Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Identify different types of patterns (linear, quadratic, geometric).
• Represent relationships between variables.
• Interpret graphs to make predictions.
• Solve problems using direct and inverse proportion.

Lesson summary

This week, we delve into the fascinating world of patterns, relationships, and representations in real-life contexts. Understanding these concepts is crucial for making informed decisions about everyday issues, from managing personal finances to interpreting national statistics. Mathematical Literacy is about applying mathematical skills to practical situations, and recognizing patterns and relationships is a cornerstone of this. In South Africa, where we face diverse economic and social challenges, the ability to analyse trends, interpret data presented in different forms, and predict future outcomes is essential for empowered citizenship and responsible participation in the economy.

Lesson notes

2.1 Types of Patterns: Linear Patterns: These patterns have a constant difference between consecutive terms. The general form is y = mx + c, where m is the constant difference (gradient) and c is the y-intercept.

Example: The cost of airtime recharges. If a recharge card costs R10 and you buy x cards, the total cost (y) follows a linear pattern: y = 10x.

Quadratic Patterns: In these patterns, the second difference between consecutive terms is constant. The general form is y = ax² + bx + c.

Example:* The total distance travelled by a car accelerating at a constant rate from a stationary position follows a quadratic pattern. If the distance is proportional to the square of the time, then after 1 second the car travels 1 meter, after 2 seconds it travels 4 meters, after 3 seconds it travels 9 meters, the distances form the pattern 1, 4,

9. The first differences are 3, 5 and the second differences are 2 (constant).

Geometric Patterns: These patterns have a constant ratio between consecutive terms. The general form is y = ar x-1 , where a is the first term and r is the common ratio.

Example:* Compound interest on a savings account. If you deposit R100 and earn 10% interest per year, the amount in your account follows a geometric pattern.

After 1 year: R110, after 2 years: R121, after 3 years: R133.

1

0. The common ratio is 1.1. 2.2 Relationships between Variables: Direct Proportion: If y is directly proportional to x, then y = kx, where k is the constant of proportionality. As x increases, y increases proportionally.

Example: The number of bricks needed to build a wall is directly proportional to the area of the wall. If 100 bricks cover 1 square meter, then b = 100a, where b is the number of bricks and a is the area.

Inverse Proportion: If y is inversely proportional to x, then y = k/x, where k is the constant of proportionality. As x increases, y decreases proportionally.

Example: The time it takes to travel a certain distance is inversely proportional to the speed. If it takes 2 hours to travel 100 km at 50 km/h, then t = 100/s, where t is time and s is speed. 2.3 Graphical Representations: Line Graphs: Used to show trends over time. They connect data points with lines.

Example:* Showing the increase in electricity prices over the last 5 years.

Bar Graphs: Used to compare different categories.

Example:* Comparing the number of unemployed people in different provinces of South Africa.

Pie Charts: Used to show the proportions of different categories that make up a whole.

Example:* Showing the percentage of the national budget allocated to different sectors (education, healthcare, etc.).

Scatter Plots: Used to show the relationship between two variables.

Example:* Plotting the relationship between the amount of rainfall and crop yield in a region. 2.4 Using Spreadsheets: Spreadsheets are powerful tools for generating patterns, analyzing relationships, and creating visual representations of data. Functions like SUM, AVERAGE, and IF can be used for calculations. Charts can be easily created to visualize data.

Example: To create a table of compound interest, you can use the formula =A2(1+B1) in cell A3, where A2 is the initial amount, and B1 is the interest rate. Dragging the formula down will automatically calculate the amount for each subsequent year. 2.5 Worked

Examples: Example 1 (Linear Pattern): A taxi charges a fixed rate of R15 plus R8 per kilometer. Write an equation to represent the cost (C) of a taxi ride of d kilometers. Calculate the cost of a 10km ride.

Solution: The equation is C = 8d +

1

5. For a 10km ride, C = 8(10) + 15 = R

9

5. Example 2 (Direct Proportion): The cost of electricity is directly proportional to the number of units used. If 50 units cost R75, what will 120 units cost?

Solution: If C = kU (where C is the cost, U is units, and k is the constant of proportionality). We know 75 = k

5

0. So k = 75/50 = 1.

5. Therefore, C = 1.5

U. When U = 120, C = 1.5120 = R

1

8

0. Example 3 (Inverse Proportion): It takes 4 workers 6 days to complete a task. How long will it take 8 workers to complete the same task, assuming all workers work at the same rate?

Solution: The number of days (d) is inversely proportional to the number of workers (w). So, d = k/w. We know that 6 = k/

4. Therefore, k =

2

4. The equation is d = 24/w. If w = 8, then d = 24/8 = 3 days. Guided Practice (With Solutions)

Question 1: Sipho invests R2000 in a fixed deposit account that earns simple interest at a rate of 7% per year. (a) Calculate the interest earned after 3 years. (b) Calculate the total amount Sipho will have in his account after 3 years. (c) Write an equation to represent the total amount (A) Sipho has after 't' years.

Solution: (a) Simple Interest = Principal Rate Time = 2000 0.07 3 = R420. (b) Total Amount = Principal + Interest = 2000 + 420 = R2420. (c) A = 2000 + (2000 0.07 t) = 2000 + 140t Question 2: A farmer sells oranges at R5 each. (a) Create a table showing the revenue (in Rands) for selling 1, 2, 3, 4, and 5 oranges.