Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Identify and classify types of algebraic expressions.
• Simplify expressions by combining like terms.
• Multiply expressions by expanding brackets.
• Factorise expressions by taking out a common factor.
• Apply these concepts to solve basic word problems.

Lesson summary

Algebraic expressions are fundamental building blocks in mathematics, crucial for understanding more advanced topics like equations, functions, and calculus. In essence, they are mathematical phrases that combine numbers, variables (represented by letters), and operations like addition, subtraction, multiplication, and division. Mastering algebraic expressions is like learning the grammar of mathematics – it empowers you to translate real-world problems into mathematical language and solve them. Why does this matter in your daily life in South Africa?

Lesson notes

2.1 Defining Algebraic Expressions: An algebraic expression is a mathematical phrase that combines numbers, variables (symbols representing unknown values, typically letters like x, y, z, a, b, c), and arithmetic operations (+, -, ×, ÷).

Terms: Parts of an algebraic expression separated by + or - signs.

Constants: Numbers that stand alone in an expression (e.g., 5, -2, 1/2).

Variables: Letters representing unknown values (e.g., x, y, a).

Coefficients: Numbers multiplying variables (e.g., in 3x, 3 is the coefficient of x).

Exponents (Powers): Indicate repeated multiplication (e.g., in x², 2 is the exponent).

Examples: `3x + 5` (two terms: 3x and 5) `2y² - 7y + 1` (three terms: 2y², -7y, and 1) `4a` (one term – a monomial) `x/2 - 9` (two terms: x/2 and -9) 2.2 Classifying Algebraic Expressions: Based on the number of terms, we classify expressions as follows: Monomial: An expression with one term (e.g., 5x, -2y², 8).

Binomial: An expression with two terms (e.g., x + 3, 2a - b).

Trinomial: An expression with three terms (e.g., x² + 2x - 1, a + b + c).

Polynomial: A general term for an expression with one or more terms. All monomials, binomials, and trinomials are also polynomials. 2.3 Simplifying Algebraic Expressions: Combining Like Terms Like terms are terms that have the same variable(s) raised to the same power(s). Only like terms can be combined. To combine like terms, add or subtract their coefficients, keeping the variable part the same.

Example 1: Simplify `3x + 5y - x + 2y` Identify like terms: `3x` and `-x` are like terms. `5y` and `2y` are like terms.

Combine like terms: `(3x - x) + (5y + 2y) = 2x + 7y` Example 2: Simplify `4a² - 2a + 7 - a² + 5a - 3` Identify like terms: `4a²` and `-a²` are like terms. `-2a` and `5a` are like terms. `7` and `-3` are like terms.

Combine like terms: `(4a² - a²) + (-2a + 5a) + (7 - 3) = 3a² + 3a + 4` 2.4 Multiplying Algebraic Expressions: Expanding Brackets (Distributive Property) The distributive property states that a(b + c) = ab + ac. This means you multiply the term outside the brackets by each term inside the brackets.

Example 1: Expand `2(x + 3)` `2(x + 3) = 2 x + 2 * 3 = 2x + 6` Example 2: Expand `-3(2y - 5)` `-3(2y - 5) = -3 2y + (-3) * (-5) = -6y + 15` Example 3: Expand `x(x - 4)` `x(x - 4) = x x - x * 4 = x² - 4x` Example 4: Expand `(x + 2)(x + 3)` This requires using the distributive property twice (often remembered as the FOIL method: First, Outer, Inner, Last): `(x + 2)(x + 3) = x(x + 3) + 2(x + 3)` `= x² + 3x + 2x + 6` `= x² + 5x + 6` 2.5 Factorising Algebraic Expressions: Taking Out a Common Factor Factorising is the reverse of expanding brackets. You find the highest common factor (HCF) of all the terms in the expression and take it out as a common factor.

Example 1: Factorise `6x + 9` Find the HCF of 6 and 9: The HCF is

3. Take out the common factor: `6x + 9 = 3(2x + 3)` Example 2: Factorise `4y² - 8y` Find the HCF of 4y² and 8y: The HCF is 4y.

Take out the common factor: `4y² - 8y = 4y(y - 2)` Example 3: Factorise `12a³b + 18a²b²` Find the HCF of 12a³b and 18a²b²: The HCF is 6a²b.

Take out the common factor: `12a³b + 18a²b² = 6a²b(2a + 3b)` Guided Practice (With Solutions)

Question 1: Simplify the expression: `5x - 3y + 2x + 7y - x` Solution: Identify like terms: `5x`, `2x`, and `-x` are like terms. `-3y` and `7y` are like terms.

Combine like terms: `(5x + 2x - x) + (-3y + 7y) = 6x + 4y`

Commentary: Remember to pay attention to the signs (+ or -) in front of each term. The order of terms does not affect the final result as long as the signs are correctly accounted for.

Question 2: Expand the expression: `4(2a - 5b + 1)` Solution: Use the distributive property: `4 2a + 4 (-5b) + 4 1` Simplify: `8a - 20b + 4`

Commentary: Each term inside the bracket must be multiplied by the term outside. It’s crucial to remember the rules of multiplying with negative numbers.

Question 3: Factorise the expression: `10p - 15q` Solution: Find the HCF of 10 and 15: The HCF is

5. Take out the common factor: `10p - 15q = 5(2p - 3q)`

Commentary: Always check if you can factorise further after taking out the initial common factor. In this case, 2p and 3q have no common factors.

Question 4: Expand and simplify: `2(x + 3) - (x - 1)` Solution: Expand the first bracket: `2(x + 3) = 2x + 6` Expand the second bracket: `-(x - 1) = -x + 1` Combine like terms: `2x + 6 - x + 1 = (2x - x) + (6 + 1) = x + 7`

Commentary: The negative sign in front of the second bracket affects every term inside the bracket. It’s a common mistake to forget to distribute the negative sign to both terms. Independent Practice (Questions Only)

Simplify: `7a + 4b - 2a - b + 5` Simplify: `3x² - 2x + 5 - x² + 4x - 1` Expand: `5(3y - 2)` Expand: `-2(4p + 1 - q)` Expand and simplify: `3(a - 2) + 2(a + 1)` Expand and simplify: `4(x + 5) - (2x - 3)` Factorise: `8m + 12n` Factorise: `9x² - 6x` Factorise: `5ab + 10ac - 15ad` Expand: (a + 4)(a - 1)