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Algebra (8Ae)

Mathematics Stage 8 Chapter Notes

What this chapter covers

Algebra - Expressions, equations and formulae
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Algebra (8Ae) notes

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In this chapter

  1. 1Expressions, Equations, and Formulae
  2. 2Substitution and Order of Operations
  3. 3Expanding and Factorising
  4. 4Forming Algebraic Expressions

1. Expressions, Equations, and Formulae

It is crucial to understand the difference between the three fundamental building blocks of algebra. An 'expression' is a combination of numbers and letters (variables) with mathematical operations, like `4x - 7` or `5a^2 + b`. It doesn't have an equals sign. An 'equation' states that two expressions are equal. It always has an equals sign and can be solved to find a specific value for the variable(s), for example, `4x - 7 = 13`. A 'formula' is a special type of equation that shows a relationship between different variables, like the formula for the area of a rectangle, `A = l × w`.

A = l × w (Formula for the area of a rectangle)

P = 2(l + w) (Formula for the perimeter of a rectangle)

Key term

Expression: A mathematical phrase containing numbers, variables, and operators, but no equals sign.

Examiner insight

Examiners reward clear understanding of these definitions. In 'show that' questions, you might form an expression first and then turn it into an equation to be solved.

Common pitfall

A common mistake is trying to 'solve' an expression. You can only solve equations. Expressions can only be simplified or evaluated if you are given the value of the variables.

Worked example 13 marks

State whether each of the following is an expression, an equation, or a formula.(a) `7x + 2`(b) `C = 2πr`(c) `7x + 2 = 23`

  1. 1

    (a) `7x + 2` has numbers, variables and operators but no equals sign. Therefore, it is an expression.

  2. 2

    (b) `C = 2πr` shows the relationship between the circumference (C) and radius(r) of a circle. It is a formula.

  3. 3

    (c) `7x + 2 = 23` has an equals sign and can be solved to find the value of x. Therefore, it is an equation.

Recap

  • An expression can be evaluated or simplified, but not solved.
  • An equation contains an equals sign and can be solved.
  • A formula is an equation that describes a relationship between variables.
  • Variables in expressions and formulae can take many values, but in an equation they represent a specific unknown value.

Quick check

  1. Is `y = mx + c` an expression, equation or formula?1 mark
  2. Write down an example of an equation.1 mark

2. Substitution and Order of Operations

Substitution is the process of replacing letters (variables) in an expression or formula with their given numerical values. Once you substitute, you must calculate the answer using the correct order of operations, often remembered by the acronym BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction). Always be extra careful when substituting negative numbers, especially with powers. For example, if `x = -3`, then `x^2` means `(-3)^2 = 9`, not `-3^2 = -9`.

Key term

Substitution: The process of replacing a variable in an algebraic expression or formula with a specific numerical value.

Examiner insight

Examiners often use questions involving substitution with negative numbers to test your attention to detail and understanding of BIDMAS. Show every step of your substitution and calculation to secure partial marks even if your final answer is wrong.

Common pitfall

The most frequent error is with negative numbers. For `x = -5`, `2x^2` is `2 × (-5)^2 = 2 × 25 = 50`. A common mistake is to calculate `(2 × -5)^2 = (-10)^2 = 100` or `2 × -25 = -50`.

Fun fact

The rules for order of operations are a global convention, ensuring that any mathematician or scientist anywhere in the world will get the exact same result from the same calculation.

Worked example 12 marks

If `p = 5` and `q = -2`, find the value of `3p - q^2`.

  1. 1

    Start with the expression: `3p - q^2`

  2. 2

    Substitute the given values: `p = 5` and `q = -2`. Use brackets for the negative number: `3(5) - (-2)^2`.

  3. 3

    Follow BIDMAS. First, calculate the Index (power): `(-2)^2 = 4`.

  4. 4

    The expression becomes: `3(5) - 4`.

  5. 5

    Next, do the Multiplication: `3(5) = 15`.

  6. 6

    The expression is now: `15 - 4`.

  7. 7

    Finally, do the Subtraction: `15 - 4 = 11`.

Worked example 23 marks

The formula for the surface area of a prism is `A = 2x^2 + 4xh`. Find the value of A when `x = 6` and `h = 2`.

  1. 1

    Write down the formula: `A = 2x^2 + 4xh`.

  2. 2

    Substitute `x = 6` and `h = 2`: `A = 2(6)^2 + 4(6)(2)`.

  3. 3

    Follow BIDMAS. Calculate the Index first: `6^2 = 36`.

  4. 4

    The formula becomes: `A = 2(36) + 4(6)(2)`.

  5. 5

    Perform the multiplications: `2(36) = 72` and `4(6)(2) = 48`.

  6. 6

    The formula is now: `A = 72 + 48`.

  7. 7

    Perform the addition: `A = 120`.

Recap

  • Always follow the order of operations (BIDMAS).
  • Use brackets when substituting negative numbers to avoid errors.
  • Remember that `(-a)^2` is positive, while `-a^2` is negative.
  • Calculate the value of terms involving powers before multiplying them by other numbers.

Quick check

  1. Find the value of `5a + 2b` when `a = 3` and `b = -4`.2 marks

3. Expanding and Factorising

Expanding and factorising are opposite processes. 'Expanding' means removing brackets by multiplying everything inside the bracket by the term outside. For example, `3(x + 5)` expands to `3 × x + 3 × 5 = 3x + 15`. This is using the distributive law. 'Factorising' is the reverse. You find the Highest Common Factor (HCF) of all the terms in an expression and 'take it out', placing it outside a new bracket. For example, to factorise `12y + 18`, the HCF of `12y` and `18` is 6. So, `12y + 18 = 6(2y + 3)`.

a(b + c) = ab + ac

Key term

Factorise: To express an algebraic expression as a product of its factors by taking out the highest common factor.

Examiner insight

The instruction 'Factorise fully' is important. Examiners are checking if you can identify the highest common factor for numbers and for each variable present.

Common pitfall

When factorising, students sometimes take out a common factor that is not the *highest* common factor. For `12x + 18`, factorising to `2(6x + 9)` or `3(4x + 6)` would not get full marks. The answer must be `6(2x + 3)`.

Worked example 12 marks

Expand and simplify `4x(2x - 5)`.

  1. 1

    To expand, multiply the term outside the bracket, `4x`, by each term inside the bracket.

  2. 2

    First term: `4x × 2x = 8x^2`.

  3. 3

    Second term: `4x × -5 = -20x`.

  4. 4

    Combine the results: `8x^2 - 20x`.

Worked example 23 marks

Factorise fully `15a^2b - 25ab^3`.

  1. 1

    Find the Highest Common Factor (HCF) of the terms `15a^2b` and `25ab^3`.

  2. 2

    HCF of numbers 15 and 25 is 5.

  3. 3

    HCF of `a^2` and `a` is `a`.

  4. 4

    HCF of `b` and `b^3` is `b`.

  5. 5

    So, the overall HCF is `5ab`.

  6. 6

    Place the HCF outside the bracket: `5ab(...)`.

  7. 7

    To find the first term inside the bracket, divide `15a^2b` by `5ab`: `15a^2b / 5ab = 3a`.

  8. 8

    To find the second term, divide `-25ab^3` by `5ab`: `-25ab^3 / 5ab = -5b^2`.

  9. 9

    Combine these to get the final answer: `5ab(3a - 5b^2)`.

Recap

  • Expanding means multiplying out brackets.
  • Factorising means putting an expression into brackets.
  • To factorise, find the highest common factor of all terms.
  • Always check your factorisation by expanding it mentally to see if you get the original expression.

Quick check

  1. Expand `7(2y - 3)`.1 mark
  2. Factorise `8x + 12`.1 mark

4. Forming Algebraic Expressions

This skill involves translating a situation described in words into the language of algebra. You represent unknown numbers with variables (letters like x, n, etc.) and build an expression based on the description. For example, if you have 'x' sweets and eat 3, you are left with `x - 3`. If a pen costs 'p' pence, then 5 pens cost `5 × p` or `5p`. Pay close attention to phrases like 'more than' (add), 'less than' (subtract), 'times' (multiply), and 'share' (divide).

Key term

Variable: A letter or symbol used in algebra to represent an unknown or changing quantity.

Examiner insight

Examiners want to see if you can both form the initial expression and then simplify it by collecting like terms. Don't stop after the first step if simplification is possible.

Common pitfall

A very common error is reversing the order of subtraction. '10 less than n' must be written as `n - 10`. Writing `10 - n` is a different expression and is incorrect.

Worked example 12 marks

A smartphone costs £C. A case for the phone costs £15. A screen protector costs half the price of the case. Write an expression for the total cost of one phone, one case, and one screen protector.

  1. 1

    Cost of the phone = `C`.

  2. 2

    Cost of the case = `15`.

  3. 3

    Cost of the screen protector is half the case price: `15 / 2 = 7.50`.

  4. 4

    Total cost is the sum of all three items: `C + 15 + 7.50`.

  5. 5

    Simplify the expression by adding the numbers: `C + 22.50`.

Worked example 23 marks

A rectangle has a width of `w` cm. Its length is 5 cm more than double its width. Write a simplified expression for the perimeter of the rectangle.

  1. 1

    Width = `w`.

  2. 2

    Length is '5 cm more than double its width'. Double the width is `2w`. 5 more than that is `2w + 5`.

  3. 3

    The formula for perimeter is `P = 2(length + width)`.

  4. 4

    Substitute the expressions for length and width: `P = 2((2w + 5) + w)`.

  5. 5

    Simplify inside the bracket first: `P = 2(3w + 5)`.

  6. 6

    Expand the bracket to get the final simplified expression: `P = 6w + 10`.

Recap

  • Read the question carefully to identify the unknown quantity and assign it a variable.
  • Break down the problem into smaller parts.
  • Translate key words into mathematical operations: 'sum' is +, 'difference' is -, 'product' is ×, 'quotient' is ÷.
  • Remember that '5 less than x' is `x - 5`, not `5 - x`.
  • Simplify your final expression by collecting like terms.

Quick check

  1. I am `y` years old. My brother is 4 years younger. Write an expression for my brother's age.1 mark
  2. A taxi journey costs a £3 flat fee plus £2 per mile (`m`). Write an expression for the total cost.1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Given `x = -3` and `y = 4`, find the value of the expression `5x^2 - xy`.3 marks
  2. Expand and simplify `5(2a - 3) - 2(a - 4)`.3 marks
  3. Factorise fully `24x^2y - 16xy^3`.3 marks
  4. The formula `F = 1.8C + 32` converts temperature from Celsius (C) to Fahrenheit (F). Find the temperature in Celsius when it is 77°F.3 marks
  5. Ben is `x` years old. Chloe is 3 years older than Ben. David is twice as old as Chloe. Write a simplified expression for the sum of their ages.4 marks
  6. A rectangle has a length of `(2x + 5)` cm and a width of `(x - 1)` cm. Write a simplified expression for its perimeter.3 marks
  7. Is `n^2 + 1 > 0` always true, sometimes true, or never true for any integer `n`? Explain your answer.2 marks
  8. Use the formula `V = πr^2h` to find the value of `h` when `V = 192π` and `r = 4`.3 marks
  9. Write an expression for the cost of `p` pencils at 35 pence each and `r` rulers at 50 pence each. Give your answer in pence.2 marks
  10. The area of the shape shown is `10x + 15` square units. It is made of a rectangle and a square. Find a possible expression for the length of the side of the square if the total width is 5.4 marks

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