Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify patterns in sequences.
• Determine the rule for a given pattern.
• Use input and output values to find a function's rule.
• Write number sentences from word problems.
• Solve simple number sentences using trial and improvement.

Lesson summary

This week, we dive deeper into patterns, functions, and simple algebraic expressions. Understanding these concepts is crucial because they form the foundation for more advanced mathematical thinking. Patterns are everywhere – from the repeating designs in traditional Ndebele art to the sequences in our financial lives (like weekly pocket money or saving for a soccer ball). Functions describe how things change in relation to each other, such as how the number of loaves of bread baked relates to the amount of flour used. Algebraic expressions are a powerful way to represent these relationships in a concise and general way.

Lesson notes

2.1 Patterns: A pattern is a regular, repeated, or recurring form or design. Patterns can be found everywhere in our world, from the shapes of snowflakes to the arrangement of bricks in a wall. In mathematics, a pattern is a sequence that repeats according to a specific rule. We can have number patterns and geometric patterns.

Number Patterns: These are sequences of numbers that follow a specific rule.

For example: 2, 4, 6, 8, 10... (each term is obtained by adding 2 to the previous term).

Example 1: Consider the sequence: 3, 7, 11, 15, ... What is the rule? What is the next term?

Solution: We observe that we add 4 to each term to get the next term. (7-3 = 4, 11-7 = 4, 15-11 = 4). So, the rule is "Add 4". The next term is 15 + 4 =

1

9. Example 2: Consider the sequence: 1, 4, 9, 16, ... What is the rule? What is the next term?

Solution: These are square numbers (1x1, 2x2, 3x3, 4x4, ...). The rule is "Square the natural numbers". The next term is 5x5 =

2

5. Geometric Patterns: These are sequences of shapes or figures that follow a specific rule. For example, a pattern could start with a square, then a triangle, then a square, then a triangle, and so on.

Example 3: Suppose we have a pattern: Circle, Square, Circle, Square, ... What is the next shape?

Solution: The pattern repeats 'Circle, Square'. So the next shape is a Circle. 2.2 Functions: A function describes a relationship between two sets of values: an input and an output. The function specifies how the input is transformed to produce the output.

Think of it as a machine: you put something in (the input), the machine does something to it according to a specific rule (the function), and then something comes out (the output).

Example 4: Consider the function: "Multiply by 3". If the input is 2, what is the output?

Solution: Input =

2. Rule = Multiply by

3. Output = 2 x 3 =

6. Example 5: Complete the following function table: | Input (x) | Output (y) | |---|---| | 1 | 5 | | 2 | 6 | | 3 | 7 | | 4 | ? | What is the rule?

Solution: We notice that the output is always 4 more than the input. So, the rule is "Add 4".

Therefore, when the input is 4, the output is 4 + 4 = 8. 2.3 Simple Algebraic Expressions: An algebraic expression is a combination of numbers, variables (letters that represent unknown values), and mathematical operations (like +, -, x, ÷). A simple algebraic expression often involves only one operation and one variable.

Example 6: Represent the following statement as an algebraic expression: "A number increased by 5".

Solution: Let the number be represented by the variable 'x'. Then the expression is x +

5. Example 7: Represent the following statement as an algebraic expression: "A number multiplied by 2, then decreased by 3".

Solution: Let the number be 'y'. The expression is 2 x y - 3 (or simply 2y - 3). 2.4 Solving Simple Number Sentences (Equations) with Trial and Improvement: A number sentence (or equation) states that two expressions are equal. Solving an equation means finding the value of the variable that makes the equation true. The trial and improvement method involves guessing a value for the variable, substituting it into the equation, and checking if the equation is true. If not, we adjust our guess and try again until we find the correct value.

Example 8: Solve the equation: x + 3 = 7, using trial and improvement.

Solution: Trial 1:* Let x =

2. Then 2 + 3 = 5. 5 is not equal to 7. (Too low)

Trial 2:* Let x =

5. Then 5 + 3 = 8. 8 is not equal to 7. (Too high)

Trial 3:* Let x =

4. Then 4 + 3 = 7. 7 is equal to

7. So, x =

4. Example 9: Solve the equation: 2 x y = 10, using trial and improvement.

Solution: Trial 1:* Let y =

3. Then 2 x 3 = 6. 6 is not equal to 10. (Too low)

Trial 2:* Let y =

6. Then 2 x 6 = 12. 12 is not equal to 10. (Too high)

Trial 3:* Let y =

5. Then 2 x 5 = 10. 10 is equal to

1

0. So, y =

5. Guided Practice (With Solutions)

Question 1: Identify the pattern and find the next term in the sequence: 5, 10, 15, 20, ...

Solution: The pattern is that each term increases by 5. (10-5 = 5, 15-10 = 5, 20-15 = 5). The rule is "Add 5". The next term is 20 + 5 =

2

5. Question 2: Complete the following function table and state the rule: | Input (a) | Output (b) | |---|---| | 2 | 4 | | 3 | 6 | | 4 | 8 | | 5 | ? | Solution: We observe that the output is double the input. (2x2 = 4, 3x2 = 6, 4x2 = 8). The rule is "Multiply by 2".

Therefore, when the input is 5, the output is 5 x 2 =

1

0. Question 3: Write an algebraic expression for the following: "A number divided by 4".

Solution: Let the number be 'n'. The algebraic expression is n ÷ 4 (or n/4).

Question 4: Solve the equation: p - 2 = 5, using trial and improvement.

Solution: Trial 1: Let p =

6. Then 6 - 2 = 4. 4 is not equal to 5. (Too low)

Trial 2: Let p =

8. Then 8 - 2 = 6. 6 is not equal to 5. (Too high)

Trial 3: Let p =

7. Then 7 - 2 = 5. 5 is equal to

5. Therefore, p = 7.