Cambridge Lower Secondary CheckpointStage 8

Algebra (8As)

Mathematics Stage 8 Chapter Notes

What this chapter covers

Algebra - Sequences, functions and graphs
ShareWhatsAppPost
Algebra (8As) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Algebra (8As) notes as text: skim, search, and jump between subtopics.

~11 min read

In this chapter

  1. 1Expressions, Equations, and Formulae
  2. 2Substitution and Order of Operations
  3. 3Expanding Single Brackets
  4. 4Factorising into a Single Bracket
  5. 5Forming Expressions from Words

1. Expressions, Equations, and Formulae

It's crucial to know the difference between three key building blocks of algebra. An 'expression' is a collection of numbers, letters (variables), and operation signs, like `5x + 3y - 2`. It doesn't have an equals sign and represents a value that can change depending on the variables. An 'equation' states that two expressions are equal, like `5x + 1 = 16`. The goal is usually to 'solve' it to find a specific, unknown value for the variable. 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 = lw`. In a formula, the variables can take many different values.

Area of a rectangle: A = lw

Circumference of a circle: C = 2πr

A linear relationship: y = mx + c

Key term

Variable: A letter or symbol used to represent an unknown or changing quantity in an algebraic expression or equation.

Examiner insight

Examiners reward precise use of terminology. Correctly identifying an item as an expression, equation, or formula shows a fundamental understanding of algebra.

Common pitfall

Confusing an expression with an equation. You can simplify or evaluate an expression, but you cannot 'solve' it as there is no equals sign.

Worked example 13 marks

For each of the following, state whether it is an expression, an equation, or a formula.(a) 2x + 7(b) V = πr²h(c) 3y - 5 = 10

  1. 1

    Step 1 (a): `2x + 7` is a collection of terms without an equals sign. Therefore, it is an expression.

  2. 2

    Step 2 (b): `V = πr²h` shows a relationship between the variables V, r, and h. It is a rule for calculating volume. Therefore, it is a formula.

  3. 3

    Step 3 (c): `3y - 5 = 10` sets two things equal to each other and contains an unknown value `y` that can be found. Therefore, it is an equation.

Recap

  • An expression is a mathematical phrase without an equals sign.
  • An equation contains an equals sign and can be solved to find a specific unknown value.
  • A formula is an equation that describes a relationship between two or more variables.
  • In a formula like `A = lw`, `A` is known as the subject of the formula.

Quick check

  1. Is `P = 2(l + w)` an expression, equation, or formula?1 mark

2. Substitution and Order of Operations

Substitution is the process of replacing variables in an expression with specific numerical values. Once you've substituted the numbers, you must evaluate the expression using the correct order of operations, commonly remembered by the acronym BIDMAS (or BODMAS).

BIDMAS stands for:

  • Brackets
  • Indices (powers and square roots)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

When substituting negative numbers, it's a very good habit to place them inside brackets to avoid errors, especially with powers.

Key term

Substitution: The process of replacing variables in an algebraic expression or formula with their given numerical values to find the value of the expression.

Examiner insight

Full marks for substitution questions require showing the initial substitution step clearly before starting calculations. This can earn method marks even if a calculation error follows.

Common pitfall

Incorrectly calculating the square of a negative number. For example, calculating `(-4)²` as `-16`. The correct calculation is `(-4) × (-4) = 16`.

Worked example 14 marks

If p = 5 and q = -4, find the value of:(a) 3p + 2q(b) p² - q

  1. 1

    Step 1 (a): Substitute the values into the expression: `3(5) + 2(-4)`.

  2. 2

    Step 2 (a): Perform the multiplications first: `15 + (-8)`.

  3. 3

    Step 3 (a): Perform the addition: `15 - 8 = 7`.

  4. 4

    Step 4 (b): Substitute the values, using brackets for the negative number: `(5)² - (-4)`.

  5. 5

    Step 5 (b): Evaluate the index (power) first: `25 - (-4)`.

  6. 6

    Step 6 (b): Perform the subtraction: `25 + 4 = 29`.

Worked example 22 marks

The formula for the surface area of a prism is `A = 2x² + 4xh`. Find the value of A when x = 3 and h = 10.

  1. 1

    Step 1: Write down the formula and substitute the given values: `A = 2(3)² + 4(3)(10)`.

  2. 2

    Step 2: Apply BIDMAS. Calculate the index (power) first: `A = 2(9) + 4(3)(10)`.

  3. 3

    Step 3: Perform the multiplications: `A = 18 + 120`.

  4. 4

    Step 4: Perform the addition to find the final answer: `A = 138`.

Recap

  • Always follow the order of operations: BIDMAS.
  • When substituting negative numbers, use brackets to avoid sign errors.
  • Calculate powers before you multiply by any coefficients (e.g., in `5x²`, square `x` first, then multiply by 5).
  • Read the question carefully to ensure you are substituting the correct values for the correct variables.

Quick check

  1. Find the value of `4a - b²` when `a = 2` and `b = -3`.2 marks

3. Expanding Single Brackets

Expanding brackets means removing them by multiplying everything inside the bracket by the term outside. This process uses the distributive law. For an expression like `a(b + c)`, you multiply `a` by `b` and then `a` by `c`. A helpful way to remember this is the 'claw' method, drawing lines from the term outside to each term inside.

a(b + c) = ab + ac

a(b - c) = ab - ac

Key term

Distributive Law: The rule which states that multiplying a term by a group of terms in a bracket is the same as doing each multiplication separately.

Examiner insight

When expanding multiple brackets, examiners look for the correct expansion of each bracket first, before any attempt to simplify. This demonstrates the method clearly.

Common pitfall

Only multiplying the first term in the bracket by the term outside. For example, expanding `4(x + 5)` to `4x + 5` instead of the correct `4x + 20`.

Worked example 13 marks

Expand the following expressions:(a) 5(2x + 3)(b) -3y(y - 4)

  1. 1

    Step 1 (a): Multiply the term outside, 5, by the first term inside, 2x. `5 × 2x = 10x`.

  2. 2

    Step 2 (a): Multiply the term outside, 5, by the second term inside, +3. `5 × 3 = 15`.

  3. 3

    Step 3 (a): Combine the results: `10x + 15`.

  4. 4

    Step 4 (b): Multiply -3y by y. `-3y × y = -3y²`.

  5. 5

    Step 5 (b): Multiply -3y by -4. `-3y × -4 = +12y` (a negative times a negative is a positive).

  6. 6

    Step 6 (b): Combine the results: `-3y² + 12y`.

Worked example 23 marks

Expand and simplify: 4(2a + 1) + 3(a - 5)

  1. 1

    Step 1: Expand the first bracket: `4 × 2a = 8a` and `4 × 1 = 4`. This gives `8a + 4`.

  2. 2

    Step 2: Expand the second bracket: `3 × a = 3a` and `3 × -5 = -15`. This gives `3a - 15`.

  3. 3

    Step 3: Write out the full expanded expression: `8a + 4 + 3a - 15`.

  4. 4

    Step 4: Collect like terms. Combine the 'a' terms: `8a + 3a = 11a`. Combine the number terms: `4 - 15 = -11`.

  5. 5

    Step 5: Write the final simplified expression: `11a - 11`.

Recap

  • To expand a bracket, multiply every term inside the bracket by the term outside.
  • Be careful with signs; a negative term outside will change the signs of the terms inside.
  • After expanding, always check if you can simplify the expression by collecting like terms.

Quick check

  1. Expand and simplify 7(x - 2) - 2(x + 3).2 marks

4. Factorising into a Single Bracket

Factorising is the reverse process of expanding. To factorise an expression, you find the Highest Common Factor (HCF) of all the terms. The HCF is the largest number, and highest power of each variable, that divides into every term. This HCF is placed outside the bracket. The terms inside the bracket are what you get when you divide the original terms by the HCF.

ab + ac = a(b + c)

Key term

Highest Common Factor (HCF): The largest factor (number and/or variable) that divides perfectly into every term of an expression.

Examiner insight

The instruction 'Factorise fully' is very important. Examiners will penalise answers that are only partially factorised, so always double-check if the terms inside your bracket have any remaining common factors.

Common pitfall

Not taking out the *highest* common factor. For example, factorising `16x + 8` as `4(4x + 2)` or `2(8x + 4)` instead of the fully factorised `8(2x + 1)`.

Worked example 12 marks

Factorise fully: 12x + 18y

  1. 1

    Step 1: Look at the numbers 12 and 18. Find their HCF. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The HCF is 6.

  2. 2

    Step 2: Look at the variables. `x` is in the first term, `y` is in the second. There are no common variables.

  3. 3

    Step 3: The overall HCF is 6. Place this outside the bracket: `6(...)`.

  4. 4

    Step 4: To find the terms inside, divide each original term by the HCF. `12x ÷ 6 = 2x`. `18y ÷ 6 = 3y`.

  5. 5

    Step 5: Place these results inside the bracket: `6(2x + 3y)`.

Worked example 23 marks

Factorise fully: 15a³b - 20ab²

  1. 1

    Step 1: Find the HCF of the numbers 15 and 20. This is 5.

  2. 2

    Step 2: Find the HCF of the variable parts. The terms are `a³b` and `ab²`. The lowest power of `a` that appears in both is `a¹` (or `a`). The lowest power of `b` that appears in both is `b¹` (or `b`). So the variable HCF is `ab`.

  3. 3

    Step 3: Combine these to get the overall HCF: `5ab`. Place this outside the bracket: `5ab(...)`.

  4. 4

    Step 4: Divide the first term by the HCF: `15a³b ÷ 5ab = 3a²`.

  5. 5

    Step 5: Divide the second term by the HCF: `-20ab² ÷ 5ab = -4b`.

  6. 6

    Step 6: Write the final answer: `5ab(3a² - 4b)`.

Recap

  • Factorising is the opposite of expanding.
  • Always look for the Highest Common Factor (HCF) of all terms.
  • The HCF includes the largest number and the highest power of any variables common to all terms.
  • You can check your answer by expanding the bracket; you should get the original expression.

Quick check

  1. Factorise fully: 18p²q - 27pq³.2 marks

5. Forming Expressions from Words

A key skill in algebra is translating real-world situations described in words into mathematical expressions. This involves identifying the unknown quantity, assigning a variable to it (like `x` or `n`), and then converting phrases into mathematical operations.

Key phrases:

  • 'More than' or 'sum of' means addition (+).
  • 'Less than' or 'difference' means subtraction (-).
  • 'Times', 'product of', or 'double/triple' means multiplication (×).
  • 'Divided by' or 'shared between' means division (÷).

Key term

Term: A single number, a variable, or numbers and variables multiplied together, separated by + or - signs in an expression.

Common pitfall

Reversing the order of terms in subtraction. For example, writing '10 less than x' as `10 - x` instead of the correct `x - 10`.

Fun fact

The word 'algebra' comes from the title of a book, 'Al-jabr w'al-muqabala', written in the 9th century by the Persian mathematician Al-Khwarizmi. 'Al-jabr' refers to the process of moving a negative term to the other side of an equation to make it positive.

Worked example 13 marks

A smartphone costs £c. A protective case costs £15 less than the phone. Write an expression for:(a) The cost of the case.(b) The total cost of the phone and the case.

  1. 1

    Step 1 (a): The phone costs `c`. The case is '15 less than c'. This translates to subtraction: `c - 15`.

  2. 2

    Step 2 (b): The total cost is the cost of the phone plus the cost of the case. This is `c + (c - 15)`.

  3. 3

    Step 3 (b): Simplify the expression by collecting like terms: `c + c - 15 = 2c - 15`.

Worked example 24 marks

The width of a rectangle is `w` metres. The length is 4 metres longer than double its width. Write a simplified expression for the perimeter of the rectangle.

  1. 1

    Step 1: Define the variables. Width = `w`.

  2. 2

    Step 2: Form an expression for the length. 'Double its width' is `2w`. '4 metres longer than' means add 4. So, Length = `2w + 4`.

  3. 3

    Step 3: Recall the formula for perimeter: `P = 2(length + width)`.

  4. 4

    Step 4: Substitute the expressions for length and width: `P = 2((2w + 4) + w)`.

  5. 5

    Step 5: Simplify inside the bracket first: `P = 2(3w + 4)`.

  6. 6

    Step 6: Expand the bracket to get the final simplified expression: `P = 6w + 8`.

Recap

  • Define your variable clearly (e.g., 'Let n be the number').
  • Break the problem down into smaller parts.
  • Translate key words into mathematical operations.
  • Be careful with subtraction: '5 less than x' is `x - 5`, not `5 - x`.
  • Simplify your final expression by collecting like terms where possible.

Quick check

  1. Ben is `y` years old. His sister is twice his age. Write an expression for the sum of their ages.2 marks

End-of-chapter exercise

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

  1. Given that a = 5, b = -2, and c = 0.5, find the value of the expression `a² - 4b / c`.3 marks
  2. Expand and simplify the expression `5(3x - 4) - 2(4x - 7)`.3 marks
  3. Factorise fully the expression `24x³y² - 16x²y³`.3 marks
  4. A plumber charges a fixed call-out fee of £40 and an hourly rate of £55. Write a formula for the total cost, C, for a job that takes h hours.2 marks
  5. The area of a trapezium is given by the formula `A = 0.5(a + b)h`. Find the value of A when a = 6.5 cm, b = 9.5 cm, and h = 4 cm.2 marks
  6. A rectangle has a length of `(3x + 2)` cm and a width of `(x - 1)` cm. Write a simplified expression for its perimeter.3 marks
  7. The temperature in Celsius (C) can be converted to Fahrenheit (F) using the formula `F = 1.8C + 32`. Find the temperature in Celsius when it is 77°F.3 marks
  8. Maya thinks of a number, `n`. She cubes it and then subtracts 5. Her friend Leo thinks of the same number `n`. He multiplies it by 3 and adds 1. Write an expression for the product of Maya's and Leo's results.3 marks
  9. Three angles in a quadrilateral are `x`, `2x + 10`, and `x + 40`. The fourth angle is 90 degrees. The sum of angles in a quadrilateral is 360 degrees. Form an equation and solve it to find the value of x.4 marks
  10. The surface area of a cone is given by `A = πr² + πrl`, where `r` is the radius and `l` is the slant height. Factorise the expression for the surface area.2 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (8Ni)Number (8Np)Number (8Nf)Algebra (8Ae)Geometry and Measure (8Gg)Geometry and Measure (8Gp)