Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and describe numerical patterns.
• Represent patterns using flow diagrams.
• Use input and output values to find a rule.
• Express a rule using a simple algebraic expression.
• Evaluate expressions by substitution.

Lesson summary

This week, we delve deeper into the fascinating world of patterns, functions, and simple algebraic expressions. Understanding these concepts is crucial because they are the building blocks for more advanced mathematical thinking and problem-solving.

Think about it: predicting stock prices, understanding how a recipe scales, or even designing a garden – all rely on recognizing and using patterns and relationships. In South Africa, understanding these concepts can help in various fields, from farming (predicting crop yields based on rainfall patterns) to business (forecasting sales based on past trends). These skills empower us to make informed decisions and solve real-world problems.

Lesson notes

2.1 Numerical Patterns and Sequences A numerical pattern is a sequence of numbers that follows a specific rule. Each number in the sequence is called a term. To understand a pattern, we need to identify the rule that connects the terms.

Example: 2, 4, 6, 8, 10… Rule: Add 2 to the previous term.

Example: 1, 4, 9, 16, 25… Rule: Square the term number (1st term is 1 squared, 2nd term is 2 squared, etc.) 2.2 Flow Diagrams and Number Sentences A flow diagram is a visual way to represent a pattern or function. It shows how an input value is transformed into an output value based on a specific rule. The input value goes into the rule, and the output value flows out.

Example: Input → Rule (+3) → Output If the input is 5, then the output is 5 + 3 = 8 A number sentence is a mathematical statement that expresses the relationship between numbers and operations. We can use number sentences to describe the rule in a flow diagram. Example (corresponding to the flow diagram above): Output = Input + 3 2.3 Finding the Rule To find the rule in a pattern, look for the relationship between the input and output values.

Example: | Input | Output | |---|---| | 1 | 5 | | 2 | 6 | | 3 | 7 | What's happening to the input to get the output? It looks like we're adding

4. Let's check if that works for all pairs. 1 + 4 = 5 (Correct) 2 + 4 = 6 (Correct) 3 + 4 = 7 (Correct) Therefore, the rule is "Add 4".

The flow diagram would be: Input → Rule (+4) → Output.

The number sentence would be: Output = Input + 4. 2.4 Simple Algebraic Expressions An algebraic expression uses letters (called variables) to represent unknown numbers. We can use variables to generalize patterns. For example, we can use the letter "n" to represent any number. If the rule is "Multiply by 2", we can write the algebraic expression as 2 x n (or simply 2n).

Example: Consider the rule "Multiply by 3 and then add 1".

The algebraic expression would be: 3n + 1 This means: Take any number 'n', multiply it by 3, and then add 1. 2.5 Evaluating Algebraic Expressions by Substitution Evaluating an algebraic expression means finding its value by substituting a given number for the variable.

Example: Evaluate the expression 2n + 5 when n =

3. Substitute: Replace "n" with 3: 2(3) + 5 Multiply: 2 x 3 = 6 Add: 6 + 5 = 11 Therefore, the value of the expression 2n + 5 when n = 3 is

1

1. Example (South African context): A farmer sells mangoes at R5 each. The total amount of money he makes can be represented by the expression 5n, where 'n' is the number of mangoes he sells. If he sells 10 mangoes, what is the total amount he makes? Substitute n = 10 into the expression: 5(10) = R50 He makes R

5

0. Guided Practice (With Solutions)

Question 1: Identify the rule in the following pattern: 5, 10, 15, 20, 25… Solution: The pattern is increasing by 5 each time.

Rule: Add 5 to the previous term.

Question 2: Complete the following flow diagram: Input: 7 → Rule: (x 4) → Output: ?

Solution: Multiply the input (7) by 4: 7 x 4 = 28 Output: 28 Question 3: Find the rule for the following input/output table: | Input | Output | |---|---| | 2 | 7 | | 4 | 9 | | 6 | 11 | Solution: Let's examine the relationship. It appears we are adding something to the input. To figure out what that something is, we can work backward. If we are adding to the input to get the output, that means subtracting the input from the output will tell us how much we added.

For example: 7-2 = 5. 9-4 = 5. 11-6=

5. So the rule is Add 5? Nope. That doesn't work. Let's try another approach. Perhaps it is multiplying by some value and adding something else. Let's assume it's "multiply by 2 and add something". Then we'd have 2*2 + x =

7. This becomes 4+x=

7. So x=

3. The rule might be "Multiply by 2 and add 3". Let's try it with all the pairs. 2*2 + 3 = 4 + 3 = 7 4*2 + 3 = 8 + 3 =

1

1. Oops. Doesn't work. 6*2 + 3 = 12 + 3 =

1

5. Also doesn't work. Let's try a different rule then. Maybe each input adds half of itself and then adds 2.5? So INPUT + (Input/2) + 2.5 = Output 2 + 1 + 2.5 = 5.

5. NOPE. Let's re-think this one... the difference between input and output changes! So the rule is not something we are simply adding or subtracting. What if the rule is multiply by 1 AND add 5? Input * 1 + 5 2*1 + 5 = 7 4*1 + 5 = 9 6*1 + 5 = 11 This WORK

S. So the rule is Input * 1 +

5. The rule is "Multiply by 1 and add 5".

Question 4: Write an algebraic expression for the following: "Multiply a number by 4 and subtract 2".

Solution: Let 'n' represent the number.

Algebraic expression: 4n - 2 Question 5: Evaluate the expression 3n + 1 if n =

5. Solution: Substitute n = 5 into the expression: 3(5) + 1 Multiply: 3 x 5 = 15 Add: 15 + 1 = 16 Therefore, the value of the expression is

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6. Independent Practice (Questions Only) What is the next number in the following pattern: 3, 6, 9, 12, ____?

Describe the rule for the pattern: 1, 3, 5, 7, 9… Complete the flow diagram: Input: 9 → Rule: (- 6) → Output: ?

Complete the flow diagram: Input: ?