Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and extend numerical and geometric patterns.
• Determine the rule for a given pattern.
• Use input and output values to find a function's rule.
• Solve simple number sentences.
• Translate word problems into algebraic expressions.

Lesson summary

This week, we delve into the exciting world of patterns, functions, and simple algebraic expressions. Understanding these concepts is crucial because patterns are everywhere around us, from the beadwork designs of the Ndebele people to the way stokvel contributions grow over time. Learning about functions helps us understand how things change in relation to each other, like the price of airtime depending on the minutes you buy. Algebra gives us the tools to describe these relationships in a concise and powerful way. Thinking algebraically is a vital skill that will help you in higher grades and in everyday problem-solving.

Lesson notes

2. 1. What are Patterns? Patterns are sequences of numbers, shapes, or other objects that follow a specific rule. Identifying the rule is key to extending the pattern. Patterns can be numerical (involving numbers), geometric (involving shapes), or even a combination of both.

Numerical Patterns: These are sequences of numbers that follow a rule, such as adding the same number each time (an arithmetic sequence) or multiplying by the same number each time (a geometric sequence).

Example 1: Arithmetic Sequence Consider the pattern: 3, 6, 9, 12, __, __, __ Explanation: We observe that each term is obtained by adding 3 to the previous term. 3 + 3 = 6, 6 + 3 = 9, 9 + 3 =

1

2. This constant addition is called the common difference.

Extending the pattern: Following the rule, the next three terms are 15, 18, and 21. (12 + 3 = 15, 15 + 3 = 18, 18 + 3 = 21).

Example 2: Geometric Sequence Consider the pattern: 2, 4, 8, 16, __, __, __ Explanation: Each term is obtained by multiplying the previous term by 2. 2 x 2 = 4, 4 x 2 = 8, 8 x 2 =

1

6. The constant multiplication is called the common ratio.

Extending the pattern: Following the rule, the next three terms are 32, 64, and 128 (16 x 2 = 32, 32 x 2 = 64, 64 x 2 = 128).

Geometric Patterns: These are sequences of shapes or figures that follow a rule.

Example 3: A pattern of triangles where each subsequent figure adds a row of triangles. Figure 1 has 1 triangle, Figure 2 has 3 triangles, Figure 3 has 6 triangles, Figure 4 has 10 triangles.

Explanation: The number of triangles increases by adding the next consecutive number. So, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 =

1

0. This is related to triangle numbers.

Extending the pattern: The next figure (Figure 5) would have 10 + 5 = 15 triangles. 2.

2. Functions and Function Machines A function is a relationship between two sets of numbers: an input and an output. We can think of a function as a machine that takes an input, applies a rule, and produces an output. The rule describes what happens to the input to get the output.

Example 4: A function machine has the rule "Multiply by 2 and add 1".

Input: 3 Rule: Multiply by 2 and add 1 Calculation: (3 x 2) + 1 = 6 + 1 = 7 Output: 7 We can write this relationship as: If input = 3, then output =

7. Finding the Rule: Sometimes you are given the input and output values and need to find the rule. Look for a pattern that connects the inputs and outputs.

Example 5: Input: 1, Output: 4; Input: 2, Output: 5; Input: 3, Output: 6 Explanation: We can see that each output is 3 more than the input. So the rule is "Add 3".

Rule: Add 3 2.

3. Simple Algebraic Expressions Algebraic expressions use letters (variables) to represent unknown numbers. This allows us to write general rules and solve problems where we don't know all the information.

Variable: A letter (like x, y, or n) that represents an unknown number.

Expression: A combination of numbers, variables, and mathematical operations (like +, -, ×, ÷).

Example 6: If n represents a number, then "3 more than the number" can be written as the algebraic expression n +

3. Example 7: If x represents the number of sweets in a packet, then "twice the number of sweets" can be written as 2x (or simply 2x). 2.

4. Solving Simple Number Sentences We can use number sentences to represent mathematical relationships. Solving a number sentence means finding the value of the unknown variable that makes the sentence true. We will use inspection and trial-and-improvement.

Inspection: Looking at the number sentence and figuring out the answer by thinking about the numbers involved.

Example 8: x + 5 = 12 Explanation: What number, when added to 5, equals 12? We know that 7 + 5 =

1

2. Solution: x = 7 Trial-and-Improvement: Guessing a value for the variable, checking if it works, and then adjusting the guess based on the result.

Example 9: 2y - 3 = 7 Trial 1: Let y =

4. Then 2(4) - 3 = 8 - 3 =

5. This is too small.

Trial 2: Let y =

5. Then 2(5) - 3 = 10 - 3 =

7. This is correct!

Solution: y = 5 Guided Practice (With Solutions)

Question 1: Extend the following numerical pattern by three terms: 5, 10, 15, 20, __, __, __ Solution: Identifying the Pattern: The pattern is increasing. We see that 10 - 5 = 5, 15 - 10 = 5, 20 - 15 =

5. So, we are adding 5 to each term.

Extending the pattern: 20 + 5 = 25, 25 + 5 = 30, 30 + 5 =

3

5. Answer: 5, 10, 15, 20, 25, 30, 35 Question 2: A function machine takes an input, multiplies it by 3, and then subtracts

2. What is the output if the input is 4?

Solution: Applying the Rule: The rule is "Multiply by 3 and subtract 2".

Calculation: (4 x 3) - 2 = 12 - 2 = 10 Answer: The output is

1

0. Question 3: Thando earns R15 per hour. Write an algebraic expression to represent how much he earns in h hours.

Solution: Identifying the Variable: h represents the number of hours.

Writing the Expression: He earns R15 for each hour, so we multiply the number of hours by

1

5. Answer: 15h or R15h.