Thinking about teaching functions in the next few weeks (to UK Year 12 ages 16-17) I realised that I could use Desmos to illustrate composite functions; the following slideshow illustrates the syntax.

We can also use Desmos to illustrate a function and its inverse. To create the page below (select the image), I started with a graph already online illustrating the general case of a quadratic function and its inverse and simplified it. f(x) and g(x) can be changed to a different function and its inverse. Note that the domain of f(x) can be changed.


Further examples: exponential function and basic quadratic (where we need to restrict the domain for an inverse function to exist).

Staying with Desmos, as I have mentioned before, the function notation is excellent for transformations:

(See this page for all Desmos slideshaows).

IFunctions two argumentst is also possible to define a function with more than one argument and use Desmos as a calculator

For some clear examples and a resource to point students to, Functions from The University of Plymouth Mathematics Support Materials is useful. The format used in this series makes the examples clear and all the exercises given have answers.

Functions - Plymouth University

Functions – Plymouth University

Other useful resources (requires Java) include the Wisweb applets, algebra arrows could be used to demonstrate functions and their inverses as shown in the following images.

This slideshow requires JavaScript.

Negative Numbers – Resources

Sometimes resources for younger students can be useful for lower secondary age students, see for example Mark Robinson’s Numberlines  from the old Ambleside Primary School site which includes an option to display a number line from -5 to 5.

For another excellent number line resource see J Barrett’s Numberline Jump Maker on I often recommend that students sketch a number line to help with addition and subtraction problems and very clear resources like these can really help. Teacher Resources on Line includes a Big Number Line under Basic Materials for display on a classroom wall

Games can be an excellent way to practise with negative numbers see for example games such as Connect 3 from Nrich and Tic Tac Go, a Wisweb applet.

Further resources include exercises from Trinity School in Nottingham (under Number) and Interactive Resources from CIMT (see unit 3, 3.3 on Negative Numbers and Unit 15, 15.1 and 15.2 for operations with negative numbers in the tutorials section).

There are many excellent resources on TES, the resource collections includes a section on Topic Specials which includes 10 of my favourite resources on Negative Numbers.