Sites with clear resources by topic – Algebra

See also: Number, Ratio, Geometry, Probability and Statistics

GCSE Content
Algebra Notation, vocabulary and manipulation
1. use and interpret algebraic notation, including:

2. substitute numerical values into formulae and expressions, including scientific formulae

3. understand and use the concepts and vocabulary of expressions, equations, formulae, identities inequalities, terms and factors

4. simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:

• collecting like terms
• multiplying a single term over a bracket
• taking out common factors
• expanding products of two or more binomials
• factorising quadratic expressions of the form x² + bx + c, including the difference of two squares; factorising quadratic expressions of the form ax² + bx + c
• simplifying expressions involving sums, products and powers, including the laws of indices

5. understand and use standard mathematical formulae; rearrange formulae to change the subject

6. know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

7. where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function’; interpret the succession of two functions as a ‘composite function’.

Graphs
8. work with coordinates in all four quadrants

9. plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient

10. identify and interpret gradients and intercepts of linear functions graphically and algebraically

11. identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square

12. recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y=1/x, with x ≠ 0 exponential functions y = k^x  for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size

13. sketch translations and reflections of a given function

14. plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

15. calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts

16. recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.

Solving equations and inequalities
17. solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph

18. solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph

19. solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph

20. find approximate solutions to equations numerically using iteration

21. translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.

22. solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph

Sequences
23. generate terms of a sequence from either a term-to-term or a position-to-term rule

24. recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( rⁿ where n is an integer, and r is a rational number > 0 or a surd) and other sequences

25. deduce expressions to calculate the nth term of linear and quadratic sequences.

Resources by topic – Algebra

• See also NCETM KS3 Secondary Assessment Materials
• Standards Unit – Mostly Algebra
• Nrich – favourite secondary resources – Algebra
• Maths GCSE Exam Questions By Topic – William Neill
• OAT Maths – Algebra
• Don Steward, Median (use topics index)
• mr barton maths Maths Topic Index Page
• MathsBot.com GCSE Resources, GCSE Exam Style Questions, GCSE Revision Grid and Practice GCSE Papers all allow a choice of topic
• Dr Austin Maths – Algebra
• Transum Maths Map for Students – Algebra
• Transum KS3 & Transum KS4 links to curriculum statements and suggested activities
• Andy Lutwyche – Algebra
• MathsHKO Algebra
• Starting Points Maths – Tasks by Topic
• White Rose Secondary Resources
• Open Middle – use the tabs at the top for topic areas by age
• Math HTML5 PhET sims by topic see PhET Sims – Index.
• Curriculum for Wales; The Foundations of Algebra. The workbook contains chapters on patterns, commutativity, distributivity & associativity.
• Dr Frost Maths KS3/KS4 Key Skills Algebra
• mathisfun Index by Year and Subject
• Section Check-in tests use OCR 6-7, for Foundation, also for Higher
• AQA Topic Tests – Algebra
• maths4everyone an extensive library of excellent resources including GCSE Examples by Topic
• NCETM Secondary Assessment Materials
• CIMT, and from Nathan Day all the CIMT practice books merged into one searchable, fully-indexed, 2100-page PDF.
• Colin Foster – Mathematical Etudes (scroll down for etudes by topic)
• Underground Mathematics Algebra Review Questions (old O/AO level).
  Some favourite tasks to challenge your GCSE students:

Resource type	Title
Review question	Two values of x that differ by 5 satisfy x2 −12x + k = 0, what is k?
Review question	How small can this triangle be?
Fluency exercise	Quadratic solving sorter
Review question	Can we solve these simultaneous equations of degree 1 and 2?
Review question	Can we simplify these algebraic fractions?
Review question	If we know two values satisfying a quadratic, can we find the quadratic?
Rich example	Quadratic grids
Building blocks	Which quadratic?
Review question	Can we find the three inequalities that define this region?
Review question	Can we solve these simultaneous equations that involve reciprocals?
Package of problems	Name that graph
Building blocks	Gradient match
Fluency exercise	Multiple manipulations
Review question	When are these quadratic inequalities true together?
Building blocks	A tangent is …
Review question	When are the coefficients of a quadratic equal to its roots?
Many ways problem	Two-way algebra

Exemplification Examples