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.

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Robocompass - simple demonstration

Robocompass – simple demonstration

Draw geometric constructions using the very attractive interface that is Robocompass.

demo commands

Robocompass is easy to use, simply type in commands. Select How to for a list of supported commands. Robocompass How To

Select the triangle image to go directly to the Robocompass file. Selecting play allows you to easily see each step. Note that you can select the page to rotate it in any direction.

Robocompass - play

Note that you can also look at individual steps,

demo 2

easily change colour or play speed:
demo 3

Reflection example

Robocompass – Reflection example

Experimenting with Robocompass made me realise that it can provide rather good demonstrations for transformations; see the above example (select the image for the file). Having set this up it is easy to change the line MN:

Reflection example 2

Or we could try a rotation.
rotation example




examplesLearn more about Robocompass by following their blog and/or follow Robocompass on Twitter.

You can learn how to use Robocompass by studying examples; note the given examples or perhaps have a look at this on a Pythagoras proof.





Plotly box plys

Plotly box plots

A very happy discovery today – Plotly, a collaborative data analysis and graphing tool. Looking at the pricing plans the free plan allows for 10 private files to be saved but unlimited public files; Plotly is free for educational use.

Plotly looks very sophisticated with many options and is certainly something I will return to in future posts as I find out more. The images here show the different types of charts which can be created.

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Plotly box plot

Plotly box plot

I was initially particularly interested in box plots as although they can be created in Excel it takes a bit of fiddling around to do so. With Plotly, simply upload your data or copy and paste it in and there are your box plots! Note the outliers shown clearly. Hovering over a plot displays the data.

Plotly Help

Plotly Help

To create the box plots illustrated here I initially started with the Math box plot example from the given examples, chose view data, deleted the given data then copied and pasted my own data (with all three columns in ascending order); I then selected all columns, changed the names to display and chose Box plot.

Choosing Help gives you the choices shown here; the instructions are very clearly set out.

If you want to code yourself, you can; see Plotly’s API Libraries. I will certainly be experimenting further with creating charts in Excel with Plotly.

See the Plotly blog for further information; see this post for example on best fit lines in Plotly or try Contour plots, error bars, chocolate, beer, meat and Facebook for a collection of examples showing the great variety, customisation and sophistication Plotly offers.

An update – Plotly Feed – just admire the beautiful graphs and presentation of information.

You can follow Plotly on Twitter or Facebook. (You do not have to be a Facebook or Twitter user to view these pages).

Update: December 2014 Plotly have put together 11 favourite graphs from 2014 

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Learn and explore Plotly – there are clear tutorials available on the Plotly site.

Thoughts This Week…

A compilation this week…

Learning and Revision
It’s still revision time with public examinations coming up and for our school we can add the younger students too as they will have school exams. Certainly resources that I will use with Year 7 (UK ages 11-12) include the spot the mistake type activity. I recently used the Transformations AfL activity for rotations successfully with Year 7 and also with Year 10 as part of their revision.

Reading Make it Stick (The Science of Successful Learning) which discusses the use of testing as a learning tool convinces me even more that mini-tests are a good idea! This week each of my three Year 10 lessons started with one which seemed to work very well and I was very pleased when two students who had to be somewhere else in the first part of one of  the lessons asked it I would send them the mini test for that day! Students need to recall information and the evidence suggests that testing is a better way of doing this than simply rereading material, a method often favoured by students. I am planning more plenary mini-tests, with the students I’ll definitely use the name I know they like, ‘Self-checks’ which I hope helps them realise they are as the authors suggest a learning tool, not something to be stressed by.

Aristotle apparently wrote “exercise in repeatedly recalling a thing strengthen the memory.”



Select image for problem on Brilliant

I do like the problems on Brilliant and looking at a problem recently, it struck me that these are an ideal sources of starters. Further details on Mathematics Starters and Plenaries.

In the news
The current UK system national curriculum levels have been removed so schools have the freedom to design their own assessments against the new national curriculum. Further details are available on the Assessment Without Levels – KS3 page.

A TES article ‘The revolution is coming, so what should you expect? suggests that the new ‘pass grade’ is to be 5 (even though a 4 is comparable to the current C grade). Equivalent grades are given by TES as follows (TES source JQC)
GCSE gradesGCSE grades chart