Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify different types of patterns.
• Represent patterns using tables, graphs, and formulas.
• Use patterns to make predictions and solve problems.
• Interpret information from tables and graphs.

Lesson summary

This week, we delve into the fascinating world of patterns, relationships, and representations. Understanding patterns allows us to make predictions, solve problems, and interpret data more effectively. These skills are crucial for making informed decisions in everyday life, from budgeting your money to understanding crime statistics in your community. Being able to represent information in different ways (tables, graphs, equations) makes it accessible and easier to understand. This is especially important in South Africa, where understanding data related to socio-economic issues, resource allocation, and development is crucial for active citizenship.

Lesson notes

2. 1. Understanding Patterns A pattern is a regular, systematic repetition or sequence. Patterns can be found in numbers, shapes, colors, sounds, and almost anything else. In mathematics, we often look for patterns in sequences of numbers or figures. There are three main types of patterns we'll focus on: Linear Patterns: These patterns have a constant difference between consecutive terms. Think of it like walking at a steady pace; you cover the same distance with each step. The general form can be represented as T n = a + (n-1)d, where T n is the nth term, a is the first term, n is the term number, and d is the common difference.

Quadratic Patterns: These patterns have 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 often linked to situations involving area or growth that accelerates. The general form is T n = an 2 + bn + c.

Geometric Patterns: These patterns have a constant ratio between consecutive terms. Think of compound interest on a savings account – the amount increases by the same percentage each period. The general form is T n = ar n-1 , where T n is the nth term, a is the first term, n is the term number, and r is the common ratio. 2.

2. Representing Patterns Patterns can be represented in several ways: Tables: A table is a structured way of organizing data into rows and columns. It shows the relationship between two or more variables.

Graphs: A graph is a visual representation of the relationship between two or more variables. Different types of graphs (line graphs, bar graphs, scatter plots) are suitable for different types of data.

Algebraic Expressions (Formulas): An algebraic expression is a mathematical phrase that combines numbers, variables, and operations. It provides a general rule for the pattern, allowing you to calculate any term in the sequence. 2.

3. Working with Tables Tables are a convenient way to organize data and identify patterns.

Let's consider an example: Example 1: Cellphone Data Costs A cellphone company charges R20 for the first 100MB of data and then R5 for every additional 50MB. Represent this information in a table and find a formula to calculate the cost for any amount of data (in 50MB increments after the initial 100MB). | Data (MB) | Cost (R) | | --------- | -------- | | 100 | 20 | | 150 | 25 | | 200 | 30 | | 250 | 35 | | 300 | 40 | Explanation: The table shows the cost for different amounts of data. We can see that the cost increases by R5 for every 50MB increment after the initial 100MB. This indicates a linear relationship. To find a formula, let x be the number of 50MB increments after the initial 100M

B. Then the cost, C, can be represented as: C = 20 + 5x This formula lets us calculate the cost for any amount of data beyond 100MB in 50MB chunks. For example, if you use 400MB (which is 6 increments of 50MB beyond 100MB, so x=6), the cost would be: C = 20 + 5(6) = 20 + 30 = R50 2.

4. Working with Graphs Graphs are powerful tools for visualizing relationships between variables.

Example 2: Taxi Fare A taxi charges a flat fee of R15 plus R8 per kilometer traveled. Plot a graph to represent the cost of a taxi ride up to 10 kilometers.

Explanation: First, we need to create a table of values: | Distance (km) | Cost (R) | | --------------- | -------- | | 0 | 15 | | 1 | 23 | | 2 | 31 | | 3 | 39 | | 4 | 47 | | 5 | 55 | | 6 | 63 | | 7 | 71 | | 8 | 79 | | 9 | 87 | | 10 | 95 | Next, we plot these points on a graph with distance on the x-axis and cost on the y-axis. The graph will be a straight line because the relationship is linear. The equation of the line is C = 15 + 8d, where C is the cost and d is the distance. The y-intercept (where the line crosses the y-axis) is 15, representing the flat fee. The slope of the line is 8, representing the cost per kilometer. This graph allows us to quickly estimate the cost of a taxi ride for any distance up to 10 kilometers. 2.

5. Working with Algebraic Expressions Algebraic expressions provide a concise and general way to represent patterns.

Example 3: Number of Seats in a School Hall A school hall has rows of seats. The first row has 10 seats, and each subsequent row has 2 more seats than the row before it. Find an algebraic expression to represent the number of seats in the nth row.

Explanation: This is a linear pattern with a first term of 10 and a common difference of

2. Using the general formula for a linear pattern, T n = a + (n-1)d, we get: T n = 10 + (n-1)2 Simplifying the expression: T n = 10 + 2n - 2 T n = 2n + 8 This formula allows us to calculate the number of seats in any row. For example, the 10th row would have: T 10 = 2(10) + 8 = 20 + 8 = 28 seats Guided Practice (With Solutions)

Question 1: A street vendor sells vetkoek. He charges R5 for one, R9 for two, R13 for three, and so on. (a) Represent this information in a table for the first 5 vetkoek.