Algebra

Sites with clear resources by topic – Algebra

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
Lyszkowski’s method

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

Activities to help learn reasoning and proof

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 = kx  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.

Simultaneous Equations – White Rose Maths Exemplar Questions

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

AQA Bridging the Gap resources

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

Resource typeTitle
Review questionTwo values of x that differ by 5 satisfy x2 −12x + k = 0, what is k?
Review questionHow small can this triangle be?
Fluency exerciseQuadratic solving sorter
Review questionCan we solve these simultaneous equations of degree 1 and 2?
Review questionCan we simplify these algebraic fractions?
Review questionIf we know two values satisfying a quadratic, can we find the quadratic?
Rich exampleQuadratic grids
Building blocksWhich quadratic?
Review questionCan we find the three inequalities that define this region?
Review questionCan we solve these simultaneous equations that involve reciprocals?
Package of problemsName that graph
Building blocksGradient match
Fluency exerciseMultiple manipulations
Review questionWhen are these quadratic inequalities true together?
Building blocksA tangent is …
Review questionWhen are the coefficients of a quadratic equal to its roots?
Many ways problemTwo-way algebra
PhET Interactive Simulations, University of Colorado Boulder, https://phet.colorado.edu

Exemplification Examples