Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and describe geometric and numeric patterns.
• Determine the rule for a given pattern or sequence.
• Represent patterns using flow diagrams and tables.
• Express the rule in words and using algebra.
• Use a rule to find any term in a sequence.

Lesson summary

Patterns, sequences, and relationships are fundamental to mathematics. Understanding them unlocks problem-solving skills across many areas of life, from predicting population growth in our country to understanding financial trends and even appreciating the intricate designs found in traditional South African art and crafts. Recognizing patterns helps us make predictions, understand how things change, and build models of the world around us. These skills are essential for success in higher mathematics and in various careers. For Grade 7 learners in South Africa, this topic builds upon previous knowledge of number patterns and introduces more complex algebraic thinking.

Lesson notes

What is a Pattern? A pattern is a recurring sequence or arrangement of elements (numbers, shapes, objects, etc.). The elements repeat in a predictable manner. We can have number patterns (sequences) and geometric patterns (visual sequences). What is a Sequence? A sequence is an ordered list of numbers (or other elements). Each number in the sequence is called a term. Sequences can be finite (ending after a certain number of terms) or infinite (continuing forever).

Types of Sequences: Arithmetic Sequence: A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.

Example: 2, 5, 8, 11, 14… (common difference = 3)

Geometric Sequence: A sequence where each term is found by multiplying the previous term by a constant. This constant is called the common ratio.

Example: 3, 6, 12, 24, 48… (common ratio = 2)

Other Patterns: Not all sequences are arithmetic or geometric. They can follow other rules, like squaring numbers or adding consecutive odd numbers.

Finding the Rule: The rule is a description of how to get from one term to the next, or how to find any term in the sequence. We express rules in words and algebraically.

Algebraic Representation: We often use 'n' to represent the term number in a sequence. The term number is the position of a term in the sequence (e.g., the 1st term, 2nd term, 3rd term, etc.). We can then write an algebraic expression that describes the value of any term based on its term number.

Flow Diagrams and Tables: Flow diagrams visually represent the rule by showing the input (term number) and the output (term value). Tables organize the term numbers and corresponding term values in rows or columns.

Worked example

Example 1: Arithmetic Sequence

Consider the sequence: 4, 7, 10, 13, …

Identify the Pattern: This is an arithmetic sequence because the difference between consecutive terms is constant (7 - 4 = 3, 10 - 7 = 3, 13 - 10 = 3).

Find the Common Difference: The common difference (d) is

3. Express the Rule in Words: To get the next term, add 3 to the previous term.

Find the Algebraic Rule:

The first term is

4.

Each subsequent term is 3 more than the previous term.

The general form of an arithmetic sequence is a + (n-1)d, where 'a' is the first term, 'n' is the term number, and 'd' is the common difference.

Therefore, the rule is: 4 + (n - 1)

3. Simplifying: 4 + 3n - 3 = 3n + 1

The nth term is 3n +

1. Verify the Rule:

For n = 1 (first term): 3(1) + 1 = 4 (Correct)

For n = 2 (second term): 3(2) + 1 = 7 (Correct)

For n = 3 (third term): 3(3) + 1 = 10 (Correct)

Find the 10th term: Using the rule, the 10th term (n = 10) is 3(10) + 1 = 31

Example 2: Geometric Sequence

Consider the sequence: 2, 6, 18, 54, …

Identify the Pattern: This is a geometric sequence because each term is multiplied by the same number to get the next term.

Find the Common Ratio: The common ratio (r) is 3 (6 / 2 = 3, 18 / 6 = 3, 54 / 18 = 3).

Express the Rule in Words: To get the next term, multiply the previous term by

3. Find the Algebraic Rule:

The first term is

2.

The general form of a geometric sequence is a r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Therefore, the rule is: 2 3^(n - 1)

Verify the Rule:

For n = 1 (first term): 2 3^(1-1) = 2 3^0 = 2 1 = 2 (Correct)

For n = 2 (second term): 2 3^(2-1) = 2 3^1 = 2 3 = 6 (Correct)