Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU Drawing graphs Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU

Thank you Sue, Steve and Paul would like to thank all the teachers and students who have been involved in the trials of these materials Some of the materials are closely linked to the ‘Making Sense of Maths’ series of books and are reproduced by the kind permission of Hodder Education

Note to Teacher This section begins with work on drawing graphs and then moves into an informal treatment of simplifying, expanding, and factorising. The ‘model’ for drawing graphs is ‘Think of a Number’, and the first section of work here relies on strategies from the previous lessons. The aim is for students to make sense of an algebraic expression by firstly reading it as ‘I think of a number….’. Selecting different numbers to ‘think of’ then leads to a table of values and a graph. Graphs of quadratic functions are introduced towards the end of the section.

Reading Expressions and equations How do you ‘read’ the following equation? (5𝑥+ 2) 3 + 1 = 10 This is reviewing ‘Think Of A Number’ from the previous section. Obviously, teachers may wish to do more revision of that idea (using some of the slides from that section) before beginning here. The focus now is on using Think of a Number to read expressions rather than solve equations. This is leading to students being able to read equations in the form Y=mx+c, so that they can complete a table of values and draw a graph.

Reading Expressions and equations And this one? 5x+1=w And this one? This is all leading to the final one here, and being able to ‘read’ the kind of functions that Foundation students will be required to draw graphs of at GCSE. Y=3x-2

x y Drawing graphs Draw the graph of y= 2x + 1 The suggestion here is that there is a table and a pair of axes (perhaps from -6 to 6) on the board. Each student is asked to ‘Think of a Number’ (input) and also to know their ‘answer’. The teacher goes round the class and, for each student, records their numbers both in the table and on the axes. At this point, the teacher is “just recording their answers”. Each student must come up with a different pair of numbers, and they must fit on the axes. This can cause hesitation but in trials, if teachers waited, ‘halves’ etc. started to appear. Enough points leads to a picture of a straight line emerging. This is quite typical of a Foundation Level GCSE question, so the following slides are just looking at the different ways in which such questions may be presented.

GCSE questions-given the values Draw the graph of y= x + 3 x -3 -2 -1 1 2 3 y So here, the ‘numbers to think of’ have been given. In trials, the use of negative numbers here proved to be the main issue

GCSE questions-Finding the values Draw the graph of 𝑦= 1 2 𝑥+2 x y In classroom trials, the issue of plotting points seemed clearly established by this point, and axes tend to be given on GCSE papers. So the focus now is primarily about using ‘Think of a Number’ to generate numbers in a table. In this question, the blank table allows student strategies to emerge (for example, ‘start with zero’, ‘start with positive numbers’, ‘look for a pattern’, etc). The next slide contains three more difficult questions and teachers may wish to practice more graphs of the form y=ax + b before moving on.

GCSE questions-Finding the values Make a table and then draw graphs of; y= 5 – x y = 3 – 2x y= 3(x-1)

y=x2 + x x y Finding values The table here is left blank; in trials, teachers selected x-values based on the level of the students. More typical GCSE questions, where values are given, follow on the next slide.

Copy and complete this table for Finding values Copy and complete this table for y=x2 - 3x x -2 -1 1 2 3 4 y A difficult question. Some teachers omitted this in trials. Draw the graph of Y= x2 - 3x

Summary Think of a Number can help you to make sense of algebraic expressions. This can be extended to allow you to work out tables of values and draw graphs for GCSE.