Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and describe numerical patterns.
• Extend patterns to find the next terms.
• Determine the rule for a pattern.
• Use flow diagrams and tables to show relationships.
• Apply these skills to solve problems.

Lesson summary

Patterns, sequences, and relationships are fundamental building blocks of mathematics and are crucial for developing logical thinking and problem-solving skills. This week, we will delve deeper into identifying, describing, and extending different types of numerical and geometric patterns. Recognizing patterns allows us to make predictions, solve problems in various contexts, and understand the underlying order in seemingly chaotic situations.

Lesson notes

What is a Pattern? A pattern is a predictable sequence of elements, such as numbers, shapes, or colors, that repeats or changes in a consistent way. Identifying the rule governing the pattern allows us to predict what comes next.

Numerical Patterns (Sequences): A numerical pattern is a sequence of numbers arranged according to a specific rule.

We will focus on two main types: Arithmetic Sequences: An arithmetic sequence is a pattern 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). Each term is obtained by adding 3 to the previous term.

Geometric Sequences: A geometric sequence is a pattern where each term is multiplied by a constant value to obtain the next term. This constant value is called the "common ratio."

Example: 3, 6, 12, 24, 48… (Common ratio = 2). Each term is obtained by multiplying the previous term by

2. Finding the Rule: To determine the rule for a numerical pattern, observe the relationship between consecutive terms.

Ask yourself: Are the terms increasing or decreasing? Is there a constant value being added or subtracted? If so, it's likely an arithmetic sequence. Is there a constant value being multiplied or divided? If so, it's likely a geometric sequence.

Representing Relationships: Flow Diagrams and Tables Flow Diagrams: A flow diagram visually represents the relationship between input and output values. You start with an input, apply a rule, and obtain the output.

Example: Input -> Rule: Multiply by 4 -> Output If the input is 2, the output is 2 x 4 =

8. Tables: A table organizes input and output values in rows and columns, making it easy to see the relationship between them.

Example: | Input (x) | Output (y) | | --------- | ---------- | | 1 | 5 | | 2 | 10 | | 3 | 15 | | 4 | 20 | In this table, the rule is: Output (y) = Input (x) x 5

Worked example

Example 1: Arithmetic Sequence

Consider the sequence: 7, 10, 13, 16, …

Identify the type of sequence: Observe that the terms are increasing.

Find the common difference: 10 – 7 = 3, 13 – 10 = 3, 16 – 13 =

3. The common difference is

3. Describe the rule: The rule is to add 3 to the previous term.

Find the next two terms: 16 + 3 = 19, 19 + 3 =

2

2. The next two terms are 19 and

2

2. Example 2: Geometric Sequence

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

Identify the type of sequence: Observe that the terms are increasing rapidly.

Find the common ratio: 6 / 2 = 3, 18 / 6 = 3, 54 / 18 =

3. The common ratio is

3. Describe the rule: The rule is to multiply the previous term by

3. Find the next two terms: 54 x 3 = 162, 162 x 3 =

4

8

6. The next two terms are 162 and

4

8

6. Example 3: Flow Diagram

Input -> Rule: Subtract 2 and then Multiply by 5 -> Output