NCETM_Mathematics_Department_Workshops_TGR

Mathematics Department Workshops Topic: Transformations of graphs Overview Objectives related to this topic are: • apply to the graph y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a) and y = af(x) for linear, quadratic, sine and cosine functions (Year 11e) • draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both of the x and y directions (Year 11e) Materials required • Resource sheets HT2.TGR.1 to HT2.TGR.7 • Data projector and internet access Suggested activities Activity 1: Getting Started As a team, discuss what function forms for the base function f(x) are most illuminating when looking at transformations. Activity 2 : Exploring transformations dynamically The link below gives an interactive graph plotter which allows you to define 3 functions. These can be in terms of the other functions, and constants which can then be varied dynamically using sliders, so they can be used to explore the effects of different transformations. Define a function f(x), and then define g(x) = af(x) [you can type it like this or use a*f(x) depending on your learners]. Then go to the parameters tab, and varying the value of a will show the effect of this transformation. You can define a third function and there are up to 3 constants available. The windows tab allows you to redefine the axes scales if necessary. http://enlvm.usu.edu/ma/nav/activity.jsp?sid=nlvm&cid=4_2&lid=109 How does this shape your view of the question in activity 1? If your institution has graph drawing software available (for example Autograph or Omnigraph) you might produce guided exploration worksheets for learners to follow. See HT2.TGR.1, HT2.TGR.2 and HT2.TGR.3 (with thanks to Alan Catley) for samples you could tailor you your software and learners. Activity 3 : Sample learning activities As a team, use the following resource sheets to explore the suggested learning activities. HT2.TGR.4 – Transformation matching A card matching activity with a range of functions and a number of transformations of each function. Discuss which set would learners find easiest and why. Can you think of ways you might vary the activity (for example, what difference is there if you give the learners all 8 transformations but only 4 graphs, or all 8 graphs and only 3 transformations?, or if you introduce a graph or transformation without a match?) HT2.TGR.5 – Trigonometric graph matching As a team discuss how you might use this set of cards with your learners – what are the learning challenges introduced by having more than one function description for a graph? HT2.TGR.6 – Follow me graph transformations (with thanks to Rachael Read) This could be used just as a card sequencing activity – start paths with the f(x) and g(x) labelled graphs, and learners find the correct order of transformations plus next graph to use all the cards. Or the learners could have www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

only the graphs, and you show a transformation for them to decide if they have an appropriate graph to move on with either the path which started at f(x) or the one starting with g(x) [offering the second transformation of a sequence before the first will challenge the learners to identify that they do not have the correct graph for either path. Try both these approaches and discuss as a team what the differences are in the reasoning required from the learners, and whether there are any other, better, ways you can see to use the activity. Has this activity changed your view of the question in activity 1? Unit A12 in The Standards Unit This resource can be found on the QIA Excellence Gateway. It includes an activity on transformations of trigonometric graphs. You can view the resources on-line or download them from http://teachingandlearning.qia.org.uk/pdf/mat_imp_02.pdf Activity 4: Summarising transformations behaviour As a team, look at the resource sheet HT2.TGR.7 and discuss how you would improve it or modify it for your learners – is the language of the summary appropriate for your learners? Are there better examples which provide clearer insights? Embedding in practice Hooks for Learning • http://www.autograph-maths.com/inaction/td/TideTable.htm shows an animation of the process of fitting a trigonometric function to tidal data. • http://www.teachers.tv/video/19119 video on quadratic fitting to the path of a projectile using data collected in the school. Action points At the end of the session, spend time recording some actions. What do you need to do: • Next day? • Next week? • Next year? www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching