Integration – Desmos & WolframAlpha

With my year 12 students (UK age 16-17) we have been looking at definite Integration. Desmos and WolframAlpha are both excellent for checking work and by using the technology we have a very clear visual representation adding to our understanding.

One of the homework questions for my students involves finding the total shaded area bounded by f(x) = x4−3x3−4x2+12x, the x-axis, the line x=−1 and the line x=3.

We could use WolframAlpha for a quick check. I like the visual representation showing students clearly that they are dealing with areas above and below the x-axis.
Definite Integral
Scrolling down the page we see that this query also returns the indefinite integral.
Indefinite Integral
For the total shaded area, students could change the limits of the query to evaluate each section.

Or we could turn to the excellent Desmos where we can very simply change the limits.

Definite Integration Definite Integration 2

If you are unfamiliar with Integration with Desmos, turn to Learn Desmos: Integrals.
Note you can explore the graph shown in the video.

We could also show students this introduction to Integration on GeoGebra.
Integration

A version of this post for students which also includes links to some useful notes and examples can be found on Mathematics for Students.
Integration notes University of Plymouth

MEI Ritangle Competition (& Technology!)

From MEI comes Ritangle, a competition for teams of students of A level Mathematics, Scottish Highers or the International Baccalaureate. This year’s Ritangle competition  launches Monday 2 October!

For the main competition, one question will be released daily for 21 consecutive weekdays, the first question will be released on 9th November 2017. I am looking forward to seeing the problems which I am sure will be of interest to many Mathematics students even if they are studying for alternative qualifications and cannot enter the competition.

Something which caught my eye is that Technology can help with some Ritangle questions which led me to some excellent Excel resources, which you can use whether or not you are participating in the competition.

Excel is a favourite for me – I use it in both my Mathematics teaching and in my Deputy Head role. Note from Nrich, we have many uses of Excel for Mathematical Investigation. I like this very clear illustration showing fraction multiplication.

Nrich Fractions multiplication

Excel Interactive Resource – Fraction Multiplication

For teaching sequences, Interactive Number patterns will be useful.

Quadratic Sequences

Nrich – Interactive Number Patterns 2

…and one of my favourites – Happy Numbers! (For the Excel spreadsheet – Nrich investigation)
Happy Numbers

Talking of technology, the Excel fractions resources here reminded me to try fractions with Graspable Math. Having tried Graspable Math on the Interactive whiteboard recently I can confirm it works perfectly.
Add Fractions

And finally, talking of Graspable Math, this led me recently to Dave Taylor’s excellent Increasingly Difficult Questions, a wonderful collection – and with an eye on the copying budget too!
Change the subject DT

Looking at the Graspable Math Teacher Resources, I see some of the Increasing Difficult Questions have been added to a canvas. Graspable Math lets you save your work so you can come back to it later or share it with others. Saving requires a google account. To share a file, use the share menu to get a link that gives others read access to a file. I decided I liked the idea of having a canvas ready for Simplifying Expressions, starting with IDQ-Simplifying Expressions 1, I opened it on my own canvas and adapted it slightly. I do like to keep all steps of the working displayed, so I have put the exercises on the left, creating a good space on the right.
Simplify Expressions IDQ

Writing Mathematics Online

I have written various posts on the available tools online for writing Mathematics and this is a topic which remains consistently popular. Time for yet another revisit and update as new possibilities are now available – all resources / links here have been checked. I have removed resources where blogs / twitter have not been updated for some considerable time as I think this is a cause for concern regarding the longevity of the resource.

Something I use a lot. I should explain my requirements – I want tools to communicate Mathematics online, for example I may wish to provide some model solutions or answer students’ questions. Writing mathematics can be a pain (and yes I know about LaTeX). Note that there are various possibilities – sometimes just a static picture is required, sometimes you may want to display how to solve a problem in stages, or perhaps you require a collaborative space. You will also need to consider if you want the examples to be permanent or whether you just want a collaborative space for discussion. A graphics tablet is essential.

My favourite method for illustrating Mathematics online (and in fact the one I use most often) when I just need a series of static displays is to turn an interactive whiteboard flipchart (or a PowerPoint) into a pdf file; the pdf file can then be sent to students or uploaded to whatever virtual learning environment or online storage your school uses. If you do not have access to interactive whiteboard software there are alternatives, one could use Windows Paint for example; there are also various free online tools available; see some of the resources below.

flipchart to pdf example

flipchart to pdf example

For sharing resources, it is possible to upload a PowerPoint or pdf file to Slideshare. There are many examples on this blog of my SlideShare slideshows – see this for example

I should mention that I find Slideshare excellent – I use the free version which offers me everything I need – it works every time – I use it a lot!
I created the PowerPoint for the slideshow above by writing on the interactive whiteboard software using my graphics tablet and taking a picture of each page using the Windows snipping tool (it’s in Accessories) – this takes seconds – the snipping tool is something I use every day! (Alternatively I could have saved the interactive whiteboard flipchart as a pdf).

There are as always several options:

Screencastomatic

Screencastomatic

If you wish to record a screencast of the moving pen / step by step solution variety and save your work, Screencast-o-matic is an excellent option. It is very easy to use to capture the screen and your recording can then be uploaded to YouTube if you wish.

Illustrating how to simplify an algebraic fraction : simplify-algebraic-fraction

Screencast-o-matic offers everything I want in this category. It is very easy to use indeed – I can write very smoothly whilst recording.

Further resources offering various solutions for writing Mathematics online:

Twiddla

Twiddla

For a collaborative board, try twiddla which seems excellent for collaboration – voice as well if you want. Twiddla offers some very sophisticated features including the ability to use mathematical formulae and upload files and images. Use of the board with all it’s features is free but you cannot save any of your work (possible with a subscription). This would be good for working online with a group of students.

doodle.ly

Doodle.ly

Doodle.ly offers a very easy way to have a quick scribble and then share your thoughts! See quadratic example (and note that I created that just to test Doode.ly years ago – a good sign that it’s still there!) Select Doodle Now, doodle, then if you want to publish you will need to sign in. You can sign in with Twitter or Facebook. I like the different pen tools available, the full screen option and the ease of publishing. Apps are available for Android and iOS.

Writing RepeaterFinally – your students may find this amusing – the Writing Repeater from ICT Games – write something and play it back – now this is a lovely tool for little ones learning to write but I’m sure we can think of some uses!

Handwriting recognition, LaTex and more!

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 MyScript. In this demonstration, handwriting is turned into LaTex (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.

MyScript

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

MyScript

MyScript

But I must confess I was just as excited to note that immediately see a graph where appropriate, powered by my favourite Desmos graphing calculator.

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

improve your scribbles!

Improve your scribbles!

MyScript now has apps for iPad and Android tablets.

GeoGebra

GeoGebra is astonishingly powerful and seems to keep just getting better. It works brilliantly on my phone and my tablet as well as on a desktop. I will be using it a great deal more in future with all the students I teach.

Time for some new pages on GeoGebra, this collection will grow, but I wanted to bookmark the tutorials and note also how to very simply use the Data Analysis tools. Sophisticated analysis is possible of course but in moments one can copy data from a spreadsheet application to GeoGebra’s spreadsheet view and see some charts.

GeoGebraThese Tutorials are an excellent place to start learning how to use GeoGebra. GeoGebra works not only on desktops but on phones and tablets as well.

The Manual is comprehensive and note the Quick Start tutorials which are very clear. You will also find manuals and much helpful documentation on the same page. The great thing about GeoGebra is that so much has already been written you can probably find what you need already online!

You can also use GeoGebra’s YouTube channel to watch demonstrations.

GeoGeoGebra is not just for Geometry, as mentioned above note how good it is for Statistics too, copy in those large data sets and get analysing! (GeoGebra Data Analysis as a pdf file or  PowerPoint: GeoGebra Data Analysis.)

The slides show the Classic application first which perhaps experienced users are most familiar with, followed by the newer Maths Calculators interface. If you are new to GeoGebra I would recommend using the Calculators which of course have the same functionality and more and will give consistency across the various platforms.
GeoGebra classic & Maths Calculators

So much is already written for GeoGebra you can use material already written. For example thinking try MEI’s very helpful advice on the Use of Technology, also on Integrating Technology into schemes of work for older students (UK A Level  age16-18). Note that tasks are also given by type of software including GeoGebra.
MEI Tasks AS

Another source where you will find GeoGebra used to help students understand and explore Mathematics is Underground Maths where many tasks have associated GeoGebra resources. An Underground Mathematics search on GeoGebra reveals the extent to which it has been made use of!

Note also some excellent examples from the ATM Conference 2017.

See also Use of Technology and Statistics in the A Level series of pages.

I would like to thank MEI for an inspirational (and free) conference recently. So many good sessions including the use of GeoGebra for statistical analysis.  A highlight had to be looking at the GeoGebra 3D graphics view with our 3D glasses!