Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Identify different types of patterns in real life.
• Represent relationships using tables, graphs, and equations.
• Interpret graphs to make predictions.
• Solve practical problems, especially financial ones.
• Critically evaluate data in different formats.

Lesson summary

This week, we delve deeper into the fascinating world of patterns, relationships, and representations. Mathematical Literacy isn't just about numbers; it's about understanding how the world works through a mathematical lens. This week's focus is on applying our knowledge of patterns and relationships to solve real-world problems. We'll be looking at how things change together, how to predict future outcomes based on observed trends, and how to use various representations (graphs, tables, equations) to make informed decisions. Why does this matter?

Lesson notes

2.1 Types of Patterns Linear Patterns: A linear pattern shows a constant difference between consecutive terms. This means the pattern increases or decreases by the same amount each time. This constant difference is known as the common difference. The general form is often represented as y = mx + c, where m is the constant difference (slope) and c is the initial value (y-intercept).

Example: The number of loaves of bread baked each day increases by

5

0. If on Monday 200 loaves were baked, on Tuesday 250 were baked, on Wednesday 300 were baked, and so on. This is a linear pattern with a common difference of

5

0. Quadratic Patterns: A quadratic pattern has a constant second difference. This means that the differences between consecutive terms are not constant, but the differences between those differences are constant. Quadratic patterns are represented by equations of the form y = ax² + bx + c.

Example: Consider the sequence 1, 4, 9, 16,

2

5. The first differences are 3, 5, 7,

9. The second differences are 2, 2,

2. Since the second difference is constant, this is a quadratic pattern.

Exponential Patterns: An exponential pattern shows a constant ratio between consecutive terms. This means the pattern increases or decreases by a constant factor (percentage) each time. The general form is often represented as y = abˣ, where a is the initial value and b is the common ratio.

Example: The number of bacteria in a culture doubles every hour. If there were initially 10 bacteria, after one hour there would be 20, after two hours 40, after three hours 80, and so on. This is an exponential pattern with a common ratio of 2. 2.2 Representations of Relationships Tables: Tables are a simple way to organise data and show the relationship between two or more variables.

Example: | Number of Hours Worked | Pay (R) | |------------------------|---------| | 1 | 50 | | 2 | 100 | | 3 | 150 | | 4 | 200 | Graphs: Graphs provide a visual representation of the relationship between variables. Different types of graphs are suitable for different types of data (e.g., line graphs for trends over time, bar graphs for comparing categories, scatter plots for showing correlations).

Example: A line graph showing the increase in electricity consumption during peak hours. The X-axis would represent the time of day, and the Y-axis would represent electricity consumption in kilowatt-hours (kWh).

Algebraic Equations: Algebraic equations provide a mathematical way to describe the relationship between variables. They allow us to make predictions and solve problems.

Example: If a taxi charges a flat rate of R20 plus R10 per kilometre, the total cost (C) can be represented by the equation C = 10k + 20, where k is the number of kilometres travelled. 2.3 Interpreting Graphs Understanding graphs is crucial for interpreting data.

Pay attention to the following: Axes: What do the X and Y axes represent? What are the units of measurement?

Scale: What is the scale of each axis? Is the scale consistent?

Trends: Is the graph increasing, decreasing, or staying constant? Are there any peaks or troughs?

Correlation: Is there a relationship between the variables? Is it a positive correlation (as one variable increases, the other increases), a negative correlation (as one variable increases, the other decreases), or no correlation?

Outliers: Are there any data points that are significantly different from the rest of the data? 2.4 Financial Applications: Simple and Compound Interest Simple Interest: Simple interest is calculated only on the principal amount.

The formula is: A = P(1 + rt), where A is the final amount, P is the principal amount, r is the interest rate (as a decimal), and t is the time in years.

Example: You invest R1000 at a simple interest rate of 8% per year for 5 years. A = 1000(1 + 0.08 5) = 1000(1 + 0.4) = 1000 1.4 = R

1

4

0

0. Compound Interest: Compound interest is calculated on the principal amount and the accumulated interest.

The formula is: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate (as a decimal), n is the number of times that interest is compounded per year, and t is the time in years.

Example: You invest R1000 at a compound interest rate of 8% per year, compounded annually for 5 years. A = 1000(1 + 0.08/1)^(15) = 1000(1.08)^5 = 1000 1.4693 = R1469.33 (approximately). Guided Practice (With Solutions)

Question 1: A cellphone company charges R1.50 per minute for calls. a) Create a table showing the cost for calls of 1, 2, 3, 4, and 5 minutes. b) Write an equation to represent the cost (C) of a call in terms of the number of minutes (m). c) Draw a graph of the relationship between the number of minutes and the cost.

Solution: a)

Table: | Minutes (m) | Cost (C) | |-------------|----------| | 1 | R1.50 | | 2 | R3.00 | | 3 | R4.50 | | 4 | R6.00 | | 5 | R7.50 | b)

Equation: C = 1.50m c) The graph would be a straight line starting at (0,0) and passing through the points listed in the table.