Algebra

Basic Skills

Algebraic Manipulation
Writing Expressions Using Algebra
Use and interpret algebraic notation (KS3)
Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors (KS3)
Know the difference between an equation and an identity (KS4)
Translate simple situations or procedures into algebraic expressions or formulae (KS4)
Simplify and manipulate algebraic expressions, including those involving surds (KS4)
Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments (KS4)
Lesson: Expressions using algebra tiles (C. Norledge)
Linking algebraic expressions, words and pictorial representations - developed using algebra tiles and these matching cards from Tim Buckton. 

Standards Unit A1: Interpreting algebraic expressions (via Craig Barton)
A fantastic card-sorting activity from the Standards box; pupils match expressions in words and algebra, then extend to diagram representations and eventually to substitution tables. There's more than enough for one lesson here, and I'd recommend revisiting this a few times while teaching algebra.

Algebraic areas (Owen134866 via TES)
Pupils write algebraic expressions to represent the area of compound shapes. Fully resourced lesson with PowerPoint and worksheet, very engaging with scope for extension.

Basic algebra worksheet (MsFattouh via TES)
Decent basic practice worksheet on simplifying straightforward algebraic expressions using correct notation.

Forming algebraic expressions for perimeter (mej via TES)
Set of three differentiated worksheets to practise writing algebraic expressions for the perimeter of 2D shapes.

Matchup activities (C. Norledge)​​

Writing algebraic expressions from contexts (C. Norledge):
Transum
  • Nincompoop: Bit of a silly name, but a great little starter requiring pupils to pair equivalent algebraic expressions and select the odd one out, with four levels of difficulty.
  • Back To School: Given values of B, T and S, form as many equations as possible. If it's not just after a holiday, Practise Your Algebra does exactly the same thing with the letters P, Y and A.
  • Algebraic notation: Good online worksheet activity for pupils to complete for homework or revision.

Flash Maths
  • Algebra keywords: Choose the keyword that matches with the description given - great for reinforcing literacy links.
  • Paper folding algebra: Write expressions for the area of compound shapes made from rectangles. Lots of scope for using this in further work on algebraic substitution too.

Perimeter expressions (NRich)
Writing algebraic expressions for perimeter using a piece of A4 paper.

Algebraic symbolism (QUIA)
Interactive repeated generator for matching word statements to algebraic notation. Some challenging vocabulary used; consider literacy level of group before using.

Algebraic expressions foldable (Equation Freak)
Templates for creating an algebraic expressions foldable for exercise books.

Multiplying algebraic terms treasure hunt (NumberLoving)
Includes an "orange" path with simple questions such as 3a x 2b, and a "green" path with more complicated problems. This could also be done in groups around a desk rather than a hunt around the room.

Area and perimeter with algebra (denningh via TES)
Beautifully presented card-based activity with perimeter and area of rectangle problems. Also includes a challenge sheet with compound shapes and shapes based on circles. Plenty to challenge even higher-attaining pupils!

Area and algebra rally coach (judsonB via TES)
Two Rally Coach activities for grades D/C and B/A at GCSE. Not used Rally Coach before? Read my blog post to find out more
Simplifying Algebraic Expressions
Simplify and manipulate algebraic expressions to maintain equivalence by collecting like terms (KS3)
Cutting and pasting to combine like terms (Math Equals Love)
Students cut out and physically group like terms, requiring appreciation that the sign "belongs" to the term to the right of it.

KS3 Strategy study module (LancsNGFL)
Contains some nice ideas and resources for teaching collecting like terms.

Increasingly Difficult Exercises (@taylorda01)
Level of difficulty increases rapidly as pupils work through problems.

Variation theory problems (Craig Barton)
Worked example and set of practice problems using variation theory.

Algebra pyramids
  • TeachIt Maths: Basic practice in the form of addition pyramids; nice mix of problems.
  • Great Maths Teaching Ideas (Easy/Hard): Two large pyramids requiring a bit of thought!

Collecting like terms (TeachingMaths_ via TES)
Selection of worksheet-based resources and a couple of matching/card-sorting activities.

Textbook exercises
  • CorbettMaths: Varied selection of problems with applications and answers.

Collecting like terms workout (Study Maths)
Question generator - good for basic repeated practice.

Exam-style questions
  • CorbettMaths: A mixture of procedural practice problems and applications using perimeters of shapes.
  • JustMaths: Simple addition/subtraction and procedural questions only.
  • KESH Takeaway: Procedural problems lifted from exam papers.

Don Steward
Collecting like terms bingo (Interactive-Maths)
Bingo generator with different levels of difficulty.
Expanding and Factorising Single Brackets
Simplify and manipulate algebraic expressions to maintain equivalence by multiplying a single term over a bracket or by taking out common factors (KS3)
Factorising linear expressions using algebra tiles (C. Norledge)
Blog post and lesson resources for introducing factorising using algebra tiles; this is my favourite way to introduce this topic, and really emphasises the idea of factorising as grouping. 

Basic practice worksheets (Worksheet Math)
Worksheet generator with lots of options, including:
Collect a joke Factorising single bracket (Numberloving)
Brilliant way to jazz up basic practice, with a selection of fairly challenging factorising problem.

Expanding and simplifying brackets workout (Study Maths)
Question generator for quick practice; expanding two brackets and simplifying results.

Don Steward
  • Not the Same: Pupils find the expressions on a spider diagram that are not equivalent to the one in the middle; great for reinforcing links for later work on factorising single brackets.
  • One Incorrect Simplification: Similar to the above activity, but with only one incorrect expression and use of fraction notation/division as well.
  • Find the Gaps (Don Steward): Challenging problems requiring pupils to fill gaps in bracket expansion problems.

Single Brackets Package (TeachingMaths_ via TES)
Collection of resources for (mostly) expanding single brackets; includes a worksheet to support use of grid method, a dominoes activity, matching pairs and spot the mistake activity - definitely check this out!

Expanding Brackets Resources (Miss Brookes Maths)
A huge collection of links to plenty of resources for expanding single brackets.

Expanding single brackets treasure hunt (Teach It Maths)
Treasure hunt activity with a hidden message; stick round the room or do in pairs/small groups on tables.

Expand and simplify matching pairs (Great Maths Teaching Ideas)
Quick card sort; pupils expand and simplify brackets to make matching pairs. Some nice tricky problems on here!

Factorising pair game (ems21 via TES)
Nice game to play in pairs to practise expanding brackets. Requires two different coloured dice and game board per pair.
Laws of Indices
Simplify expressions involving sums, products and powers, including the laws of indices (KS4)
​Jo Morgan (Resourceaholic): Some fantastic ideas and resources for planning and teaching lessons on indices.

Rules of indices introduction worksheet (Labrown20 via TES)
Nice little worksheet to introduce the laws of indices; pupils do the calculations, then try to spot the laws for themselves. Includes multiplying, dividing and brackets, with an extension using coefficients.

Variation theory problems (Craig Barton)
Worked example and set of practice problems using variation theory.

Basic practice worksheets (Worksheet Math)
Excellent worksheet generator with options to select numerical, algebraic or a mixture of problems, and automatic answer generation:
Textbook exercises
  • CorbettMaths: Varied selection of problems with applications and answers.

Simplifying indices codebreaker (Teachit Maths)
Codebreaker activity using all laws of indices. Includes negative powers, with two versions (one with indices written as simple negatives, one with indices written as fractions).

Laws of indices worksheet (HWB via TES)
Nicely formatted standard worksheet for basic laws of indices.

Exam-style questions
  • CorbettMaths: Procedural problems applying laws of indices - mostly algebraic.
  • KESH Takeaway: Procedural problems lifted from old specification exam papers.
  • JustMaths: Must/Should/Could practice with exam questions

Indices Dominoes (Teachit Maths)
Domino activity that spells a sentence when matched correctly; great for practice of simple index laws (multiplication, division and brackets).
Algebraic Fractions
Simplify and manipulate algebraic fractions (KS4 Higher only)
Coming soon!
Proof (H)
​Use algebra to support and construct proofs (KS4 Higher only)
Coming soon!
Quadratic Expressions
Expanding Quadratics and other Binomials
Simplify and manipulate algebraic expressions to maintain equivalence by expanding products of two or more binomials (KS3)
Basic practice worksheet (Worksheet Math)
Worksheet generator with options to select sign variations of coefficients.
Factorising Quadratics (A = 1)
Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form x² + bx + c, including the difference of two squares (KS4)
Factorising double brackets (Worksheet Math)
Worksheet generator with options to select sign variations of coefficients.

Sum Products (Flash Maths)
Use in reverse - give pupils the sum and product and see if they can work out what the missing numbers are.

Two Numbers (Transum): Quick starter similar to the activity above.
Factorising Quadratics (A not 1) (H)
Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax² + bx + c, including the difference of two squares (KS4 Higher only)
Factorising harder quadratics (Jo Morgan)
A good read with lots of ideas for different ways to teach factorising more difficult quadratics.

Cross method for factorising tricky quadratics (The Storeroom)
Some thoughts about using the cross method to organise working out when factorising more difficult quadratics.

Don Steward:
  • Tap Top: Some great puzzles leading nicely into factorising more tricky quadratics - this has worked really well for me as a "separate" activity or starter, as pupils don't realise they are practising the skills needed to factorise more difficult quadratics
  • Quadratic expression puzzles: The last two sets of puzzles of this set focus on quadratic expressions; great for a bit of extra challenge!
Completing the Square
Using algebra tiles:
Practise the basics activities:
Don Steward:
Tricks and Tips: Quadratics (Jo Morgan)
Four methods for finding the turning point of a quadratic - have a look at the symmetry method (#4) for an alternative approach for those pupils struggling with completing the square.

Standards C1 Linking properties and forms of quadratic graphs (via Craig Barton)
This is in the A Level section (and I use it every year when teaching Core 1), but is a great card sort linking all forms of quadratics, and particularly encouraging pupils to see the link between completing the square and finding the turning points of quadratic graphs.

Completing the square (Kate Nowak)
Teaching completing the square by spotting patterns in complete squares.

Proof without words (Illuminations)
Interactive demonstration of completing the square.

Completing the square (Worksheet Math)
Standard worksheet generator with answers.

Completing the square worksheet (phildb via TES)
Standard worksheet with decent variety of problems.

Completing the square differentiated question cards (headofslytherin via TES)
Tagged as an A-Level Core 1 revision resource, but contains some excellent completing the square problems.

Completing the square codebreaker (alutwyche via TES)
Excellent codebreaker activity requiring pupils to solve quadratic equations by completing the square; includes some questions requiring rearrangement. 

CIMT exercises (Link)
Textbook chapter on quadratics; see pg13-14 for exercises on completing the square, including context-based problems.

Workout (Study Maths)
Question generator for simple CTS problems - good for basic repeated practice.

Exam-style questions (Corbett Maths)
Revision booklet with exam-style questions (1MAO) on completing the square. There's also a workout sheet available.

101 uses of a quadratic equation (PlusMaths)
​Article detailing the importance of quadratic equations.

Completing the square dominoes (Teachit Maths)
Domino loop; pupils match quadratic with completed square.

Minimum/maximum points card match activity (Nuffield)
Match quadratic, completed square and graph; challenging activity which should stretch even the highest achievers!

Completing the square calculator (Wolfram Alpha)
Useful for checking answers when creating resources or for pupils to check independent work.

Geogebra demonstration (Link)
Visual demonstration of standard completing the square formula.
Substitution and Formulae
Substitution
Substitute numerical values into formulae and expressions, including scientific formulae (KS3)
Elementary substitution shade-out grid (alexwoodforth1 via TES)
Great puzzle for basic practice; includes a mixture of negatives and power calculations.​

Algebra Substitution Interactive (via Teacher Led)
Lovely little activity practising simple substitution; includes use of Bidmas and fraction notation. Can be reused by changing the value that pupils need to substitute in.

Substitution Grids (Flash Maths)
Infinite question generator with options to allow negatives and fractions - great with mini--whiteboards or for quick skills practice.

Substituting into formulae crime scene (annah03 via TES)
Excellent murder mystery activity using formulae for area, BMI, Fahrenheit to Celsius conversion and speed. Getting pupils to write and then mark each other's crime scene report using the school's literacy marking policy is a great way of getting a bit of cross-curricular literacy into the lesson as well.

Substitution package (TeachingMaths_ via TES)
Nice collection of resources, including three four-in-a-row games and some simple worksheets.
Formulae
​​Understand and use standard mathematical formulae (KS3)
Translate simple situations or procedures into algebraic expressions or formulae (KS4)
Coming soon!
Changing the Subject of a Formula
Rearrange formulae to change the subject (KS3)
Preparation for A Level Booklet (Aquinas Maths Dept)
​Good examples and worksheets.

Rearranging Real Life Equations (ABCWpresents via TES)
​Superb resource with a complete A-Z of real-life and mostly realistic rearrangement problems.

Grouping Formulae by Changing the Subject (TeachIt Maths)
​Group the cards by finding the four arrangements of each formula. There are eight in total, including fractions and square roots.

Equations and Inequalities

Linear Equations and Inequalities
Solving Two-step Linear Equations 
Use algebraic methods to solve linear equations in one variable including all forms that require rearrangement (KS3)
Derive an equation, solve the equation and interpret the solution (KS4)
Teaching solving equations (The Storeroom)
Blog post on my thoughts and strategies for teaching solving equations.

Solving two-step equations using algebra tiles (The Storeroom )
Full lesson using algebra tiles to develop understanding of solving simple two-step linear equations. Find out more about algebra tiles here.

Modelling algebra (Math Playground)
Fab interactive with a mixture of scale balancing and algebra tiles - could be a useful support tool or a step towards written methods.

CIMT activities (Link)
A collection of quick activities including codebreaking work to support solving linear equations.

Basic practice worksheets (Math Drills)
A huge selection of basic practice problems; select by problem type. Great for quick skills work or adapting into other resources. Quick links to:
Basic practice worksheets (Worksheet Math)
Excellent worksheet generators with lots of customisable options:
Forming equations using perimeter of 2D shapes (HWB via TES)
Pupils form, simplify and solve two-step equations using perimeter of 2D shapes; great for practising solving equations in context.

Master the Basics (The Storeroom)
Transum
  • Think of a Number: Four problems; could be used as a way to introduce the topic or as quick revision for forming and solving.
  • Think Back: Using inverse operations to solve a problem.
  • PYA (Practise Your Algebra): Given three letters/values, create as many equations as possible.

Equation generator (Flashy Maths)
Brilliant interactive for creating and solving equations; emphasises doing the same operation to both sides.

Exploring algebra (Math Interactives)
Great little game for pupils to play independently.
Equations with Brackets and Unknowns Both Sides
Use algebraic methods to solve linear equations in one variable including all forms that require rearrangement (KS3)
Coming soon!
Representing and Solving Linear Inequalities
Solve linear inequalities in one variable (KS4)
Represent the solution set on a number line (KS4)
Coming soon!
Simultaneous Equations
Two Linear
Solve two linear simultaneous equations in two variables (KS4)
Find approximate solutions using a graph (KS4)
Derive two simultaneous equations, solve the equations and interpret the solution (KS4)
Worksheet Generator (Helping with Math)
Generates problems in form ax + by = c, integer solutions only.

Worksheet Generators (Active Maths)
Solving a pair of simultaneous equations by elimination with five difficulty options. Requires subscription.

The Storeroom (C. Norledge)
  • ​Simultaneous shapes 1:  Find solutions to one equation with two unknowns.
  • Simultaneous shapes 2: Find solutions to two equations with two unknowns. This can be done informally by trial and error/reasoning or by forming and solving a pair of simultaneous equations.
  • Simultaneous factors: Systems of three simultaneous equations, one always of the form abc = integer. Good for practising factor listing and basic algebra skills.
  • Three clues - Simultaneous equations: Solve a set of three simultaneous equations , one always of the form abc = integer. Could lead to discussion about how many "clues" (equations) are needed to solve for three unknowns.

Transum
Quick problems that can be solved using a pair of simultaneous equations:
​Solving Simultaneous Equations: French Wine (METAL maths & DeSTRESS stats via YouTube)
Fantastic clip with explanation of profit and loss modelling leading to a pair of simultaneous equations; don't play full program as notation gets offputting after two minutes in. There's a similar video on labour markets that's also worth a look.

Break Even Point (Study.com)
Video looking at break-even points on a lemonade stand; simple explanations.

Matchless (NRich)
Problem-solving with system of five simultaneous equations. Printable version via Teacher's Resources. There are a further four problems like this available from Don Steward.

Systems of Linear Equations Sample Chapter (Big Ideas Math)
Includes a nice break-even problem for a bed-and-breakfast enterprise. 
One Linear, One quadratic and Inequalities (H)
Solve two simultaneous equations, one linear and one quadratic, in two variables
Solve linear inequalities in two variables (KS4 Higher only)
Coming soon!
Quadratic Equations and Inequalities
Solving Quadratic Equations by Factorising
Solve quadratic equations algebraically by factorising (KS4)
Solving quadratics by factorising (Worksheet Math)
Worksheet generator with options to select coefficient of x² and sign variations of all coefficients.
Quadratics using the Formula or Completing the Square (H)
Solve quadratic equations, including those that require rearrangement, by completing the square and by using the quadratic formula (KS4 Higher only)
Solving Equations by Completing the Square (Kuta Software)
Basic worksheets with lots of problems with answers.

Basic Practice Worksheets (Worksheet Math)
Excellent worksheet generator with a variety of options and automatic answer generation:
Solving Quadratic Inequalities (H)
Solve quadratic inequalities in one variable
Represent the solution set on a number line, using set notation and on a graph (KS4 Higher only)
Coming soon!
Iteration
Iteration
xx
Coming soon!

Sequences, Functions and Graphs

Coordinates
Coordinates
Work with coordinates in all four quadrants (KS3)
Coordinating Classroom Coordinates (Idea from NRich)
A good way to introduce coordinates using pupils as points. Could be done in the classroom or outside.

Worksheets from greatschools.org
Cartesian Alien (Flashy Maths)
Aim and shoot the coordinate displayed on the screen to destroy the alien there; cute five-minute time filler.

T-Rex Coordinate Picture Plot (Maths is Fun)
Straightforward picture plot to practise coordinates; I usually use this with Year 7 at the start of the year to good effect! Uses coordinates in all four quadrants
Linear Graphs
Drawing Linear Graphs
Recognise, sketch and produce graphs of linear functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane (KS3)
Coming soon!
Interpreting and Using Linear Graphs
Reduce a given linear equation in two variables to the standard form y = mx + c (KS3)
Calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically (KS3)
Use linear graphs to estimate values of y for given values of x (KS3)
Coming soon!
Equation of a Line and Parallel Lines
Use the form y = mx + c to identify parallel lines
Find the equation of the line through two given points, or through one point with a given gradient ​
Graphing lines (PHeT)
Interactive demonstration of finding the gradient of a line joining two points.
Perpendicular Lines (H)
Use the form y = mx + c to identify perpendicular lines (KS4 Higher only)
Coming soon!
Graphs of Inequalities
Solve linear inequalities in one or two variables
Represent the solution set using set notation and on a graph (KS4 Higher only)
Coming soon!
Non-Linear Graphs
Quadratic Graphs
Recognise, sketch and produce graphs of quadratic functions (KS3)
Use quadratic graphs to estimate values and to find approximate solutions of simultaneous linear equations (KS3)
Identify and interpret roots, intercepts and turning points of quadratic functions graphically (KS4)
​Road tunnel (Nuffield FSMA)
Pupils model a problem about road tunnel heights and curvatures leading to graphical solutions of quadratic equations.

Projectile motion (PHeT)
Interactive demonstrations of a variety of projectiles and the graphs they create.
Cubic and Reciprocal Graphs
Recognise, sketch and interpret graphs of simple cubic functions, and the reciprocal function y = 1/x
Coming soon!
Exponential Graphs
Recognise, sketch & interpret graphs of exponential function y = k^x for positive values of k
Plot & interpret exponential graphs in real contexts to find approximate solutions to problems (Both KS4 Higher only)
Graphing Exponential Functions (MathAids)
​Worksheet generator for graph sketching and finding the equation from the graph. Includes fractions.

Exponential Growth (ProfChrisBishop via YouTube)
Demonstration of an exponential reaction using mouse traps and ping-pong balls.

Moore's Law and The Secret World Of Ones And Zeroes (SciShow via YouTube)
Predicted exponential growth in computer processing speed.
 
Exponential Growth and Decay (CMSP)
Lots of materials for exploring exponential growth and decay in the context of financial mathematics.

Bacterial Growth (dan izzo via YouTube)
A 15 second video clip showing bacterial growth.

Exponents in the Real World (Passy's WoM)
​Loads of real-life examples of exponential growth and decay, which could easily be adapted into examples or problems for pupils to tackle.

Exponential Function Fitting (Don Steward)
​Fit a function in form k(n^x) to the graph.

Radioactive Decay and Carbon Dating (UCE)
Exploring radioactive decay in depth, leading to work on Euler's number (e).
Trigonometric Graphs (H)
Recognise, sketch and interpret graphs of the trigonometric functions y = sin x, y = cos x and y = tan x for angles of any size (KS4 Higher only)
Coming soon!
Transformations of Graphs (H)
Sketch translations and reflections of the graph of a given function (KS4 Higher only)
Coming soon!
Equation of a Circle (H)
​Recognise and use the equation of a circle with centre at the origin (KS4 Higher only)
Find the equation of a tangent to a circle at a given point (KS4 Higher only)
Coming soon!
Real Life Graphs
Graphs from Real Life Contexts
Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs (KS3)
Plot and interpret graphs (including reciprocal graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems (KS4)
Coming soon!
Distance-Time and Velocity-Time Graphs
Plot and interpret graphs to find approximate solutions to simple kinematic problems involving distance, speed and acceleration (KS4)
Coming soon!
Gradients and Areas
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 (KS4 Higher)
Coming soon!
Sequences and Functions
Linear Sequences
Generate terms of a sequence from either a term-to-term or a position-to-term rule (KS3) 
Recognise arithmetic sequences and find the nth term (KS3
Visual Patterns Gallery (Link)
A huge gallery of growing patterns; also includes links to some fantastic ideas from teachers for using these in the classroom.
Recognising other sequences
Recognise geometric and other sequences (KS3)
Recognise and use sequences of triangular, square and cube numbers, Fibonacci type sequences, quadratic sequences, and simple geometric progressions where n is an integer, and r is a positive rational number (KS4) 
Coming soon!
Nth term of a quadratic sequence (H)
Deduce expressions to calculate the nth term of quadratic sequences (KS4 Higher only)
Coming soon!
Geometric Progressions with Surds (H)
Recognise and use simple geometric progressions where n is an integer, and r is a surd (KS4 Higher only) 
Coming soon!
Simple Functions
Where appropriate, interpret simple expressions as functions with inputs and outputs (KS4)
Bridging the Gap materials (AQA)
Lesson ideas and activities specifically for bridging the gap from old KS3 to new KS4 programme of study; some nice ideas for discussion and interesting activities.

Function machine interactive (MathsPad)
Practise substitution into functions using correct notation; four pages ramping up in difficulty, with the final page working from outputs to inputs.
Inverse and Composite Functions
Interpret the reverse process as the ‘inverse function’
Interpret the succession of two functions as a ‘composite function’ (KS4 Higher only)
Coming soon!