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.

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

Math is fun!


A favourite site – Mathisfun! is a treasure trove of resources.
There are clear indices; the Index from the Home page takes you to the very useful Index by Year and Subject. Explanations and examples are very clear and each section has additional questions to try with solutions. Look for example at the Algebra Index and choose any topic to see the very clearly presented information and activities.

Polynomial Long Division
Polynomial Long Division


You can also use the following pages to search :

I often refer to this site and know that students like the clear presentation. I have used everything from Balance when Adding and Subtracting with younger students to working out the inverse of a 3×3 Matrix with the Sixth Form and lots in between! The Dictionary too is very helpful.
Algebra Balance

3x3 Matrices - Inverses

Thinking about my current teaching, with Year 12 we are looking at the Binomial Expansion, we can see some clear illustrations here including a rather nice illustration for (a+b)3.

With Year 10, having met Surds for the first time this year, some additional examples will be helpful; as mentioned above – each section has a quiz
Surds Quiz

My Further Mathematicians are studying Complex Numbers, some rather nice illustrations can be found here for them.
Complex Numbers
..and on the subject of Complex Numbers, John and Betty can provide an introduction!
John & Betty

Spot the Mistake

…and other updates.

Edexcel Model Answers example
Updating the page on Edexcel’s Teaching and Learning materials (part of the A Level (16+) Resources series) I have included their now complete set of GCSE to A Level Transition worksheets and also exemplar answers with examiner comments, a particularly valuable resource. These booklets look at questions from the AS and A level Sample Assessment Materials, which was used in the trial undertaken in summer 2017. Real student responses are shown together with commentary showing how the examining team apply the mark schemes. The commentary includes always useful notes on common errors. Noting that these could be used in class and students asked to find errors reminded me of some more excellent resources – time for an update of the Spot the Mistake collection.

Erica's Errors - A Lutwyche

Particularly excellent resources come from Andy Lutwyche, look at his excellent Erica’s Errors series for Spot the Mistake activities.

For more resources – see the Spot the Mistake collection.

MEI Making Sense of Information
Another updated page in the A Level series is on Statistics; this includes links to all the large data sets used by the examination boards as well as suggestions and resources for teaching. Note the September/October 2017 edition of MEI’s very helpful M4 magazine which has a focus on the teaching of Statistics and includes information and examples of updates on the large data sets for all the examination boards. The PowerPoint resource could also be used with younger students to get them thinking about the presentation and interpretation of data.

Other checked and updated posts include



Open Middle

Open Middle

I was happily distracted this morning by this lovely problem, Create a System of Two Equations by Daniel Luevanos on Open Middle, accessible for students yet such a great task for mathematical thinking. We could discuss inequalities here as well as simultaneous equations.

Noting the link to a Desmos page as a suggested answer I couldn’t resist creating a more general Desmos page.
Open Middle Problem

Graspable Math created a canvas for this problem, in class we could have this available as well as Desmos. Note the scrub feature.
Sim Eq Opn Middle Problem 1
Sim Eq Opn Middle Problem

Now we could systematically change one variable at a time and start talking about inequalities….

If you are not familiar with Open Middle do explore these excellent problems; you can read more about the type of problems you will find on the site on the About page.

About Open Middle

Open Middle

Note the various ways you can search; note the drop-down menus for each grade.


Top Tools for Learning 2017

Jane Hart, of the Centre for Learning and Performance Technologies, has published her annual list of Top Tools for Learning. Like last year, she has three sub-lists including Top 100 Tools for Education. Note that you can easily see the rankings for each sub-list using these rankings. Jane Hart’s analysis includes details of the contributors as well as her commentary on trends.

Looking at the Top 100 Tools for Education I see my favourites WordPress (Blogging and website tool) and Evernote (Personal information system) in there, these are also popular in all the lists.

Continuing to look at the Top 100 Tools for Education Excel is quite rightly highly placed. I regularly use Excel resources; just a few examples of some favourites:

Normal Trainer

Excel Files – Mike Hadden


STEM Centre – Descriptive Statistics

See STEM Learning, part of the A Level (16+) Resources series.

A few more for investigation …
KahootKahoot (Classroom response tool) is very easy to use and free for teachers and students. In a few minutes I created a quiz on Directed Numbers … (not very exciting – just a test, very easy to create.) Kahoot qn

Another popular quizzing tool I know some of my colleague’s use is Quizlet.

Seeing Padlet (online discussion board) on the list reminded me of this very easy to use tool. I shall try this with Year 7!
Padlet equations

Unsplash – beautiful free photos to do anything you like with! Perhaps not surprising that this has moved rapidly up the lists!

Photo by Marivi Pazos on Unsplash

Apart from illustrating Unsplash, I can have pictures of flowers to illustrate a mathematical connection! From Science News reading Fibonacci’s Missing Flowers we discover that the most common number of petals is five and whilst there are many flowers with the number of petals a Fibonacci number there are also flowers with four, six, seven or nine petals!

GrammaryI added a comma in a couple of places in this post thanks to Grammarly which has jumped up the Top 100 Tools list by 70 places. Very easy to use, Grammarly lets you check for 250 types of grammatical, spelling, and punctuation mistakes.