May 30 2014

Functions

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).

I Functions two arguments t 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

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.

Feb 13 2014

Happy Valentine’s Day

Click on the image and move that slider!

In what happily seems to have become an annual tradition Desmos have provided you with the means to send a math-o-gram to the mathematicians in your life!
Desmos Valentine instructions

This year you can even use the Desmos API …

Elsewhere – express your feelings for WolframAlpha!

and here’s a logic starter from Transum for Valentine’s Day!

Wishing Mathematicians everywhere a happy day and if you are a UK teacher about to start half term – have a lovely week!

Feb 1 2014

Transformations with the Desmos Graphing Calculator

This week Year 10 (UK age 14-15) have been exploring different graph types and also transformations and graphs.

For homework I asked them to draw just a small number of graphs by hand but wanted them to check their work and explore further graphs using the Desmos graphing calculator. Early in the week I made sure they could all use Desmos including the use of tables so in an IT room they used the slideshow here and created several graphs of their own.

Once all the students were confident to use Desmos to create various lines and curves I asked them to explore a series of graphs so that this coming week we can discuss transformations and graphs. Using Desmos allowed them to explore many graphs in a short space of time and several students chose to take screenshots and make notes for themselves.

Desmos – simple transformations example

Having used sliders they were able to create this type of graph page.

I have created the slideshow below to use in class to summarise our work and act as a revision aid for them.

These slideshows are both available here for students.

Dec 21 2013

Desmos, WolframAlpha & handwriting recognition!

December 2014 – Note that MyScript (formerly Vision Objects) now has apps for iPad and Android tablets.

Reviewing popular posts and checking for correct links caused me to revisit my post on handwriting recognition and LaTeX. The original post has now been corrected (in case people have it bookmarked). I have also reproduced the post below.

Web Equation by Vision Objects

A consistently popular post on this blog is that on online whiteboards. If I want to communicate mathematics online to answer a student query for example I find it quicker to use a graphics tablet and an online whiteboard.

I do keep an eye on various LaTex generators, one that has come to my attention is Web Equation; from MyScript. What I like about this is that handwriting is turned into LaTex (only one line at a time). The handwriting recognition is impressive and I found it easy using my graphics tablet to enter expressions accurately; see the quadratic formula below for example.

Web Equation by MyScript

So you scribble an expression and it get turns into LaTex for you – it works:

$x^{2}+y^{2}=25$

$\dfrac {-b\pm \sqrt {b^{2}-4ac}} {2a}$

Note that if you want to copy / edit the Latex then just select the LaTex expression and you will have the opportunity to copy or edit it.

But I must confess I was just as excited to note that now you can immediately see a graph where appropriate, powered by my favourite Desmos graphing calculator (it appears the scales are fixed) and by the option to ‘Compute with WolframAlpha’! (see above image).

..and finally if you wish to be distracted by some more fun applications there are some other great demonstrations from VisionObjects. Try Web Shape or example and turn your sketches into vectorized shapes. This should work well on the interactive whiteboard.

Improve your scribbles!

Nov 23 2013

In class this week ….

Year 7 (age 11-12)
…had an Independent homework this week and I was really pleased with the variety of choices made. Some chose to use online resources / their text to practise a topic tested recently, others worked further on some investigations we have looked at in class, some chose to do a Junior Maths Challenge. A couple of students used the Diagnostic Questions site to practice their chosen topic. One of the investigations we have looked at recently was on the number of winning lines in a noughts and crosses game – the 2 dimensional game was fairly quickly sorted out – so moving to 3 dimensions is the natural extension.

Nought & Crosses

Some were fascinated by this and determined to find the number of winning lines in a game of 3D noughts and crosses – two students so keen that they made a model of their own to help them answer the question successfully! They made some counters as well!

I have read some great posts on feedback recently, see Alex Quigley’s Improving Written Feedback and Verbal Feedback Given….. on Shaun Allison’s ‘Class Teaching’. Such a good idea, I couldn’t resist – so I bought myself the stamper and decided to try this with Year 7 – early days yet, but I love the idea so far..

Another student in the same Year 7 class mentioned above was determined to find the number of winning lines in a game of 3D noughts and crosses which she did successfully. She drew some clear sketches of the different groups of winning lines and after a discussion with me was the recipient of my first stamp! I was very enthusiastic about her written work on this as you can see from her response! I had explained to the class that if I discuss their work with them, then give them a stamp – they have to write down that feedback as Shaun suggests in his post.

verbal feedback v3

Year 10 (age 14-15)
Year 10 and I have been looking at inequalities – using algebra and graphs. After a ‘self-check’ (somehow that sounds so much better than test and the emphasis is very much that it is useful to see what they can recall) in class, the feedback from the students was that most of the questions were fine but any problems seemed to be on working out the equation of a line when trying to describe a given region. The wonderful Desmos graphing calculator to the rescue – this page allows students to experiment with several variables. (Post for students here.)
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,Update – since writing this post, along came Desmos…

and created a superior version! This shows many of the awesome Desmos features. I really must explore these further – note that where a folder is used selecting the arrow shows or hides the contents. This is a brilliant feature for creating pages to share with students.

Gradient of a straight line by Desmos

I mentioned feedback from the students – a simple feedback idea – I asked this class as the first part of a homework to email me with any outstanding queries from the mini test, having looked at the worked solutions on our homework blog. I have been impressed by their understanding and articulation of what they initially found tricky.

Year 12 (age 16-17)

zweigmedia – numerical integration utility

In Year 12 we have been studying integration. Teaching the trapezium rule I remembered some applets I used last year which demonstrate this well.

Graphs on WolframAlpha

The Year 12 Further Maths class are currently studying Decision Mathematics and a student asked what WolframAlpha can do?! Looking at the examples we can see that describing connections between nodes will generate images of a graph. I must explore further!

Normal Trainer – Mike Hadden

Year 13 (age 17-18)
Studying the Normal Distribution I showed students Mike Hadden’s excellent normal trainer which generates an endless supply of examples. I really like the step by step approach and the very clear page on the backwards problems.

With the Year 13 Further Mathematicians I was reminded of the huge variety of Tarsia puzzles around and that these are not just for younger classes. In Year 13 we use a Tarsia on De Moivre’s theorem.