Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and describe number patterns.
• Determine the rule for a simple function using flow diagrams and tables.
• Write simple algebraic expressions.
• Substitute values into simple algebraic expressions.

Lesson summary

This week, we're diving into the exciting world of patterns, functions, and simple algebraic expressions! Understanding these concepts is crucial because they form the building blocks for more advanced mathematics and are used every day, whether you realise it or not. From calculating the cost of airtime to understanding how a recipe scales up or down, patterns and functions are everywhere. Algebraic expressions help us describe these patterns in a concise and powerful way. Imagine you are helping your family budget for groceries – understanding patterns in pricing and using simple expressions to calculate total costs becomes invaluable!

Lesson notes

a)

Number Patterns: A number pattern is a sequence of numbers that follows a specific rule. These rules often involve adding, subtracting, multiplying, or dividing by a constant value. We need to identify this rule to predict the next numbers in the pattern.

Constant Difference (Arithmetic Sequence): The difference between consecutive terms is always the same.

Example: 2, 5, 8, 11, ... (The constant difference is +3)

Constant Ratio (Geometric Sequence): Each term is multiplied by the same value to get the next term.

Example: 3, 6, 12, 24, ... (The constant ratio is x2)

Example 1 (Arithmetic): Consider the sequence: 7, 12, 17, 22, ... To find the rule, calculate the difference between consecutive terms: 12 - 7 = 5 17 - 12 = 5 22 - 17 = 5 The constant difference is +

5. Therefore, the rule is "add 5 to the previous term." The next two terms in the sequence would be: 22 + 5 = 27 and 27 + 5 =

3

2. Example 2 (Geometric): Consider the sequence: 4, 12, 36, 108, ... To find the rule, try dividing consecutive terms: 12 / 4 = 3 36 / 12 = 3 108 / 36 = 3 The constant ratio is x

3. Therefore, the rule is "multiply the previous term by 3." The next two terms in the sequence would be: 108 x 3 = 324 and 324 x 3 = 972. b) Functions and Relationships (Flow Diagrams and Tables): A function describes a relationship between two sets of numbers: an input and an output. We can represent functions using flow diagrams and tables. The rule specifies how to transform the input into the output.

Flow Diagram: A visual representation of the function, showing the input, the rule, and the output. Input → Rule → Output Table: Organizes the input and corresponding output values in rows or columns. | Input (x) | Output (y) | |-----------|------------| | 1 | 4 | | 2 | 5 | | 3 | 6 | Example 3 (Function): Rule: Multiply by 2 and add

1. Flow Diagram: 3 → x2 + 1 → 7 5 → x2 + 1 → 11 Table: | Input (x) | Output (y) | |-----------|------------| | 1 | 3 | | 2 | 5 | | 3 | 7 | | 4 | 9 | To find the output for an input of 4, we apply the rule: (4 x 2) + 1 = 9 c)

Simple Algebraic Expressions: An algebraic expression uses variables (letters that represent unknown quantities) and constants (numbers) combined with mathematical operations (+, -, x, /).

Variable: A symbol (usually a letter) that represents an unknown number. For example, x, y, n.

Constant: A fixed number. For example, 5, -3, ½.

Example 4: "A number increased by 7" can be written as x + 7 "Twice a number" can be written as 2n "A number divided by 3" can be written as y / 3 or y ÷ 3 d)

Substituting into Algebraic Expressions: To find the value of an algebraic expression, we substitute (replace) the variables with their given numerical values and then perform the operations.

Example 5: Find the value of the expression 3a + 2b if a = 4 and b =

5. Substitute a with 4 and b with 5: 3(4) + 2(5)

Perform the multiplication: 12 + 10 Perform the addition: 22 Therefore, the value of the expression is

2

2. Example 6: Sipho earns R rand per hour. Write an expression for how much he earns in 8 hours. Then, calculate his earnings if R =

2

5. Expression: 8 R or 8R Substitution: 8 25 Calculation: 200 Sipho earns R200 in 8 hours. Guided Practice (With Solutions)

Question 1: Complete the following number pattern: 3, 8, 13, __, __. What is the rule?

Solution: Find the difference between consecutive terms: 8 - 3 = 5, 13 - 8 =

5. The constant difference is +

5. The rule is "add 5 to the previous term." The next two terms are: 13 + 5 = 18 and 18 + 5 =

2

3. Answer: 3, 8, 13, 18,

2

3. The rule is "add 5." Question 2: Use the following function rule to complete the table: Input x 3, then subtract 2. | Input (x) | Output (y) | |-----------|------------| | 5 | | | 8 | | | 10 | | Solution: Apply the rule to each input value: x = 5: (5 x 3) - 2 = 15 - 2 = 13 x = 8: (8 x 3) - 2 = 24 - 2 = 22 x = 10: (10 x 3) - 2 = 30 - 2 = 28 Complete the table: | Input (x) | Output (y) | |-----------|------------| | 5 | 13 | | 8 | 22 | | 10 | 28 | Question 3: Write an algebraic expression for: "A number decreased by 9." Then, find the value of the expression if the number is

1

5. Solution: Let the number be represented by the variable n. "Decreased by 9" means subtract

9. The algebraic expression is: n - 9 Substitute n with 15: 15 - 9 Calculate: 6 Answer: The expression is n -

9. The value is

6. Question 4: Write an algebraic expression for: "Half of a number, plus 4." Solution: Let the number be represented by the variable y. "Half of a number" means divide by 2: y / 2 "Plus 4" means add

4. The algebraic expression is: y / 2 + 4 Independent Practice (Questions Only)

Complete the number pattern: 1, 4, 9, 16, __, __. Describe the rule for the following number pattern: 2, 6, 18, 54, … Use the following function rule to complete the table: Input + 5, then divide by 2.