Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify, describe and extend patterns
• Represent patterns using tables, diagrams and rules
• Determine a general rule for a pattern
• Solve problems involving patterns

Lesson summary

Patterns, sequences, and relationships form the foundation for understanding the world around us. From the arrangement of bricks in a wall to the growth of a savings account, patterns are everywhere. In South Africa, recognizing patterns can help us understand social trends, predict weather patterns for farming, manage finances effectively, and even appreciate the intricate designs in traditional art and crafts like beadwork. This topic isn’t just about numbers; it's about developing critical thinking and problem-solving skills applicable to everyday life. Understanding patterns allows us to make informed decisions, predict future outcomes, and appreciate the beauty and order in the world.

Lesson notes

What is a Pattern? A pattern is a repetitive or predictable arrangement of elements. These elements can be numbers, shapes, colours, or any other objects. We focus mainly on numerical and geometric patterns in Grade

7. What is a Sequence? A sequence is an ordered list of numbers (or other elements). Each number in a sequence is called a term. Sequences often follow a specific pattern or rule.

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 value. This constant value is called the common ratio.

Example: 3, 6, 12, 24, 48... (Common ratio = 2)

Other Sequences: Sequences can also follow other rules, such as squaring, cubing, or using more complex formulas. These aren't strictly arithmetic or geometric.

Representing Patterns and Sequences: Number Lines: You can visually represent sequences on a number line, showing the position of each term.

Tables: A table can be used to show the relationship between the term number (position in the sequence) and the term value. | Term Number (n) | Term Value | | :--------------: | :--------: | | 1 | 2 | | 2 | 5 | | 3 | 8 | | 4 | 11 | Flow Diagrams: A flow diagram uses boxes and arrows to show the rule that generates the sequence.

Example: ``` Input (Term Number) --> [x 3] --> [+ 1] --> Output (Term Value) 1 -> 3 -> 4 2 -> 6 -> 7 3 -> 9 -> 10 ``` Algebraic Expressions (Formulas): An algebraic expression provides a general rule for finding any term in the sequence based on its position. This is also called the nth term rule. If the nth term is T(n), where n is the term number.

Example: For the sequence 2, 5, 8, 11..., the nth term rule is T(n) = 3n - 1 Finding the nth Term Rule: Identify the type of sequence: Is it arithmetic, geometric, or something else?

For arithmetic sequences: Find the common difference (d). The nth term rule will be in the form T(n) = dn + c, where 'c' is a constant you need to find. Substitute n = 1 and the value of the first term into the formula to solve for 'c'.

Example: Sequence: 4, 7, 10, 13... Common difference (d) = 3 T(n) = 3n + c When n = 1, T(1) =

4. So, 4 = 3(1) + c => c = 1 Therefore, the nth term rule is T(n) = 3n + 1 For geometric sequences: Find the common ratio (r). The nth term rule will be in the form T(n) = a r^(n-1), where 'a' is the first term.

Example: Sequence: 2, 6, 18, 54... Common ratio (r) = 3 T(n) = 2 3^(n-1)

Worked example

Example 1: Arithmetic Sequence

Consider the sequence: 7, 10, 13, 16…

What is the common difference? The common difference is 3 (10 - 7 = 3, 13 - 10 = 3, etc.)

Find the next two terms. The next two terms are 19 (16 + 3) and 22 (19 + 3).

Find the nth term rule. The rule will be in the form T(n) = 3n + c. When n=1, T(1) =

7. So, 7 = 3(1) + c. This gives us c =

4. Therefore, T(n) = 3n +

4.

Find the 10th term. Using the rule, T(10) = 3(10) + 4 =

3

4. Example 2: Geometric Sequence

Consider the sequence: 1, 4, 16, 64...

What is the common ratio? The common ratio is 4 (4 / 1 = 4, 16 / 4 = 4, etc.)

Find the next two terms. The next two terms are 256 (64 4) and 1024 (256 4).

Find the nth term rule. The rule will be in the form T(n) = a r^(n-1), where a=1 and r=

4. Therefore, T(n) = 1 4^(n-1) = 4^(n-1).

Find the 6th term. T(6) = 4^(6-1) = 4^5 = 1024.