Algebraic notation and simplifying expressions
Writing and interpreting algebraic notation, substituting values, collecting like terms, and expanding or factorising a single bracket
About three lessons of 45 to 60 minutes
Think of a number... the trick that always works
Think of any number. Add 5. Double the result. Subtract 10. Divide by 2. You will always end up back at your original number, whichever number you started with. Try it with 3, then try it with 100, it works every single time.
This isn't magic, it's algebra. Writing the trick as an expression, and letting the starting number be x, the whole trick is (2(x + 5) - 10) / 2. Expanding the bracket and simplifying step by step: 2(x + 5) = 2x + 10, then subtracting 10 gives 2x, and dividing by 2 gives back x. Algebraic notation lets you write down a calculation for EVERY possible number at once, and simplifying an expression is how you prove a trick like this always works, not just check a handful of examples.
- 3y written instead of y + y + yalgebraic notation is just a shorter way to write a repeated addition or multiplication
- The 'think of a number' trick: (2(x + 5) - 10) / 2simplifies to exactly x, however many different numbers you try
- A rectangle 3 units by (x + 2) unitsits area, 3(x + 2), expands to 3x + 6, the same total either way
- 5x + 3x simplifying to 8xcollecting like terms combines equivalent quantities, the same way 5 apples plus 3 apples makes 8 apples
What students will be able to do
Students will use and interpret algebraic notation (ab, 3y, a², a/b), substitute numerical values into expressions, simplify expressions by collecting like terms, and expand a single term over a bracket or factorise an expression by taking out the highest common factor.
- I can write a repeated addition or multiplication using algebraic notation.
- I can interpret algebraic notation such as ab, a², and a/b.
- I can substitute a numerical value into an algebraic expression to find its value.
- I can simplify an expression by collecting like terms.
- I can expand a single term over a bracket, e.g. 3(x + 4).
- I can factorise an expression by taking out the highest common factor.
Standards this unit teaches
- KS3 Maths: AlgebraUK National Curriculum (England)Algebra
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Algebra" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "use and interpret algebraic notation, including: ab in place of a × b; 3y in place of y + y + y and 3 × y; a² in place of a × a, a³ in place of a × a × a; a²b in place of a × a × b; a/b in place of a ÷ b; coefficients written as fractions rather than as decimals; brackets"; "substitute numerical values into formulae and expressions"; "understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors"; "simplify and manipulate algebraic expressions to maintain equivalence by: collecting like terms; multiplying a single term over a bracket; taking out common factors".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 exponents & algebraic expressions teaching unitthe same-age unit on writing algebraic expressions from words and evaluating simple exponents
- Variable in the glossary
- Greatest common factor in the glossarythe same highest-common-factor technique used to factorise an expression
- Factor in the glossary
Words to teach and display
- Expression
- a mathematical phrase built from numbers, variables and operations, with no equals sign, such as 3x + 5
- Term
- a single part of an expression, separated by + or -, e.g. 3x and 5 are the 2 terms in 3x + 5
- Like terms
- terms that have exactly the same variable (or combination of variables), such as 3x and 5x, which can be combined by adding or subtracting their coefficients
- Coefficient
- the number multiplying a variable in a term, e.g. the coefficient of 3x is 3
- Factor (algebra)
- a number or expression that divides exactly into every term of an expression
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Algebraic notation and substitution
ConcreteAlgebra has its own shorthand for writing calculations. When a letter stands for an unknown number, the multiplication sign is dropped (a x b is written ab), repeated addition becomes a coefficient (y + y + y becomes 3y), and repeated multiplication becomes an index (a x a becomes a²).
The division sign is also dropped in favour of a fraction: a ÷ b is written a/b. Once an expression is written in algebraic notation, substituting a number for the letter turns it back into an ordinary calculation.
To substitute, replace every occurrence of the letter with the given value (in brackets, to keep the sign clear), then calculate as normal, following the order of operations.
If x = 5, find the value of 3x + 4.
- Substitute x = 5 into the expression: 3(5) + 4.
- Calculate: 3 x 5 = 15, then 15 + 4 = 19.
Answer: 3x + 4 = 19 when x = 5.
- Write a x a x a using index notation.
- If m = 7, what is the value of m²?
2. Simplifying expressions: collecting like terms
PictorialLike terms have exactly the same letter (or letters) and can be combined, the same way 5 apples and 3 apples combine to 8 apples, but apples and oranges cannot. 5x and 3x are like terms because they share the letter x; 5x and 3y are NOT like terms, because x and y are different unknowns.
To collect like terms, add or subtract their coefficients and keep the letter the same: 5x + 3x = 8x, because 5 of something plus 3 more of the SAME thing is 8 of that thing.
An expression can have several different letters and a constant (a plain number) all mixed together; collect each letter's terms separately, and the constant stays on its own: 4x + 3y + 2x - y + 5 simplifies to 6x + 2y + 5.
Simplify: 5x - 2x + 3.
- Collect the x terms: 5x - 2x = 3x.
- The constant term, 3, has no x, so it stays on its own.
Answer: 5x - 2x + 3 simplifies to 3x + 3.
- Are 4a and 4b like terms? Explain why or why not.
- Simplify 6y + 2y - 3y.
3. Expanding a bracket and factorising
AbstractMultiplying a single term over a bracket (expanding) means multiplying EVERY term inside the bracket by the term outside: 3(x + 4) means 3 lots of x plus 3 lots of 4, so 3(x + 4) = 3x + 12. Factorising is the reverse: writing an expression as a term multiplied by a bracket, by taking out the highest common factor of every term.
This is the exact same distributive idea used for ordinary numbers: 3 x 7 can be split as 3 x (5 + 2) = 3x5 + 3x2 = 15 + 6 = 21. Expanding a bracket in algebra works identically, just with an unknown x instead of a known number.
To factorise 6x + 9, find the highest common factor of 6 and 9, which is 3. Dividing both terms by 3 gives 2x and 3, so 6x + 9 = 3(2x + 3). Expanding 3(2x + 3) gets back 6x + 9, which checks the factorising is correct.
Expand 4(x - 3), then factorise 10x + 15 by taking out the highest common factor.
- Expand: 4 x x = 4x, and 4 x (-3) = -12, so 4(x - 3) = 4x - 12.
- Factorise: the highest common factor of 10 and 15 is 5. 10 / 5 = 2 and 15 / 5 = 3, so 10x + 15 = 5(2x + 3).
Answer: 4(x - 3) = 4x - 12, and 10x + 15 = 5(2x + 3).
- Expand 2(x + 6).
- What is the highest common factor of 8 and 12, and how would you use it to factorise 8x + 12?
Common misconceptions and how to address them
Misconceptiona x a is written as 2a.
Why it happens: Students confuse repeated MULTIPLICATION (which uses an index, or power) with repeated ADDITION (which uses a coefficient): a + a is 2a, but a x a is a².
How to address it: Ask what operation is actually happening. a + a means 2 lots of a added, so it is 2a. a x a means a multiplied by itself, so it is a², a completely different value: if a = 5, 2a = 10 but a² = 25.
Misconception3y means 3 + y.
Why it happens: Students read the juxtaposition of a number and a letter as if a symbol were simply missing, defaulting to the more familiar operation, addition.
How to address it: In algebra, a number written directly next to a letter always means MULTIPLICATION: 3y means 3 x y, i.e. y + y + y, never 3 + y. Substituting a value makes the difference obvious: if y = 4, 3y = 12, but 3 + y = 7.
Misconception3x + 2y simplifies to 5xy (or 5(x + y)).
Why it happens: Students try to combine EVERY term in an expression, even when the terms have different letters and are not like terms.
How to address it: Only terms with the exact same letter (or combination of letters) can be combined. 3x and 2y have different letters, so 3x + 2y is already fully simplified, it cannot be combined into a single term.
MisconceptionExpanding 3(x + 4) only multiplies the FIRST term inside the bracket, giving 3x + 4.
Why it happens: Students distribute the outside term to just the first term inside the bracket and forget the second.
How to address it: Every single term inside the bracket must be multiplied by the term outside: 3(x + 4) = (3 x x) + (3 x 4) = 3x + 12, not 3x + 4. Checking by substituting a number catches this: if x = 2, 3(2 + 4) = 3 x 6 = 18, but 3x + 4 with x = 2 gives 6 + 4 = 10, a different (wrong) answer.
Guided practice (with answers)
1. Write p × p × p × p using index notation.
Answer: p⁴, because p multiplied by itself 4 times is p to the power of 4.
2. If n = 6, find the value of 2n - 5.
Answer: 7, because 2 x 6 = 12, and 12 - 5 = 7.
3. Simplify: 7a + 2a - 3.
Answer: 9a - 3, because 7a + 2a = 9a, and the constant -3 stays on its own.
4. Simplify: 5x + 4y - 2x + y.
Answer: 3x + 5y, because 5x - 2x = 3x, and 4y + y = 5y.
5. Expand: 6(x + 2).
Answer: 6x + 12, because 6 x x = 6x, and 6 x 2 = 12.
6. Factorise: 12x + 18.
Answer: 6(2x + 3), because the highest common factor of 12 and 18 is 6, and 12 / 6 = 2, 18 / 6 = 3.
Independent practice worksheets
Practise algebraic notation and substitution, collecting like terms, and expanding/factorising with computed, never-wrong answer keys.
Differentiation
- Keep a reference card of notation conversions visible: a x b = ab, y + y + y = 3y, a x a = a².
- For substitution, always write the value in brackets in place of the letter before calculating, to keep the sign correct.
- Colour-code like terms (e.g. all the x terms in blue, all the y terms in green) before collecting them.
- Practise expanding with numeric distributive-law examples (e.g. 4 x (10 + 3)) before moving to an unknown x.
- Investigate: does a² always equal 2a? Test with several values of a to show they are only equal when a = 0 or a = 2.
- Expand and simplify an expression with 2 brackets added together, e.g. 3(x + 2) + 2(x + 5).
- Factorise an expression where the highest common factor is greater than 1 and appears differently in each term, e.g. 4x + 4.
- Prove the 'think of a number' trick from the hook algebraically for a different sequence of operations invented by the student.
Assessment: exit ticket
A three-question exit ticket sampling notation and substitution, collecting like terms, and expanding a bracket.
1. If x = 8, find the value of 5x - 3.
Answer: 37, because 5 x 8 = 40, and 40 - 3 = 37.
2. Simplify: 9x - 4x + 6.
Answer: 5x + 6, because 9x - 4x = 5x, and the constant 6 stays on its own.
3. Expand: 5(x - 2).
Answer: 5x - 10, because 5 x x = 5x, and 5 x (-2) = -10.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 notation and substitution (section 1), Lesson 2 collecting like terms (section 2), Lesson 3 expanding and factorising plus the exit ticket (section 3).
- This unit assumes comfort with the Grade 6 foundational work on writing expressions from words and evaluating simple exponents (see the linked prior-knowledge unit); it moves into the UK's specific notation conventions (ab, a², a/b) and the manipulation skills (collecting like terms, expanding, factorising) the KS3 programme of study lists as a single connected bullet.
- Language to keep repeating: a letter next to a number always means multiply; only terms with the exact same letter can be collected; expanding multiplies EVERY term inside the bracket; factorising is expanding in reverse.
- The array-split figure in section 3 deliberately reuses the same distributive-law diagram already used elsewhere on the site for numeric multiplication (e.g. 3 x 7 = 3x5 + 3x2), so students see expanding a bracket as the SAME idea they already know, just applied to an unknown.
- Use Student view to project this lesson. Print saves the full teacher unit, including answers and teacher notes; use the linked independent-practice worksheets for student handouts.