a Relationship? A relationship describes how two or more things are connected. In mathematics, we often focus on 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.

Example: The relationship between the number of airtime minutes you buy (x) and the cost in Rands (y). 2.2 Independent and Dependent Variables Independent Variable: This is the input variable, the one we control or choose. It's the 'cause'. We typically represent it with x.

Dependent Variable: This is the output variable, the one that depends on the independent variable. It's the 'effect'. We typically represent it with y.

Example: In the airtime example, the number of minutes you buy (x) is the independent variable, and the cost (y) is the dependent variable because the cost depends on how many minutes you purchase. 2.3 Representing Relationships We can represent relationships in several ways: Verbal Description: Using words to describe the relationship.

Example: "The cost of data is R10 per gigabyte." Table of Values: A table showing corresponding values of the independent and dependent variables. | Gigabytes (x) | Cost (y) | |---------------|----------| | 1 | 10 | | 2 | 20 | | 3 | 30 | | 4 | 40 | Flow Diagram: A diagram showing the operations performed on the input to get the output. Input (x) → Multiply by 10 → Output (y)

Algebraic Equation (Formula): A mathematical equation that expresses the relationship.

Example: y = 10x 2.4 Linear Relationships and Graphs A linear relationship is a relationship where the graph is a straight line. The equation for a linear relationship is often in the form y = mx + c, where m is the slope (gradient) and c is the y-intercept (the point where the line crosses the y-axis). In grade 8, we'll focus on relationships where c=0 (y=mx).

How to Draw a Linear Graph: Create a Table of Values: Choose a few values for x (e.g., -2, -1, 0, 1, 2) and calculate the corresponding y values using the equation.

Plot the Points: Plot the (x, y) pairs on a Cartesian plane (a graph with x and y axes).

Draw a Straight Line: Use a ruler to draw a straight line through the points.

Example 1: Let's consider the relationship y = 2x.

Table of Values: | x | y = 2x | |------|--------| | -2 | -4 | | -1 | -2 | | 0 | 0 | | 1 | 2 | | 2 | 4 | Plot the Points: Plot (-2, -4), (-1, -2), (0, 0), (1, 2), and (2, 4) on a graph.

Draw a Straight Line: Draw a straight line through the plotted points. The line will pass through the origin (0,0).

Example 2: A taxi charges R15 per kilometer. Represent this relationship using a table, equation and graph.

Verbal Description: Cost = R15 per kilometer Table of Values: | Kilometers (x) | Cost (y) | |---------------|----------| | 1 | 15 | | 2 | 30 | | 3 | 45 | | 4 | 60 | Algebraic Equation: y = 15x Graph: (Plot the points and draw a straight line) 2.5 Interpreting Graphs Graphs provide a visual representation of the relationship between variables.

We can use graphs to: Find y for a given x: Locate the value of x on the x-axis, move vertically until you reach the line, and then move horizontally to the y-axis to read the corresponding y value.

Find x for a given y: Locate the value of y on the y-axis, move horizontally until you reach the line, and then move vertically to the x-axis to read the corresponding x value.

Understand the Trend: Determine if the relationship is increasing (the line goes upwards) or decreasing (the line goes downwards). In Grade 8, we mostly focus on increasing relationships where m in y=mx is positive. Guided Practice (With Solutions)

Question 1: A shop sells sweets for R2 each.

Represent this relationship with: a) A table of values for 1 to 5 sweets. b) An algebraic equation. c) A graph.

Solution: a)

Table of Values: | Sweets (x) | Cost (y) | |------------|----------| | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 | | 5 | 10 | b)

Algebraic Equation: y = 2x c)

Graph: (Plot the points (1,2), (2,4), (3,6), (4,8), (5,10) and draw a straight line through them). Note the axes should be labelled "Number of sweets" (x-axis) and "Cost (R)" (y-axis).

Question 2: Use the equation y = 3x to find the value of y when x =

4. Solution: Substitute x = 4 into the equation: y = 3 4 Calculate: y = 12 Therefore, when x = 4, y =

1

2. Question 3: If y = 5x, what is the value of x when y = 25?

Solution: Substitute y = 25 into the equation: 25 = 5x Divide both sides by 5: 25 / 5 = x Calculate: x = 5 Therefore, when y = 25, x =

5. Question 4: Thando saves R5 per week. Represent this with an equation. Identify the dependent and independent variables.

Solution: Equation: y = 5x (where x is the number of weeks and y is the total savings)

Independent Variable: x (number of weeks)

Dependent Variable: y (total savings) Independent Practice (Questions Only) Complete the table for the equation y = 4x: | x | y | |------|------| | -1 | | | 0 | | | 1 | | | 2 | | Draw a graph for the equation y = 1.5x.