Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Simplify expressions by combining like terms.
• Expand expressions by multiplying terms.
• Factorise using the Highest Common Factor (HCF).
• Factorise using the Difference of Two Squares.
• Apply the laws of exponents.

Lesson summary

Algebraic expressions and factorisation are fundamental building blocks in mathematics. They are not just abstract concepts but are essential tools for problem-solving in various real-life scenarios. For South African learners, understanding these concepts can help in budgeting, understanding financial transactions, calculating areas for building projects (like RDP houses), and even in understanding basic programming. For example, understanding how to factorise can help you understand how to divide up resources fairly or how to optimize costs when buying in bulk. Without these skills, learners will struggle with more advanced mathematical concepts in later grades and beyond.

Lesson notes

2.1 What are Algebraic Expressions? An algebraic expression is a mathematical phrase that contains numbers, variables (represented by letters like x, y, a, b), and operation symbols (+, -, ×, ÷). For example, 3x + 5, 2a - b, and x 2 + 4x + 3 are algebraic expressions. A term is a single number or variable, or numbers and variables multiplied together. In 3x + 5, the terms are 3x and 5. 2.2 Like Terms: Like terms are terms that have the same variable(s) raised to the same power(s). Only like terms can be combined. For instance, 3x and 5x are like terms, but 3x and 5x 2 are not. Combining like terms involves adding or subtracting their coefficients (the number in front of the variable).

Example: Simplify 4a + 2b - a + 6b.

Solution: Group like terms: (4a - a) + (2b + 6b).

Simplify: 3a + 8b. 2.3 Expanding Algebraic Expressions: Expanding an algebraic expression involves removing brackets by multiplying each term inside the brackets by the term outside.

The distributive property is crucial here: a(b + c) = ab + ac.

Example: Expand 3(x + 2).

Solution: 3(x + 2) = 3 x + 3 2 = 3x +

6. Example: Expand -2y(3y - 1).

Solution: -2y(3y - 1) = -2y 3y - (-2y 1) = -6y 2 + 2y. Remember that multiplying a negative by a negative gives a positive. 2.4 Factorisation: Factorisation is the reverse process of expansion. It involves writing an algebraic expression as a product of its factors. 2.4.1 Highest Common Factor (HCF): The highest common factor (HCF) is the largest factor that divides into all terms of an expression. To factorise using the HCF, find the HCF of all the terms and then divide each term by the HC

F. Example: Factorise 6x +

1

2. Solution: The HCF of 6x and 12 is

6. Divide each term by 6: 6x/6 + 12/6 = x +

2. Therefore, 6x + 12 = 6(x + 2).

Example: Factorise 15a 2 - 10a.

Solution: The HCF of 15a 2 and 10a is 5a.

Divide each term by 5a: 15a 2 / (5a) - 10a / (5a) = 3a -

2. Therefore, 15a 2 - 10a = 5a(3a - 2). 2.4.2 Difference of Two Squares: An expression in the form a 2 - b 2 is called the difference of two squares. It factorises as (a + b)(a - b).

Example: Factorise x 2 -

9. Solution: Here, a = x and b = 3 (since 9 = 3 2 ).

Therefore, x 2 - 9 = (x + 3)(x - 3).

Example: Factorise 4y 2 -

2

5. Solution: Here, a = 2y (since 4y 2 = (2y) 2 ) and b = 5 (since 25 = 5 2 ).

Therefore, 4y 2 - 25 = (2y + 5)(2y - 5). 2.5 Laws of Exponents: These laws are essential for simplifying expressions with exponents: x m x n = x m+n (Product of powers) x m / x n = x m-n (Quotient of powers) (x m ) n = x mn (Power of a power) (xy) n = x n y n (Power of a product) (x/y) n = x n /y n (Power of a quotient) x 0 = 1 (Any non-zero number to the power of 0 equals 1) x -n = 1/x n (Negative exponent)

Example: Simplify a 3 a 2 .

Solution: a 3 a 2 = a 3+2 = a 5 .

Example: Simplify (2x 2 ) 3 .

Solution: (2x 2 ) 3 = 2 3 (x 2 ) 3 = 8x 6 . Guided Practice (With Solutions)

Question 1: Simplify the expression: 5x + 3y - 2x + y.

Solution: Group like terms: (5x - 2x) + (3y + y).

Combine like terms: 3x + 4y.

Commentary: This question tests the understanding of identifying and combining like terms. The key is to ensure that only terms with the same variable are added or subtracted.

Question 2: Expand the expression: 4a(2a - 3).

Solution: Apply the distributive property: 4a 2a - 4a

3. Simplify: 8a 2 - 12a.

Commentary: This question checks the ability to expand an algebraic expression using the distributive property. Pay close attention to signs when multiplying.

Question 3: Factorise the expression: 9x 2 - 15x.

Solution: Find the HCF of 9x 2 and 15x. The HCF is 3x.

Divide each term by 3x: (9x 2 )/(3x) - (15x)/(3x) = 3x -

5. Write the factorised expression: 3x(3x - 5).

Commentary: This tests the understanding of factorisation by finding the highest common factor. Ensure you identify the largest factor common to all terms, including variables.

Question 4: Factorise the expression: x 2 -

1

6. Solution: Recognise that the expression is a difference of two squares: x 2 - 4 2 . Apply the difference of two squares formula: (x + 4)(x - 4).

Commentary: This tests the recognition and application of the difference of two squares factorisation. Identify the terms being squared.

Question 5: Simplify the expression: (2x 3 ) 2 / x 4 Solution: Apply power of a product: (2x 3 ) 2 = 2 2 (x 3 ) 2 = 4x 6 Divide powers with the same base: 4x 6 / x 4 = 4x 6-4 = 4x 2

Commentary: This question tests understanding of both the power of a product rule and division of powers with the same base. Independent Practice (Questions Only)

Simplify: 7y - 4x + 2y + 6x.

Expand: -5b(4b + 2).

Factorise: 12p 2 + 18p.

Factorise: a 2 -

4

9. Simplify: (3m 2 ) 3 / m 5 .

Expand: 2(x+3) - 4(x-1).

Factorise: 25x 2 - 1 Simplify: (4a 2 b 3 )(2ab -1 )

Expand: (x + 2)(x -3)

Factorise: 3x 2 + 6x