• Coles, A. and B. Laurinda (1998) ‘Developing Algebra: A case study of the first lessons from the beginning of year 7’, Bristol, BSRLM

This case study looks at one mixed ability year 7 class investigating what algebra is taught and what algebra is learnt over the period of their first term at secondary school. The motivation for the study is to explore the challenge inherent in the quote: 'Can we develop a school algebra culture in which pupils find a need for algebraic symbolism to express and explore their mathematical ideas?' (Sutherland 1991).

• Winter, J. (2000) Pupils doing algebra: interviews with Year 7 pupils in an ESRC project, Bristol, BSRLM.

Four teachers work with their Y7 classes to develop an algebraic activity and their pupils are interviewed before and after taking part in these activities.

• Hewitt, D. (1998) Approaching Arithmetic Algebraically (PDF), MT163, ATM.

Dave begins this article with a refutation of Vygotsky’s claim that “… algebra is harder than arithmetic” by asserting that “Arithmetic is impossible without algebra”. He then goes on to analyse what it means to work algebraically, how algebraic structures can support arithmetic and how you can help students develop their own structures.
[N.B. you need to be a member of ATM to access this article]

• Watson, A. (2009) Key Understandings in mathematics learning, Paper 6: Algebraic reasoning (PDF), London: Nuffield Foundation

In this detailed and systematic review of how students learn algebra, Anne Watson describes understandings and misconceptions in algebra and offers a series of recommendations; about mathematical learning, for teaching and for research.

• Little, C. (2008) The role of context in linear equations questions: utility or futility?, Southampton, The British Society for Research into Learning Mathematics (BSRLM)

This paper considers the role of context in four linear equation questions, concluding that the purpose of the context is not utility, but concept formation and abstraction. Paper presented at the British Society for Research into Learning Mathematics day conferences held at Southampton University, 26 and 27 June 2008

• Ronda, E. (2009) Growth Points in Students’ Developing Understanding of Functions in Equation Form, University of the Philippines, Mathematics Education Research Journal, Vol. 21, No. 1, 31-53

This article is about functions in Key Stages 3 and 4 and about how students shift from seeing functions as a collection of data points to seeing them as objects that can be transformed.

Originally written for a research audience, this paper is presented within the NCETM ‘Accessing Research’ micro-site and is accompanied by a study module (module 7) in the form of a set of PowerPoint slides to support and guide you through the paper.

• Gould, L., Ikhinmwin, S., Walley, A., Clark-Wilson, A., & Hoyles, C. (2013). Embedding dynamic technologies in the key stage 3 curriculum – The Cornerstone Maths approach. Mathematics Teaching, 235, 44-47.

In this article, the authors describe some of the outcomes of the ‘Cornerstone Maths’ project. Cornerstone Maths is a USA/UK collaboration that offers teachers of Key Stage 3 mathematics some technology-enhanced units of work. The first of the Cornerstone units on linear functions is ‘Sand Circle Mobile Games’ in which students are put into the role of a mobile game software designer for which they need to understand the different ways that linear motion can be described. The article gives a snapshot of the ‘Cornerstone’ approach and 3 project teachers give an account of their experiences.

Icosogram

Draw a pentagon and draw in all the diagonals (lines joining the vertices not including the edges of the pentagon), how many diagonals is this?

This is called a pentagram. How many diagonals are there in a hexagram, heptagram, octagram, nonagram, decagram etc. up to a 20 sided shape.

Does any pattern occur when looking at the number of diagonals?
Why is this so?
Can you find a formula?

Flowerbeds

The council wants to create 85 flowerbeds and surround them with hexagonal paving slabs according to the pattern shown. In this example, 18 paving slabs surround four flowerbeds.

Interpreting Algebraic Expressions A1

Pupils learn to translate between words, symbols, tables, and area representations of algebraic expressions, recognise the order of operations, recognise equivalent expressions and understand the distributive laws of multiplication and division over addition (expansion of brackets).

Creating and Solving Equations A2

Pupils learn to create and solve their own equations, where the unknown appears once. Building equations is easier than solving them because it postpones the second difficulty and so is an easier place to start.

Evaluating Algebraic Expressions A4

Pupils learn to distinguish between and interpret equations, inequations and identities and substitute into algebraic statements in order to test their validity in special cases.

Interpreting Distance-Time Graphs with a Computer A5

Pupils interpret linear and non-linear distance-time graphs using the computer programme Traffic. This program provides a simple yet powerful way of helping learners to visualise distance–time graphs from first principles. The program generates situations involving traffic moving up and down a straight section of road. It then allows the user to take ‘photographs’ of this situation at one-second intervals, places these side-by-side, and then gradually transforms this sequence of pictures into a distance–time graph. In this way, direct correspondences between speeds and gradients are obtained.

Interpreting Distance-Time Graphs A6

Pupils learn to interpret and construct distance–time graphs; relating speeds to gradients of the graphs and accelerations to changes in these speeds.

Interpreting Functions, Graphs and Tables A7

Pupils learn to understand the relationship between graphical, algebraic and tabular representations of functions, the nature of proportional, linear, quadratic and inverse functions and doubling and squaring.

Developing an Exam Question: Generalising Patterns A8

Pupils will learn to: use past examination papers creatively, explore, identify, and use pattern and symmetry in algebraic contexts, investigate whether a particular case can be generalised further, understand the importance of counter-examples, develop the ability to generalise from geometric patterns and devise and explore their own questions in this context.

Example A

1. (i) The first five terms of an arithmetic sequence are

   4

   11

   18

   25

   32

   Find, in terms of n, an expression for the nth term of the sequence.
   Jane says that 697 is a term in the arithmetic sequence,
   
   (ii) Is Jane correct?
   You must justify your answer.
2. The nth term of a different number sequence is 4n + 5
   Work out the first three terms of this number sequence.
3. Here are some patterns made from matchsticks:
   

   Pattern number 1	Pattern number 2	Pattern number 3

   (i) Draw Pattern number 4.
   
   (ii) Complete this table for the pattern sequence

   Pattern number	1	2	3	4	5		…	100
   Number of matchsticks	4	7	10

   
   (iii) Find, in terms of n, an expression for the nth term of the sequence

Example B

1. Eggs are sold in boxes.
   A small box holds 6 eggs.
   
   Hina buys x small boxes of eggs.
   
   Write down, in terms of x, the total number of eggs in these small boxes.
    
2. Mr Smith owns minibuses and coaches.
   
   Each minibus has 12 seats.
   
   (i) Write an expression, in terms of m, for the number of seats in m minibuses.
   
   Each coach has 48 seats
   
   (ii) Write an expression, in terms of
   m and c, for the number of seats in m minibuses and c coaches.

Example C

Simplify these expressions:

1. 3a + 2b + 2a – b
2. 4x + 7 + 3x – 3 – x
3. 3(x + 5)
4. 12 – (n – 3)
5. m(n – p)
6. 4(a + 2b) – 2 (2a + b)
7. 3(x – 2) – 2(4 – 3x)
8. (n + 1) 2 – (n + 1) + 1

Factorise:

1. 4a + 10b
2. p² + 6p
3. 6x² - 9xy

Example D

1. Solve the equation 5x = 30
2. Solve x + 2x = 12
3. Solve 2y – 1 = 13
4. Solve 4(x + 3) = 6

Example E

1. P = 3a + 5b
   
   a = 5.8
   b = –3.4
   
   Work out the value of P.
2. S = 2p + 3q
   
   p = – 4
   q = 5
   
   (i) Work out the value of S.
   
   T = 2m + 30
   T = 40
   
   (ii) Work out the value of m.
3. P = x² - 7x
   
   Work out the value of P when x = - 5
4. P = 𝜋r + 2r + 2a
   P = 84
   r = 6.7
   
   Work out the value of a.
   Give your answer correct to 3 significant figures.

Example F

1. Expand and simplify (x + 7)(x – 4)
2. Expand y(y³ + 2y)
3. Expand p(q – p²)
4. Expand and simplify 5(3p + 2) – 2(5p – 3)

Example G

1. Rearrange the equation y = 4x - 5 to find x in terms of y
2. Make t the subject of the formula v = 5t + u
3. Make c the subject of the formula a = 3c – 4

Example H

Harry left school at 3 30 pm.
He walked home.
On the way home, he stopped to talk to a friend.

His sister, Sophie, left the same school at 3 45 pm.
She cycled home using the same route as Harry.

Here are the distance-time graphs for Harry’s and Sophie’s complete journeys.

1. Find the distance Harry walked during the first 10 minutes of his journey.
2. Find the total time that Harry stopped to talk to his friend.
3. Write down the distance that Harry had walked when Sophie cycled past him.

Example I

Here are five graphs labelled A, B, C, D and E.

Each of the equations in the table represents one of the graphs A to E.

Write the letter of each graph in the correct place in the table.

Example J

1. 
   
   Diagram NOT accurately drawn
   
   The diagram shows two straight lines intersecting at point A.
   The equations of the lines are
   
   y = 4x – 8
   y = 2x + 3
   
   Work out the coordinates of A.
2. Solve the simultaneous equations
   
   5a + 3b = 9
   2a – 3b = 12
3. Solve the simultaneous equations
   
   6x – 2y = 33
   4x + 3y = 9
4. The graphs of the straight lines with equations 3y + 2x = 12 and y = x – 1 have been drawn on the grid.
   
   
   
   Use the graphs to solve the simultaneous equations
   3y + 2x = 12
   y = x –1

Example K

1. A straight line passes through the points (0, 5) and (3, 17).
   Find the equation of the straight line.
2. Find the gradient of the straight line with equation 5y = 3 – 2x.

Juggling with Algebra

This is a Teachers’ TV programme from the series KS3/4 Maths. In this episode, Year Seven students at St Edward's school get an inspiring introduction to number sequences and algebra courtesy of one teacher's juggling skills.

Seven Squares

This is a video showing how three people draw (and therefore see) this matchstick pattern of seven squares:

… and how different ways of seeing might lead to different generalisations and formulae for how many match sticks are needed for 25, 100, n squares.

Building on rich starting points

This video shows the start of a mathematics lesson at Kingsfield School where pupils are presented with data giving hints regarding a function which they then make conjectures about.

Factorising with Multilink

This is slightly beyond the scope of the KS3 Programme of Study statement about factorising by taking out a common factor. However, this video of Charlie and Alison from NRICH showing how expressions like x2 + 7x + 12 can be factorised using Multilink is very clever and engaging and will hopefully prompt you and your students to ask “could I do this with another example?”

Quadratic Equations

This Teachers’ TV video is part of the ‘Hard to teach’ series and show how, at Henry Cort Community College it is not just the maths department that teaches quadratic functions.

The PE, design technology, science and IT departments all work together with their maths counterparts to teach the subject in a way that interests the students.

Straight line graphs

This video resource from Teachers TV features a lesson on straight line graphs to a year 9 class. Using a set of axis drawn on the floor of the classroom, practical activities help students to understand the concepts of gradient and intercept.