Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and describe numerical sequences.
• Extend sequences by finding the next terms.
• Determine the rule that generates a sequence.
• Represent patterns using flow diagrams and tables.
• Solve problems using patterns.

Lesson summary

Patterns, sequences, and relationships are fundamental to mathematics and are all around us. From the arrangement of tiles in a township house to the growth of plants in a garden, recognising and understanding patterns helps us make predictions, solve problems, and appreciate the order in the world. In South Africa, understanding these mathematical concepts can help learners analyze trends in data like population growth, predict resource consumption, or even understand financial planning and budgeting. This week, we'll delve deeper into identifying, extending, and describing various types of numerical patterns and relationships.

Lesson notes

What are Sequences? A sequence is an ordered list of numbers, objects, or events. Each element in a sequence is called a term. In this lesson, we'll focus on numerical sequences, which are sequences of numbers. The key to understanding sequences is identifying the pattern rule that governs how the terms are related. Types of Numerical Sequences Arithmetic Sequences: In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference.

Example: 2, 5, 8, 11, 14, ... In this sequence, the common difference is 3 (5-2 = 3, 8-5 = 3, and so on).

Geometric Sequences: In a geometric sequence, the ratio between consecutive terms is constant. This constant ratio is called the common ratio.

Example: 3, 6, 12, 24, 48, ... In this sequence, the common ratio is 2 (6/3 = 2, 12/6 = 2, and so on). Finding the Pattern Rule To find the pattern rule, examine the relationship between consecutive terms.

Ask yourself: Are the terms increasing or decreasing? Is there a constant difference between terms? If so, it's likely an arithmetic sequence. Is there a constant ratio between terms? If so, it's likely a geometric sequence. Is the pattern more complex, involving squares, cubes, or other operations? Flow Diagrams and Tables Flow diagrams and tables are useful for representing patterns and relationships visually.

Flow Diagrams: A flow diagram shows the input, the rule applied to the input, and the output.

Example: ``` Input --> Rule: Multiply by 2 and add 1 --> Output ``` If the input is 3, the output is (3 x 2) + 1 =

7. Tables: A table organizes inputs and corresponding outputs in rows and columns.

Example: | Input (x) | Output (y) | |---|---| | 1 | 4 | | 2 | 7 | | 3 | 10 | In this table, the rule is likely: y = 3x + 1

Worked example

Example 1: Arithmetic Sequence

Sequence: 7, 11, 15, 19, ...

Identify the type of sequence: The terms are increasing. Let's find the difference between consecutive terms.

11 - 7 = 4

15 - 11 = 4

19 - 15 = 4

The difference is constant (4), so this is an arithmetic sequence.

Find the common difference: The common difference is

4. Determine the next two terms: Add the common difference to the last term:

19 + 4 = 23

23 + 4 = 27

Therefore, the next two terms are 23 and

2

7. Example 2: Geometric Sequence