Lesson Notes By Weeks and Term v5 - Grade 5

Lesson Video

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Performance objectives

• Identify and describe number patterns.
• Extend patterns by finding the rule.
• Use flow diagrams to represent patterns.
• Describe the relationship between input and output values.
• Solve simple problems involving patterns.

Lesson summary

In Grade 5 Mathematics, understanding patterns, functions, and relationships is like learning to read a secret code! It helps us predict what comes next, see connections between different things, and solve problems in a logical way. This week, we'll focus on finding rules in number patterns and using these rules to extend the patterns. This is important because patterns are everywhere, from the repeating designs in our traditional Ndebele art to the growth of plants in our gardens, and even the scheduling of loadshedding (understanding when it's likely to occur). Knowing how patterns work makes us better problem-solvers in everyday life.

Lesson notes

What are Number Patterns? A number pattern (also called a sequence) is a list of numbers that follow a specific rule or order. The rule tells us how to get from one number in the pattern to the next.

Examples of Number Patterns: 2, 4, 6, 8, ... (The rule is: Add 2) 1, 3, 5, 7, ... (The rule is: Add 2) 5, 10, 15, 20, ... (The rule is: Add 5) 20, 18, 16, 14, ... (The rule is: Subtract 2) Constant Difference (Arithmetic Sequences) A common type of number pattern is one with a "constant difference." This means the same number is added (or subtracted) each time to get the next number. We call these arithmetic sequences.

Example: 3, 7, 11, 15, ... (The constant difference is +4)

How to Find the Rule: Look at the difference between consecutive numbers: Subtract the first number from the second number. Then, subtract the second number from the third number. If the difference is the same each time, you've found the constant difference. Determine if you are adding or subtracting: If the numbers are increasing, you are adding. If they are decreasing, you are subtracting.

Write the rule: State clearly what you are adding or subtracting.

Example 1: Find the rule and the next two numbers in the pattern: 6, 9, 12, 15, ...

Difference: 9 - 6 = 3; 12 - 9 = 3; 15 - 12 =

3. The difference is

3. Adding or Subtracting: The numbers are increasing, so we are adding.

Rule: Add

3. Next two numbers: 15 + 3 = 18; 18 + 3 =

2

1. The pattern is: 6, 9, 12, 15, 18, 21, ... Flow Diagrams A flow diagram is a way to represent a number pattern visually. It shows the "input" (the starting number), the "rule" (what you do to the number), and the "output" (the result).

Example 2: Input: 1 -> Rule: Add 5 -> Output: 6 Input: 2 -> Rule: Add 5 -> Output: 7 Input: 3 -> Rule: Add 5 -> Output: 8 Input: 4 -> Rule: Add 5 -> Output: 9 We can write this as: | Input | Output | |---|---| | 1 | 6 | | 2 | 7 | | 3 | 8 | | 4 | 9 | Example 3: A More Complex Pattern Consider the pattern generated by the rule: Multiply by 2, then add

1. Input: 1 -> Rule: Multiply by 2, then add 1 -> Output: (1 * 2) + 1 = 3 Input: 2 -> Rule: Multiply by 2, then add 1 -> Output: (2 * 2) + 1 = 5 Input: 3 -> Rule: Multiply by 2, then add 1 -> Output: (3 * 2) + 1 = 7 Input: 4 -> Rule: Multiply by 2, then add 1 -> Output: (4 * 2) + 1 = 9 | Input | Output | |---|---| | 1 | 3 | | 2 | 5 | | 3 | 7 | | 4 | 9 | Why is This Important? These patterns are used everywhere. Imagine you are helping your family budget for groceries. You know that each loaf of bread costs R

1

5. The cost of the bread follows a pattern: 1 loaf = R15, 2 loaves = R30, 3 loaves = R45, and so on. You can use this pattern to calculate the cost of any number of loaves. Or consider the seating arrangement in a theatre; it may follow a specific pattern to maximize the available space. Guided Practice (With Solutions)

Question 1: Find the rule and the next three numbers in the pattern: 4, 8, 12, 16, ...

Solution: Difference: 8 - 4 = 4; 12 - 8 = 4; 16 - 12 =

4. The difference is

4. Adding or Subtracting: The numbers are increasing, so we are adding.

Rule: Add

4. Next three numbers: 16 + 4 = 20; 20 + 4 = 24; 24 + 4 =

2

8. The pattern is: 4, 8, 12, 16, 20, 24, 28, ...

Question 2: Complete the flow diagram: Input: 2 -> Rule: Multiply by 3 -> Output: ?

Input: 5 -> Rule: Multiply by 3 -> Output: ?

Input: 8 -> Rule: Multiply by 3 -> Output: ?

Solution: Input: 2 -> Rule: Multiply by 3 -> Output: 2 3 = 6 Input: 5 -> Rule: Multiply by 3 -> Output: 5 3 = 15 Input: 8 -> Rule: Multiply by 3 -> Output: 8 3 = 24 | Input | Output | |---|---| | 2 | 6 | | 5 | 15 | | 8 | 24 | Question 3: Describe the relationship between the input and output values in the following table: | Input | Output | |---|---| | 1 | 5 | | 2 | 6 | | 3 | 7 | | 4 | 8 | Solution: The output is always 4 more than the input. (Rule: Add 4)

Question 4: What are the next two terms of the pattern: 1, 4, 9, 16, ...

Solution: This isn't an arithmetic sequence.

Let's look closer: 1 = 1 * 1 = 1 2 4 = 2 * 2 = 2 2 9 = 3 * 3 = 3 2 16 = 4 * 4 = 4 2 The rule is: Square the consecutive natural numbers.

So the next terms are: 5 2 = 25 6 2 = 36 The pattern is: 1, 4, 9, 16, 25, 36, ... Independent Practice (Questions Only) Find the rule and the next three numbers in the pattern: 10, 20, 30, 40, ... Find the rule and the next three numbers in the pattern: 25, 20, 15, 10, ...

Complete the flow diagram: Input: 3 -> Rule: Add 7 -> Output: ?

Input: 6 -> Rule: Add 7 -> Output: ?

Input: 9 -> Rule: Add 7 -> Output: ?

Complete the flow diagram: Input: 1 -> Rule: Multiply by 4, then subtract 1 -> Output: ?

Input: 2 -> Rule: Multiply by 4, then subtract 1 -> Output: ?

Input: 3 -> Rule: Multiply by 4, then subtract 1 -> Output: ? Describe the relationship between the input and output values in the following table: | Input | Output | |---|---| | 2 | 4 | | 4 | 8 | | 6 | 12 | | 8 | 16 | A taxi charges a base fee of R10 and then R5 per kilometer.