Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Identify different types of patterns (linear, quadratic, exponential).
• Represent relationships using tables, graphs, and equations.
• Interpret graphs and tables to draw conclusions.
• Determine the equation of a linear relationship.
• Use patterns to solve practical problems.

Lesson summary

This week, we delve into the crucial topic of patterns, relationships, and representations in real-life contexts. Mathematical Literacy isn't just about numbers; it's about understanding the world around us and making informed decisions. In South Africa, being able to identify patterns in data, understand relationships between different variables, and represent information effectively is essential for everything from managing personal finances and understanding news reports to participating in community development and interpreting statistics related to social issues. Understanding these concepts empowers you to critically analyze information and solve problems you encounter every day.

Lesson notes

This section provides a detailed exploration of patterns, relationships, and representations. 2.1 Patterns: A pattern is a predictable regularity or arrangement. In mathematics, patterns often involve sequences of numbers, shapes, or events.

Linear Patterns: A linear pattern has a constant difference between consecutive terms. The general form is y = mx + c, where m is the constant difference (slope) and c is the starting value (y-intercept).

Quadratic Patterns: A quadratic pattern has a constant second difference between consecutive terms. The general form is y = ax² + bx + c.

Exponential Patterns: An exponential pattern involves a constant ratio between consecutive terms. The general form is y = abˣ, where a is the initial value and b is the common ratio.

Example 1: Linear Pattern (Cell Phone Data Costs) A cell phone company charges R10 for the first 100MB of data and then R0.50 for each additional M

B. Pattern: The cost increases linearly with each additional MB beyond the initial 100M

B. Representation: Table: | Data (MB) | Cost (R) | | --------- | -------- | | 100 | 10 | | 101 | 10.50 | | 102 | 11.00 | | 103 | 11.50 | Equation: Cost = 10 + 0.50(Data - 100)* (for Data > 100)

Example 2: Quadratic Pattern (Tile Arrangement) Imagine you are arranging tiles in a square pattern. A 1x1 square requires 1 tile. A 2x2 square requires 4 tiles. A 3x3 square requires 9 tiles.

Pattern: The number of tiles is the square of the side length of the square.

Representation: Table: | Side Length | Number of Tiles | | ----------- | --------------- | | 1 | 1 | | 2 | 4 | | 3 | 9 | | 4 | 16 | Equation: Number of Tiles = (Side Length)² or y = x² Example 3: Exponential Pattern (Compound Interest) You invest R1000 in a savings account that pays 5% compound interest per year.

Pattern: The amount increases exponentially each year.

Representation: Table: | Year | Amount (R) | | ---- | ---------- | | 0 | 1000 | | 1 | 1050 | | 2 | 1102.50 | | 3 | 1157.63 | Equation: Amount = 1000 (1.05)ᵞᵉᵃʳ 2.2 Relationships: A relationship describes how two or more variables are connected. We often represent relationships using tables, graphs, and equations.

Independent Variable: The variable that is manipulated or changed (e.g., the amount of data used).

Dependent Variable: The variable that is affected by the independent variable (e.g., the cost of data). 2.3 Representations: Tables: Organize data in rows and columns, showing the relationship between variables.

Graphs: Visual representations of data.

Common types include: Scatter Plots: Show the relationship between two variables as points.

Line Graphs: Connect data points with lines, showing trends over time.

Bar Graphs: Compare quantities across different categories.

Pie Charts: Show proportions of a whole.

Equations: Mathematical formulas that describe the relationship between variables. 2.4 Analyzing Graphs: Trend: Is the graph increasing, decreasing, or staying constant?

Slope: The steepness of a line, indicating the rate of change. (Rise / Run)

Intercepts: Points where the graph crosses the x-axis (x-intercept) or the y-axis (y-intercept).

Correlation: The strength and direction of a linear relationship (positive, negative, or no correlation). Guided Practice (With Solutions)

Question 1: A taxi charges a fixed call-out fee of R25 and then R8 per kilometer. a) Create a table showing the cost for distances of 0 km, 1 km, 2 km, 3 km, and 4 km. b) Write an equation to represent the relationship between distance and cost. c) Draw a graph of the relationship.

Solution: a)

Table: | Distance (km) | Cost (R) | | ------------- | -------- | | 0 | 25 | | 1 | 33 | | 2 | 41 | | 3 | 49 | | 4 | 57 | b)

Equation: Cost = 25 + 8 Distance c)

Graph: (Imagine a line graph with Distance on the x-axis and Cost on the y-axis. The line starts at (0, 25) and has a slope of

8. Learners would need to draw this on graph paper or use graphing software.)

Commentary: This question reinforces linear relationships and their representation in tables, equations, and graphs. We first created the table by applying the rule (R25 + R8 per km). Then, we translated this rule into an equation. Finally, the graph visually represents this linear relationship.

Question 2: The number of bacteria in a culture doubles every hour. Initially, there are 50 bacteria. a) Create a table showing the number of bacteria after 0, 1, 2, and 3 hours. b) Write an equation to represent the number of bacteria as a function of time.

Solution: a)

Table: | Time (hours) | Number of Bacteria | | ------------ | ------------------ | | 0 | 50 | | 1 | 100 | | 2 | 200 | | 3 | 400 | b)

Equation: Number of Bacteria = 50 2ᵀⁱᵐᵉ

Commentary: This question illustrates an exponential relationship. The doubling every hour is the key indicator. The initial amount (50) is multiplied by 2 raised to the power of the number of hours.

Question 3: The table below shows the relationship between the number of hours worked and the amount earned.