Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Identify different types of patterns in real-world contexts.
• Represent relationships using tables, graphs, and equations.
• Analyze and interpret data to make predictions.
• Solve problems involving direct and indirect proportion.
• Use patterns to make informed financial decisions.

Lesson summary

This week, we delve into the fascinating world of patterns, relationships, and representations and how they apply directly to our lives as South Africans. Understanding these concepts allows us to make informed decisions about finances, understand trends in our communities, and even predict future outcomes based on current data. From analysing cellular data costs to interpreting crime statistics, this week’s content will equip you with crucial skills for navigating the world around you. These skills build upon the foundation of pattern recognition and data analysis from earlier grades, expanding into more complex scenarios and representations.

Lesson notes

2.1 Types of Patterns: Linear Patterns: These show a constant rate of change. This means the difference between consecutive terms is always the same.

Example: 2, 4, 6, 8... (Adding 2 each time). In a graph, this appears as a straight line. The equation takes the form y = mx + c, where 'm' is the constant rate of change (slope) and 'c' is the starting value (y-intercept).

Quadratic Patterns: The second difference between consecutive terms is constant.

Example: 1, 4, 9, 16... (The differences are 3, 5, 7... and the differences of those are all 2). These patterns form parabolas when graphed. The general equation is y = ax² + bx + c.

Exponential Patterns: In these patterns, each term is multiplied by a constant factor to get the next term.

Example: 2, 4, 8, 16... (Multiplying by 2 each time). Exponential patterns grow very quickly and represent things like population growth (under ideal conditions) or compound interest. The equation takes the form y = a b^x, where 'a' is the initial value and 'b' is the growth/decay factor. 2.2 Representing Relationships: Tables: Useful for organizing data and showing the relationship between two or more variables. They provide a clear and structured way to present information.

Graphs: Visual representations of data.

Scatter Plots: Used to show the relationship between two variables. Each point represents a pair of data values. They are useful for identifying trends (positive, negative, or no correlation).

Line Graphs: Used to show changes in a variable over time. Data points are connected by lines, making it easy to see trends and fluctuations.

Equations: Mathematical statements that describe the relationship between two or more variables. They allow us to make predictions and solve problems. 2.3 Proportional Relationships: Direct Proportion: Two quantities are directly proportional if one increases, the other increases proportionally. The relationship can be written as y = kx, where 'k' is the constant of proportionality. For example, the distance you travel is directly proportional to the time you travel at a constant speed.

Indirect Proportion (Inverse Proportion): Two quantities are indirectly proportional if one increases, the other decreases proportionally. The relationship can be written as y = k/x, where 'k' is the constant of proportionality. For example, the time it takes to complete a job is inversely proportional to the number of workers. 2.4 Worked

Examples: Example 1: Cellular Data Costs A South African cell phone company offers the following data packages: 500MB for R50 1GB for R90 2GB for R160 5GB for R350 a) Represent this data in a table and a scatter plot. b) Is there a linear relationship between data and cost? Explain. c) Estimate the cost of 3GB of data using the given information.

Solution: a)

Table: | Data (MB) | Cost (R) | | --------- | -------- | | 500 | 50 | | 1000 | 90 | | 2000 | 160 | | 5000 | 350 | Scatter plot: (A scatter plot would have data (MB) on the x-axis and Cost (R) on the y-axis. Plot the points (500, 50), (1000, 90), (2000, 160), and (5000, 350)). b) To determine if the relationship is linear, check if the rate of change is constant.

From 500MB to 1GB: Increase of 500MB, Cost increase of R

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0. From 1GB to 2GB: Increase of 1000MB, Cost increase of R

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0. From 2GB to 5GB: Increase of 3000MB, Cost increase of R

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0. Since the cost per MB is not constant, the relationship is not linear. c) To estimate the cost of 3GB (3000MB), we can observe the pattern. The increase from 2GB to 5GB is R190 for 3000MB. A linear interpolation is not ideal since the whole set is not linear. A reasonable estimate would consider that the price increase from 1 GB to 2 GB is R

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0. If we assume that a further 1 GB would cost roughly the same again, that is about R70, then the cost of 3GB would be the cost of 2 GB plus R

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0. R160 + R70 = R

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0. A more accurate approach would involve regression to find the equation of the curve of best fit, but this is beyond the scope of the curriculum for this week.

Example 2: Population Growth The population of a small town in the Eastern Cape is currently

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0. It is growing at a rate of 3% per year. a) Write an equation to represent the population growth over time. b) What will the population be in 5 years? c) How long will it take for the population to double?

Solution: a) This is an exponential growth scenario.

The equation is: Population = Initial Population * (1 + Growth Rate)^Time P = 5000 * (1 + 0.03)^t P = 5000 * (1.03)^t b) To find the population in 5 years, substitute t = 5: P = 5000 * (1.03)^5 P ≈ 5796.37 Therefore, the population will be approximately 5796 in 5 years. c) To find when the population doubles, we want P = 10000: 10000 = 5000 * (1.03)^t 2 = (1.03)^t To solve for t, we can use logarithms (although this is not always explicitly taught in Grade 11 Mathematical Literacy, it provides the most accurate result).