Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and extend geometric patterns.
• Represent patterns using flow diagrams and tables.
• Write number sentences (equations) for simple relationships.
• Solve simple equations by inspection.
• Identify variables and constants in expressions.

Lesson summary

This week, we delve deeper into patterns, functions, and simple algebraic expressions. Understanding patterns is crucial because it helps us predict and understand the world around us. From predicting the weather to understanding how prices change in the market, patterns are everywhere! Algebraic expressions provide a shorthand way to represent these patterns, allowing us to solve problems more efficiently. In South Africa, understanding financial patterns (like loan repayments or salary increases) and geometric patterns (like those found in traditional art) is extremely useful.

Lesson notes

2.1 Geometric Patterns: Increasing and Decreasing Sequences Geometric patterns are sequences where each term is obtained by multiplying or dividing the previous term by a constant value (the common ratio).

However, in Grade 6, we will focus on patterns that increase or decrease consistently, using addition or subtraction.

Increasing Sequence: Each number is greater than the one before it. Usually involves addition.

Decreasing Sequence: Each number is smaller than the one before it. Usually involves subtraction.

Example 1: Increasing Sequence Consider the sequence: 3, 7, 11, 15, ... What is the rule for this sequence? What are the next two numbers?

Finding the Rule: The difference between consecutive terms is constant: 7 - 3 = 4, 11 - 7 = 4, 15 - 11 =

4. The rule is to add 4 to each term to get the next term.

Next Two Numbers: 15 + 4 = 19, 19 + 4 =

2

3. The next two numbers are 19 and

2

3. Example 2: Decreasing Sequence Consider the sequence: 50, 45, 40, 35, ... What is the rule for this sequence? What are the next two numbers?

Finding the Rule: The difference between consecutive terms is constant: 45 - 50 = -5, 40 - 45 = -5, 35 - 40 = -

5. The rule is to subtract 5 from each term to get the next term.

Next Two Numbers: 35 - 5 = 30, 30 - 5 =

2

5. The next two numbers are 30 and 25. 2.2 Flow Diagrams and Tables Flow diagrams and tables are visual tools that help us represent the relationship between input and output values in a pattern. A flow diagram shows the rule as an operation that transforms the input into the output. A table organizes input and output values in rows and columns.

Example 3: Flow Diagram and Table Consider the rule: Multiply by 2 and add

1. Flow Diagram: Input --> [x2] --> [+1] --> Output Table: | Input (x) | Output (y) | | --------- | ---------- | | 1 | 3 | | 2 | 5 | | 3 | 7 | | 4 | 9 | | 5 | 11 | Here, if the input is 'x', the output 'y' can be represented as: y = (x * 2) +

1. Explanation: The table shows the result (Output) when different numbers (Input) are put through the rule stated in the flow diagram. Understanding flow diagrams helps to establish the connection between input and output. 2.3 Number Sentences (Equations) A number sentence or equation expresses the relationship between numbers and variables using mathematical symbols. A variable is a letter or symbol that represents an unknown number.

Example 4: "A number plus 5 equals 12." This can be written as the equation: x + 5 = 12 Here, 'x' is the variable representing the unknown number. 2.4 Solving Simple Equations by Inspection Solving an equation by inspection means finding the value of the variable simply by looking at the equation and using mental math.

Example 5: Solve for x: x + 3 = 7 By inspection, we can see that x must be 4 because 4 + 3 =

7. Example 6: Solve for y: y - 2 = 5 By inspection, we can see that y must be 7 because 7 - 2 =

5. Example 7: Solve for z: 2 * z = 10 By inspection, we know that z must be 5 because 2 * 5 = 10. 2.5 Variables and Constants Variable: A symbol (usually a letter) that represents a quantity that can change or vary. (e.g., x, y, n, a). Its value is unknown or can take on different values.

Constant: A number that has a fixed value and does not change. (e.g., 2, 5, -3, ½).

Example 8: In the expression 3x + 5: 'x' is the variable. Its value can change. '3' and '5' are constants. Their values are fixed.

Explanation: The term 3x means 3 multiplied by whatever the value of x is.

Therefore, as 'x' changes, '3x' will also change. '5', however, always remains

5. Guided Practice (With Solutions)

Question 1: Extend the pattern: 2, 6, 10, 14, ___, ___ Solution: Identify the Rule: The difference between consecutive terms is 4 (6-2=4, 10-6=4, 14-10=4). The rule is to add 4 to the previous term.

Extend the Pattern: 14 + 4 = 18, 18 + 4 =

2

2. The next two terms are 18 and

2

2. Question 2: Complete the table for the rule: Output = Input x 3 - 2 | Input | Output | | ----- | ------ | | 2 | | | 5 | | | 8 | | Solution: Apply the Rule: Input = 2: Output = (2 x 3) - 2 = 6 - 2 = 4 Input = 5: Output = (5 x 3) - 2 = 15 - 2 = 13 Input = 8: Output = (8 x 3) - 2 = 24 - 2 = 22 Completed Table: | Input | Output | | ----- | ------ | | 2 | 4 | | 5 | 13 | | 8 | 22 | Question 3: Solve for 'n': n + 7 = 15 Solution: Inspection: What number added to 7 equals 15?

Answer: n = 8, because 8 + 7 =

1

5. Question 4: Identify the variable and constant in the expression: 5y - 3 Solution: Variable: y Constant: 5 and -3 Explanation: The term '5y' represents 5 multiplied by a changing value 'y', so 'y' is the variable. '5' scales the value of y, while '-3' subtracts a fixed amount.