Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and describe different types of numerical patterns.
• Extend patterns to find the next terms.
• Determine the rule that governs a pattern.
• Analyse patterns using flow diagrams and tables.

Lesson summary

Patterns, sequences, and relationships are fundamental building blocks of mathematics. They help us understand order, predict what comes next, and make sense of the world around us. This week, we delve deeper into recognizing, describing, and extending various types of patterns. Understanding these concepts is crucial not only for success in mathematics but also for problem-solving in everyday situations. In South Africa, recognizing patterns is vital in understanding societal trends, analyzing data related to resource allocation, and even appreciating the artistic beauty of traditional designs.

Lesson notes

What is a Pattern? A pattern is a predictable regularity or arrangement in a set of numbers, objects, or events. We can describe patterns using rules or relationships. These rules tell us how to get from one element in the pattern to the next.

Types of Numerical Patterns: 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... (The common difference is +3)

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... (The common ratio is x2)

Other Sequences: Some sequences may not be strictly arithmetic or geometric, but they still follow a definable rule.

Example: 1, 4, 9, 16, 25... (The rule is squaring consecutive whole numbers: 1 2 , 2 2 , 3 2 , 4 2 , 5 2 ...)

Flow Diagrams: A flow diagram shows the relationship between input and output values. It often involves an "input", a "rule", and an "output".

Identifying the Rule: To find the rule for a pattern, look for a relationship between the terms.

Ask yourself: What is being added or subtracted to get from one term to the next? (Arithmetic sequence) What is being multiplied or divided to get from one term to the next? (Geometric sequence) Is there some other operation (squaring, cubing, etc.) being performed on the term number? (Other sequences) Representing Patterns with Tables and Flow Diagrams: Tables and flow diagrams are useful tools for organizing and visualizing patterns.

Tables: A table lists the term number and the corresponding term value. This can help you see the relationship between them.

Flow Diagrams: Help visualise the steps involved in getting from an input number to an output number based on a specified rule.

Worked example

Example 1: Arithmetic Sequence

Consider the sequence: 7, 12, 17, 22, ...

Identify the type of sequence: The difference between consecutive terms is constant (12 - 7 = 5, 17 - 12 = 5, 22 - 17 = 5). This is an arithmetic sequence.

Determine the common difference: The common difference is +

5. Extend the sequence: To find the next term, add 5 to the last term: 22 + 5 =

2

7. The sequence continues: 7, 12, 17, 22, 27, ...

Express the rule in words: "Start at 7 and add 5 to each term to get the next term."

Example 2: Geometric Sequence

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

Identify the type of sequence: The ratio between consecutive terms is constant (6 / 2 = 3, 18 / 6 = 3, 54 / 18 = 3). This is a geometric sequence.

Determine the common ratio: The common ratio is x

3. Extend the sequence: To find the next term, multiply the last term by 3: 54 x 3 =

1

6

2. The sequence continues: 2, 6, 18, 54, 162, ...

Express the rule in words: "Start at 2 and multiply each term by 3 to get the next term."