Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and describe different types of patterns.
• Extend patterns to find the next terms.
• Determine the rule for a sequence or table.
• Represent patterns using flow diagrams and tables.
• Solve real-life problems involving patterns.

Lesson summary

Patterns, sequences, and relationships are the building blocks of mathematics and are essential for problem-solving. In our everyday lives in South Africa, we see patterns everywhere, from the tiles on our roofs to the arrangement of shops in a mall. Understanding these patterns helps us predict future events, make informed decisions, and appreciate the beauty and order around us. This week, we'll focus on identifying, describing, and extending different types of patterns and using them to solve problems. Think about the patterns in the growth of maize crops, the payment plans for phones, or even the schedule of taxis - patterns are everywhere.

Lesson notes

What is a Pattern? A pattern is a predictable or regular arrangement of objects, numbers, or events. These arrangements can repeat (repeating patterns) or grow or shrink according to a specific rule (growing or shrinking patterns).

Types of Patterns: Geometric Patterns: These patterns consist of shapes that are arranged in a specific order. The shapes can change in size, orientation, or type.

Example: A geometric pattern could be triangles arranged in a row, each one getting bigger than the last: △, △△, △△△, … Number Patterns: These patterns involve a sequence of numbers that follow a specific rule. We often call these "sequences".

Example: A number pattern could be 2, 4, 6, 8, … (adding 2 each time)

Patterns in Tables: These patterns show the relationship between two or more sets of numbers or values.

Example: A table could show the number of hours worked and the amount earned: | Hours Worked | Amount Earned (R) | | :----------- | :---------------- | | 1 | 50 | | 2 | 100 | | 3 | 150 | | 4 | 200 | In this table, the pattern is that for every additional hour worked, R50 is earned. Sequences A sequence is a list of numbers or objects that follow a specific rule or pattern. Each number or object in a sequence is called a term.

Arithmetic Sequence: An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference.

Example: 3, 7, 11, 15, … (The common difference is 4)

Geometric Sequence: A geometric sequence has a constant ratio between consecutive terms. This ratio is called the common ratio.

Example: 2, 6, 18, 54, … (The common ratio is 3) Finding the Rule To understand a pattern, we need to find its rule. This rule describes how to get from one term to the next.

Here's how to find the rule: Look for a Constant Difference or Ratio: Check if the difference or ratio between consecutive terms is the same. This will help you determine if it is an arithmetic or geometric sequence.

Consider Multiplication and Addition: Sometimes, the rule involves both multiplication and addition (or subtraction).

Use Trial and Error: If the pattern is not straightforward, try different operations until you find the rule that works for all terms. Flow Diagrams and Tables Flow diagrams and tables are useful tools for representing patterns.

Flow Diagram: A flow diagram shows the input (the starting number), the rule (the operation applied), and the output (the resulting number).

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

Worked example

Example 1: Find the next three terms in the sequence: 1, 4, 7, 10, …

Solution:

Find the difference between consecutive terms: 4 - 1 = 3, 7 - 4 = 3, 10 - 7 =

3. The common difference is

3. Therefore, the rule is "add 3 to the previous term".

The next three terms are: 10 + 3 = 13, 13 + 3 = 16, 16 + 3 =

1

9.

So, the sequence is: 1, 4, 7, 10, 13, 16, 19, …

Example 2: Consider the following geometric pattern:

▢, ▢▢, ▢▢▢, …

Draw the next two shapes in the sequence.

Solution: The pattern is increasing by one square each time.

The next shape will have 4 squares: ▢▢▢▢

The shape after that will have 5 squares: ▢▢▢▢▢

Example 3: A spaza shop owner buys cool drinks for R5 each and sells them for R8 each. Create a table to show the profit made for selling 1 to 5 cool drinks.

Solution:

The profit per cool drink is R8 - R5 = R3

The table would look like this:

| Number of Cool Drinks Sold | Profit (R) |

| :------------------------- | :--------- |

| 1 | 3 |

| 2 | 6 |

| 3 | 9 |

| 4 | 12 |

| 5 | 15 |

Guided Practice (With Solutions)

Reference guide

• and ensures the stability and aesthetics of structures. The repetitive nature of bricklaying is a good example of a pattern.
• Agriculture: Farmers can use patterns to optimize planting arrangements and predict crop yields. For example, if a farmer plants maize in rows with a specific spacing, they can analyze the yield over several seasons and adjust the spacing to maximize production. Understanding the cyclical patterns of seasons and weather is essential for successful farming.
• Differentiation, Remediation and Extension
• Remediation:
• Simplified Examples: Provide learners with simpler patterns and sequences to start with, using smaller numbers and fewer steps. For example, start with adding 1 or 2 to a number.
• Visual Aids: Use concrete objects or drawings to represent patterns visually. For geometric patterns, use shapes or blocks to help learners see the relationship between terms.