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, diagrams, and equations.
• Interpret information from tables and graphs.
• Solve real-world problems using patterns.
• Calculate the nth term of a linear sequence.

Lesson summary

This week, we delve into the exciting world of patterns, relationships, and representations. Understanding how patterns work, recognizing the relationships between different quantities, and representing these relationships in various ways (tables, graphs, and equations) are crucial skills. This isn't just about numbers; it's about understanding the world around you. From predicting the cost of your electricity bill based on usage to understanding how your savings will grow over time, these skills are essential for making informed decisions in your daily life as a South African.

Lesson notes

2.1 Understanding Patterns A pattern is a regular, predictable arrangement of numbers, shapes, or objects. We can often describe patterns using a rule. These rules can be represented numerically, visually, or algebraically.

Linear Patterns: These patterns increase or decrease by a constant amount (called the common difference) each time. For example, 2, 4, 6, 8,... (adding 2 each time) is a linear pattern. The rule here is "add 2 to the previous term." Quadratic Patterns: The difference between consecutive terms is not constant, but the difference between those differences is constant. This sounds complicated, but you'll see it's not too bad! For example, 1, 4, 9, 16,... (the squares of the natural numbers). The difference between 1 and 4 is 3, between 4 and 9 is 5, between 9 and 16 is

7. The differences between these differences (5-3 and 7-5) is 2, a constant.

Geometric Patterns: These patterns are formed by multiplying (or dividing) by a constant amount (called the common ratio) each time. For example, 3, 6, 12, 24,... (multiplying by 2 each time) is a geometric pattern. 2.2 Representing Patterns Tables: A table organizes data into rows and columns, showing the relationship between two or more variables.

Example: A cellphone contract charges R50 per month plus R0.50 per minute of call time.

We can represent this in a table: | Minutes of Call Time | Monthly Cost (R) | |----------------------|------------------| | 0 | 50 | | 10 | 55 | | 20 | 60 | | 30 | 65 | Flow Diagrams: A flow diagram shows the step-by-step process of a pattern. It visually represents the rule or relationship.

Example: To generate terms in the linear sequence that starts at 5 and adds 3 each time, we can draw the following flow diagram: `Input (Term Number) --> Multiply by 3 --> Add 2 --> Output (Term Value)` So, the 1st term is (1 3) + 2 =

5. The 2nd term is (2 3) + 2 = 8, and so on.

Algebraic Equations (Formulas): An algebraic equation expresses the relationship between variables using symbols and mathematical operations. This allows us to calculate any term in a sequence without having to calculate all the terms before it.

Example: For the cellphone contract above, the monthly cost (C) can be represented as: `C = 50 + 0.50m`, where 'm' is the number of minutes of call time.

The nth term of a linear sequence: The nth term (T n ) of a linear sequence is given by the formula: T n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. 2.3 Interpreting Tables and Graphs Tables and graphs are visual representations of data. It's crucial to understand how to read and interpret them.

Tables: Look for headings, units, and trends within the data. Identify relationships between rows and columns.

Graphs: Identify the axes, scales, and type of graph (e.g., bar graph, line graph, pie chart). Look for trends (increasing, decreasing, constant), turning points, and relationships between the variables on the axes. Line graphs are particularly useful for visualizing changes over time.

Worked example

Example 1: Linear Pattern - Calculating Cellphone Costs

A cellphone contract charges R80 per month plus R0.80 per minute of call time.

(a) Create a table showing the monthly cost for 0, 10, 20, and 30 minutes of call time.

Solution:

| Minutes of Call Time | Monthly Cost (R) |

|----------------------|------------------|

| 0 | 80 |

| 10 | 88 |

| 20 | 96 |

| 30 | 104 |

Explanation: For each 10 minutes of call time, the cost increases by R0.80 10 = R8.

(b) Write an algebraic equation to represent the monthly cost (C) in terms of the number of minutes of call time (m).

Solution: C = 80 + 0.80m

Explanation: The fixed cost is R80, and the variable cost is R0.80 per minute.

(c) Use the equation to calculate the monthly cost for 55 minutes of call time.

Solution: C = 80 + 0.80(55) = 80 + 44 = R124

Explanation: Substitute m = 55 into the equation.

Example 2: Finding the nth term of a linear sequence

Consider the sequence: 5, 8, 11, 14, ...

(a) Determine the common difference.

Solution: 8 - 5 = 3, 11 - 8 = 3, 14 - 11 =

3. The common difference (d) is 3.

(b) Write the formula for the nth term of this sequence.

Solution: T n = a + (n-1)d, where a = 5 (the first term) and d =

3. Therefore,

T n = 5 + (n-1)3

T n = 5 + 3n - 3

T n = 3n + 2

(c) Calculate the 20th term of the sequence.

Solution: T 20 = 3(20) + 2 = 60 + 2 = 62