GCSE Maths · Algebra Basics · Higher · Summary

GCSE Maths Algebra Basics (Higher): summary and key words

Part of the GCSE Maths algebra basics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page

Chapter summary

Skill	Method
Notation	3y = 3 × y; ab = a × b; a² = a × a; a ÷ b = ab; write fractions, not decimals, as coefficients
Vocabulary	Expression (no =), equation (true for some values), formula (a rule), identity (true for all values, ≡), term, coefficient, constant, factor
Collect like terms	Group terms with the same letters and powers and add the coefficients. 4x + 3y − x = 3x + 3y
Index laws	a^m × aⁿ = a^m+n; a^m ÷ aⁿ = a^m−n; (a^m)ⁿ = a^mn; a⁰ = 1
Multiply and divide terms	Multiply or divide the numbers, then add or subtract the powers of each letter
Expand a single bracket	a(b + c) = ab + ac. Every term inside is multiplied; watch the signs
Expand two brackets	Every term times every term (grid or F-O-I-L), then collect like terms. (x + a)(x + b) = x² + (a + b)x + ab
Expand three brackets (Higher)	Expand two, collect, then multiply by the third
Common factor	Take out the highest common factor of all terms: 6x² + 9x = 3x(2x + 3)
x² + bx + c	Two numbers that multiply to c and add to b: x² + 5x + 6 = (x + 2)(x + 3)
Difference of two squares	a² − b² = (a − b)(a + b)
ax² + bx + c (Higher)	Multiply a × c, find two numbers that multiply to it and add to b, split the middle term, factorise in pairs
Show that and proof	Work from one side (expand, simplify, factorise) to the other; use 2n (even), 2n + 1 (odd), n, n + 1 (consecutive); finish with a sentence
Counter-example	One example that makes a statement false

Can you…?

I can write algebra using the correct notation (ab, 3y, a², fractions).
I know the meaning of expression, equation, formula, identity, term, coefficient and factor.
I can simplify an expression by collecting like terms.
I can multiply and divide algebraic terms and use the laws of indices.
I can expand single brackets, including with a negative or a letter outside.
I can expand and simplify two brackets, including squares.
I can take out a common factor and factorise fully.
I can factorise x² + bx + c and the difference of two squares.
I can tell the difference between an equation and an identity.
I can show that two expressions are equal, and use algebra to prove simple statements about numbers.
(Higher) I can expand three brackets, factorise ax² + bx + c, and prove harder statements about integers.

Key terms

Term: One part of an expression, with the sign in front of it, for example 3x or −5.
Coefficient: The number multiplying the letters in a term. The coefficient of 4x² is 4.
Constant: A term that is just a number, such as +7.
Variable: A letter that stands for a number that can change or is not yet known.
Expression: Numbers, letters and operations with no equals sign, such as 3x + 2.
Equation: Two expressions joined by =, true only for particular values of the letter.
Formula: A rule that links quantities, for example A = lw.
Identity: An equality that is true for every value of the letters. Written with ≡.
Inequality: A statement comparing two expressions with <, >, ≤ or ≥.
Factor: Something that is multiplied; a number or expression that divides exactly into another.
Common factor: A factor that divides exactly into every term of an expression.
Highest common factor (HCF): The biggest common factor of a set of terms.
Index (power): The small raised number that says how many copies of a letter are multiplied: x³.
Like terms: Terms with the same letters raised to the same powers. They can be added or subtracted.
Expand: Multiply out brackets so that none are left.
Factorise: Write an expression as a product, using brackets.
Quadratic: An expression whose highest power is 2, such as x² + 5x + 6.
Binomial: An expression with two terms, such as x + 3.
Difference of two squares: An expression a² − b², which factorises as (a − b)(a + b).
Integer: A whole number: positive, negative or zero.
Consecutive: Following one after another, such as 7, 8, 9 or 2n, 2n + 2.
Proof: A logical argument that shows a statement is always true, using algebra rather than examples.
Counter-example: One example that shows a statement is false.