Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Identify linear and non-linear relationships.
• Translate information between tables, graphs, and equations.
• Use equations to model real-life situations.
• Interpret the meaning of slope and y-intercept.
• Solve problems involving direct and inverse proportion.

Lesson summary

This week, we delve into the heart of Mathematical Literacy: understanding patterns, relationships, and representations in contexts we encounter daily. This is crucial because these skills enable you to critically analyze information, make informed decisions, and solve practical problems in your personal and professional lives. Whether it's budgeting your weekly expenses, interpreting data on crime statistics in your community, or understanding how interest rates affect your future financial stability, this topic provides the tools you need to navigate the world effectively.

Lesson notes

2.1 Linear Relationships: A linear relationship is a relationship between two variables that can be represented by a straight line on a graph. The general equation of a linear relationship is: y = mx + c Where: y is the dependent variable (the value that depends on x) x is the independent variable m is the slope (gradient) of the line, representing the rate of change of y with respect to x. c is the y-intercept (the value of y when x = 0).

Understanding Slope (m): The slope tells us how much y changes for every one unit increase in x.

It is calculated as: m = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)* Understanding Y-intercept (c): The y-intercept is the point where the line crosses the y-axis. It represents the value of y when x is zero. In a real-world context, it's often the starting value or initial condition. 2.2 Non-Linear Relationships: A non-linear relationship is a relationship that cannot be represented by a straight line. These relationships can take many forms, including: Quadratic: y = ax² + bx + c (Forms a parabola)

Exponential: y = a^x (Shows rapid growth or decay)

Inverse Proportion: y = k/x (As one variable increases, the other decreases proportionally) This week we will focus mainly on Inverse Proportion. 2.3 Inverse Proportion: In inverse proportion, as one quantity increases, the other quantity decreases, and vice versa. The product of the two quantities remains constant.

The equation for inverse proportion is: y = k/x Where: y and x are the two variables that are inversely proportional. k is the constant of proportionality.

Example 1: Linear Relationship - Cellphone Data Costs A Vodacom contract charges R50 per month plus R10 per gigabyte (GB) of data used. Write an equation to represent the total monthly cost (C) in terms of the data used (D). C = 10D + 50 Create a table showing the total cost for 0 GB, 1 GB, 2 GB, and 3 GB of data. | Data (GB) | Cost (R) | | --------- | -------- | | 0 | 50 | | 1 | 60 | | 2 | 70 | | 3 | 80 | Draw a graph representing this relationship. (A graph should be drawn with Data (GB) on the x-axis and Cost (R) on the y-axis. The graph should be a straight line passing through (0, 50) and (1, 60).) What is the slope of the line, and what does it represent? The slope is

1

0. It represents the cost per gigabyte of data (R10/GB). What is the y-intercept, and what does it represent? The y-intercept is

5

0. It represents the fixed monthly cost (R50) even if no data is used.

Example 2: Inverse Proportion - Construction Time A construction company estimates that it will take 12 workers 30 days to complete a building project. Assuming all workers work at the same rate, how many days would it take 15 workers to complete the same project? Since the number of workers and the time taken are inversely proportional, we have: Workers \* Time = Constant (k) Initially, 12 * 30 =

3

6

0. So, k =

3

6

0. If there are 15 workers, then 15 * Time = 360 Time = 360 / 15 = 24 days. It would take 15 workers 24 days.

Example 3: Identifying Linear and Non-Linear Relationships Consider the following tables: Table A: | x | y | |---|---| | 1 | 3 | | 2 | 5 | | 3 | 7 | | 4 | 9 | Table B: | x | y | |---|---| | 1 | 2 | | 2 | 4 | | 3 | 8 | | 4 | 16| Table A represents a linear relationship. The difference between consecutive y-values is constant (2). We can express this as y = 2x +

1. Table B represents a non-linear (exponential) relationship. Each y-value is double the previous y-value. We can express this as y = 2^x. Guided Practice (With Solutions)

Question 1: A taxi charges a fixed fare of R15 plus R8 per kilometer traveled. a) Write an equation to represent the total fare (F) in terms of the distance traveled (d). b) What is the fare for a 5km trip? c) If the fare is R71, how far did the taxi travel?

Solution: a) F = 8d + 15 b) F = 8(5) + 15 = 40 + 15 = R55 c) 71 = 8d + 15 56 = 8d d = 7 km Question 2: The time (t) it takes to complete a task is inversely proportional to the number of people (n) working on it. If 4 people can complete the task in 6 hours, how long will it take 3 people to complete the same task?

Solution: Since the relationship is inversely proportional, t = k/n. First, find k: 4 * 6 = 24, so k =

2

4. Now, find t when n = 3: t = 24/3 = 8 hours.

Question 3: Identify which of the following equations represents a linear relationship: a) y = 3x² + 2 b) y = 5x - 1 c) y = 2/x d) y = √x Solution: The correct answer is (b) y = 5x -

1. This equation fits the form y = mx + c, which is the general form of a linear equation. The other equations represent quadratic, inverse proportion, and square root functions, respectively. Independent Practice (Questions Only) A shop sells airtime. They charge R12 for every R10 worth of airtime. Represent this relationship with an equation. Create a table showing the amount charged for R10, R20, R30, and R40 worth of airtime. The number of loaves of bread a bakery can produce is directly proportional to the number of bakers working.