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 patterns using tables, graphs, and equations.
• Use patterns to make predictions and solve problems.
• Interpret information presented in different formats.
• Understand the limitations of predictions.

Lesson summary

This week, we delve into the critical skill of identifying, analyzing, and representing patterns and relationships that exist all around us. Mathematical Literacy isn't just about numbers; it's about understanding the world through numbers. Whether it's predicting your cellphone data usage, budgeting for groceries based on price trends, or understanding how interest rates affect loan repayments, recognizing and interpreting patterns is essential for informed decision-making. In South Africa, with its diverse socio-economic landscape, the ability to understand and use patterns can empower individuals to make better choices and navigate everyday challenges.

Lesson notes

What are Patterns? A pattern is a predictable sequence or regularity. In mathematics, patterns often involve numerical or geometrical relationships that can be described using rules.

Types of Patterns: Linear Patterns: These patterns increase or decrease by a constant amount each time. They can be represented by equations of the form y = mx + c, where m is the constant difference (slope) and c is the starting value (y-intercept).

Quadratic Patterns: These patterns have a constant second difference. They can be represented by equations of the form y = ax² + bx + c. The graph of a quadratic pattern is a parabola.

Exponential Patterns: In these patterns, each term is multiplied by a constant ratio to get the next term. Exponential patterns are often associated with growth or decay.

Representations of Patterns: Tables: Organize data into rows and columns, making it easy to see the relationship between variables.

Graphs: Visual representations of data that show the relationship between variables.

Common types include: Line graphs: Used to show trends over time.

Scatter plots: Used to show the relationship between two variables when the data is not necessarily linear.

Bar graphs/Histograms: Useful for comparing different categories.

Equations: Mathematical expressions that describe the relationship between variables.

Example 1: Cellphone Data Usage (Linear Pattern) Sipho has a data bundle of 5GB. He uses 500MB of data each day. Let's represent his remaining data as a function of the number of days.

Table: | Day | Data Remaining (GB) | |-----|----------------------| | 0 | 5 | | 1 | 4.5 | | 2 | 4 | | 3 | 3.5 | | 4 | 3 | Equation: Let y be the data remaining (in GB) and x be the number of days. Since Sipho uses 0.5GB (500MB) per day, the equation is: y = 5 - 0.5x Graph: (A line graph would show a straight line decreasing from (0, 5) with a slope of -0.5)

Explanation: The equation allows us to predict how much data Sipho will have left after any given number of days. The graph provides a visual representation of the data consumption over time. This helps Sipho understand how long his data will last.

Example 2: Taxi Fare (Linear Pattern) A taxi charges a fixed call-out fee of R25 plus R8 per kilometer traveled.

Table: | Kilometers (km) | Fare (R) | |-----------------|----------| | 0 | 25 | | 1 | 33 | | 2 | 41 | | 3 | 49 | Equation: Let y be the total fare (in Rand) and x be the number of kilometers traveled.

The equation is: y = 8x + 25 Graph: (A line graph would show a straight line increasing from (0, 25) with a slope of 8)

Explanation: This example highlights how linear patterns are used in everyday pricing scenarios. You can easily calculate the cost of a taxi ride based on the distance.

Example 3: Population Growth (Exponential Pattern - Simplified) Suppose a rabbit population doubles every month. If we start with 2 rabbits, let’s see how the population grows.

Table: | Month | Rabbit Population | |-------|-------------------| | 0 | 2 | | 1 | 4 | | 2 | 8 | | 3 | 16 | | 4 | 32 | Equation: Let y be the rabbit population and x be the number of months.

The equation is: y = 2 * 2 x (or y = 2 x+1 )

Graph: (An exponential curve would be shown increasing rapidly).

Explanation: This is a simplified example to demonstrate exponential growth. Real-world population growth is often more complex, but this illustrates the basic principle.

Example 4: Distance traveled when dropping an object (Quadratic) An object is dropped from a height. The approximate distance it falls in t seconds is given by the formula d = 4.9t 2 where d is in metres.

Table: | Time(t) in seconds | Distance(d) in metres | |--------------------|------------------------| | 0 | 0 | | 1 | 4.9 | | 2 | 19.6 | | 3 | 44.1 | Equation: d = 4.9t 2 Graph: A parabola opening upwards with the vertex at the origin.

Explanation: The distance fallen increases at an increasing rate over time due to the acceleration of gravity. This example links the concept of patterns to scientific principles. Guided Practice (With Solutions)

Question 1: A street vendor sells pap and vleis. The cost of ingredients is R15 per portion, and he wants to make a profit of R10 per portion. Create a table showing his potential income for selling 0, 5, 10, and 15 portions. Write an equation to represent his total income, and explain what type of pattern this represents.

Solution: Table: | Portions Sold | Cost of Ingredients (R) | Income (R) | |---------------|-------------------------|------------| | 0 | 0 | 0 | | 5 | 75 | 125 | | 10 | 150 | 250 | | 15 | 225 | 375 | (Explanation: Each portion costs R15 + R10 = R

2

5. Income = portions sold * R25)

Equation: Let y be the total income (in Rand) and x be the number of portions sold.

The equation is: y = 25x Type of Pattern: This represents a linear pattern because the income increases by a constant amount (R25) for each additional portion sold.

Question 2: A borehole drilling company charges R500 for setup and R200 per meter drilled.