Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and describe number and geometric patterns.
• Find the rule for a pattern and use it to find missing terms.
• Use flow diagrams and tables to show relationships.
• Understand and use variables in simple algebraic expressions.
• Write and find the value of algebraic expressions.

Lesson summary

This week, we delve into the fascinating world of patterns, functions, and simple algebraic expressions. Understanding these concepts is crucial for building a strong foundation in mathematics and for solving everyday problems. Patterns are everywhere around us, from the design of traditional Zulu beadwork to the layout of houses in a township. Functions help us understand how things change in relation to each other (e.g., how the price of airtime increases with the amount you buy). Algebraic expressions are like secret codes that use letters to represent unknown quantities, allowing us to solve complex problems in a simplified manner.

Lesson notes

Number Patterns (Sequences): A number pattern (or sequence) is an ordered list of numbers that follow a specific rule. We need to identify this rule to predict the next numbers in the sequence.

Increasing Patterns:* The numbers get larger. The rule often involves addition or multiplication.

Decreasing Patterns:* The numbers get smaller. The rule often involves subtraction or division.

Geometric Patterns:* These patterns involve repeating shapes or figures.

Example 1 (Increasing): 2, 4, 6, 8, ___ , ___ The rule is to add 2 to the previous number. The next two numbers are 10 and

1

2. Example 2 (Decreasing): 20, 17, 14, 11, ___ , ___ The rule is to subtract 3 from the previous number. The next two numbers are 8 and

5. Example 3 (Geometric): Triangle, Square, Pentagon, Hexagon, ___ The pattern is that each shape adds one side. The next shape is a Heptagon (7 sides). Functions and Relationships (Flow Diagrams and Tables): A function describes the relationship between an input (the number we start with) and an output (the number we get after applying a rule). We can represent functions using flow diagrams and tables.

Flow Diagram: Input --> (Rule: + 5) --> Output If the input is 3, the output is 3 + 5 =

8. If the input is 10, the output is 10 + 5 =

1

5. Table: | Input | Output | |-------|--------| | 1 | 6 | | 2 | 7 | | 3 | 8 | | 4 | 9 | The rule for this table is to add 5 to the input.

Example 4: A taxi charges R15 as a call-out fee and then R5 per kilometre. Let's represent this relationship using a table: | Distance (km) | Cost (R) | |-----------------|----------| | 0 | 15 | | 1 | 20 | (15 + 5*1) | 2 | 25 | (15 + 5*2) | 3 | 30 | (15 + 5*3)

Algebraic Expressions: An algebraic expression is a mathematical phrase that combines numbers, variables (letters representing unknown numbers), and operations (addition, subtraction, multiplication, division).

Variable:* A variable is a letter (usually x, y, or z) that represents an unknown number.

Coefficient:* The number in front of a variable is called the coefficient. For example, in the expression 3x, the coefficient is

3. Constant:* A number without a variable is called a constant. For example, in the expression 2x + 5, the constant is

5. Writing Algebraic Expressions: "A number increased by 7" can be written as x + 7. "Three times a number" can be written as 3x. "A number divided by 2" can be written as x/2 or x ÷ 2. "Five less than a number" can be written as x -

5. Evaluating Algebraic Expressions (Substitution): To evaluate an algebraic expression, we substitute a given value for the variable and then perform the operations.

Example 5: Evaluate the expression 2x + 3 when x =

4. Substitute x = 4: 2(4) + 3 Multiply: 8 + 3 Add: 11 Example 6: Evaluate the expression 5y - 10 when y =

2. Substitute y = 2: 5(2) - 10 Multiply: 10 - 10 Subtract: 0 Guided Practice (With Solutions)

Question 1: Find the next two terms in the following number pattern: 3, 7, 11, 15, ___, ___ Solution: Identify the pattern: The pattern is to add 4 to the previous term.

Find the next two terms: 15 + 4 = 19, 19 + 4 = 23 Answer: 19, 23

Commentary:* We identified the constant difference between terms (4) to extrapolate the next values.

Question 2: Complete the following flow diagram: Input --> (Rule: x 3 - 2) --> Output Input: 4, Output: ?

Input: 7, Output: ?

Solution: For Input 4: 4 x 3 - 2 = 12 - 2 =

1

0. Output =

1

0. For Input 7: 7 x 3 - 2 = 21 - 2 =

1

9. Output =

1

9. Answer: Input 4, Output 10; Input 7, Output 19

Commentary:* We carefully followed the order of operations (multiplication before subtraction) when applying the rule.

Question 3: Write an algebraic expression for "The sum of a number and twice that number." Solution: Let the number be x. Twice the number is 2x. The sum of the number and twice that number is x + 2x.

Answer: x + 2x (which can be simplified to 3x)

Commentary:* We broke down the verbal phrase into its component parts before translating it into an algebraic expression. Simplifying is a good habit!

Question 4: Evaluate the expression 4a - 6 when a =

5. Solution: Substitute a = 5: 4(5) - 6 Multiply: 20 - 6 Subtract: 14 Answer: 14

Commentary:* Again, attention to order of operations (multiplication before subtraction) is critical.

Question 5: A vendor at the market sells papayas. Each papaya costs R

8. Write an expression for the total cost of buying 'p' papayas. If someone buys 6 papayas, what will they pay?

Solution: The cost of one papaya = R8 If 'p' = the number of papayas purchased, then total cost of papayas is 8 p or 8p If p = 6, then total cost will be 86 = 48 Answer: 8p; if p = 6, total cost = R48 Independent Practice (Questions Only) Find the next two terms in the following number pattern: 5, 10, 15, 20, ___, ___ Find the next two terms in the following number pattern: 30, 26, 22, 18, ___, ___ Complete the following flow diagram: Input --> (Rule: ÷ 2 + 1) --> Output Input: 10, Output: ?

Input: 24, Output: ?