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 to draw conclusions and make predictions.
• Use mathematical models (equations) to solve practical problems.
• Critically evaluate the limitations of these models.

Lesson summary

This week, we delve into the crucial topic of patterns, relationships, and representations in real-life contexts. Understanding these concepts is fundamental to making informed decisions about everyday situations we face as South Africans. From budgeting our limited income to interpreting statistical data in news reports about employment or the spread of disease, recognising patterns and relationships allows us to predict outcomes, analyze trends, and solve problems more effectively. This empowers us to navigate complex information and participate actively in our communities and the broader economy.

Lesson notes

2. 1.

Types of Patterns: Linear Patterns: These patterns show a constant rate of change. If you add (or subtract) the same amount each time, it's likely a linear pattern. This is best described using equations in the form y = mx + c, where 'm' is the constant rate of change (gradient) and 'c' is the initial value (y-intercept).

Quadratic Patterns: These patterns have a changing rate of change. They often involve a squared term. The second difference between consecutive terms will be constant. They are generally described by equations in the form y = ax² + bx + c.

Exponential Patterns: These patterns show growth or decay that increases or decreases at a rapidly increasing rate. The terms are multiplied or divided by the same factor at each step (common ratio). Exponential patterns are often described by equations in the form y = a b^x, where 'a' is the initial value and 'b' is the growth/decay factor. If b > 1, it's growth; if 0 5km, Fare (F) = 20 + 5(d-5)

Example Question: What is the fare for a 12km trip? Since 12km > 5km, use the second part of the equation: F = 20 + 5(12-5) = 20 + 5(7) = 20 + 35 = R55 Guided Practice (With Solutions)

Question 1: The price of maize meal increases by R2 each year. If a bag of maize meal costs R30 this year, what will it cost in 5 years?

Solution: This is a linear pattern. The increase each year is constant (R2).

We can use the equation: Cost = 30 + 2years.

After 5 years: Cost = 30 + 25 = 30 + 10 = R40

Commentary: This question tests understanding of linear patterns and using an equation to make predictions.

Question 2: A savings account earns 8% interest per year, compounded annually. If you deposit R1000, how much will you have after 3 years?

Solution: This is an exponential pattern (compound interest).

The equation for compound interest is: A = P(1 + r)^n, where A = amount, P = principal, r = interest rate, n = number of years. A = 1000(1 + 0.08)^3 = 1000(1.08)^3 ≈ 1000 1.2597 ≈ R1259.71

Commentary: This question tests understanding of exponential patterns and compound interest calculations. Be sure to understand what "compounded annually" means.

Question 3: The following table shows the number of houses built in a new development each year: | Year | Number of Houses | |---|---| | 1 | 5 | | 2 | 10 | | 3 | 17 | | 4 | 26 | Describe the pattern and predict how many houses will be built in year

5. Solution: Let's find the first differences: 10-5 = 5, 17-10 = 7, 26-17 =

9. Let's find the second differences: 7-5 = 2, 9-7 =

2. The second difference is constant, so it's a quadratic pattern. The next first difference will be 9+2 =

1

1. Therefore, the number of houses built in year 5 will be 26 + 11 =

3

7. Commentary: This question requires identifying a quadratic pattern using differences. Independent Practice (Questions Only) A bakery sells loaves of bread. The profit they make is R5 per loaf. If their fixed costs are R200 per day, write an equation to represent their daily profit. What profit do they make if they sell 60 loaves? The number of unemployed people in a city decreases by 2% each year. If there are currently 50,000 unemployed people, how many will be unemployed in 4 years? A car depreciates in value by 15% each year. If the car was initially bought for R150,000, what will its value be after 6 years? The table below shows the height of a plant over several weeks: | Week | Height (cm) | |---|---| | 1 | 2 | | 2 | 4 | | 3 | 8 | | 4 | 16 | Describe the pattern. How tall will the plant be in week 6? A store is having a sale where all items are 20% off. Write an equation to represent the sale price of an item. If an item originally costs R250, what is the sale price? The distance a taxi travels and the cost are recorded below: | Distance (km) | Cost (R) | |---|---| | 3 | 40 | | 5 | 60 | | 8 | 90 | | 11 | 120 | Determine the fixed cost and rate per km for the taxi fare. Develop an equation to model this. A car hire company charges a daily rate plus a fee per kilometre. If the cost for 100km is R550 and the cost for 250km is R925, calculate the daily rate and the fee per kilometre. John invests R5000 into an account earning simple interest. After 3 years, he has R

5

9

0

0. What is the annual simple interest rate?