Top >10 Mathematics Websites for Students

It struck me that it might be useful to think about my top recommendations for students. Using some categories again gives me the excuse to mention more than 10! All these resources are free to use.

Desmos Art

Francesco Bondi’s art work on Desmos. Click the image to see the graph on Desmos.

For an online graph plotter try the excellent Desmos graphing calculator, it is very easy to use and allows you to save your graphs if you sign up. (Facebook is one option you can use to sign in to Desmos). You can see more examples of Desmos graphs here and there is a helpful user manual you can download from Desmos. There are many creative users of Desmos, have a look at the selection of art work! Make sure you get Desmos on your phone and/or tablet too.


normal proabilities

Calculators

For checking your work WolframAlpha is so useful, it is free to use for checking answers for as many queries as you want (step by Step solutions require a subscription). The set of  slideshows here show you the syntax for a variety of queries.

For more excellent calculators and tools for checking your work, try this collection.


Plymouth University workbook

Calculus workbook from Plymouth University

There are many sites with useful notes and examples online for all ages, you will find several on the Notes pagethis Evernote shared notebook, Mathematics notes includes many links, several universities have very helpful resources which they have made available to all students. You do not have to be an Evernote user (though I’d recommend it highly), just select ‘View’ to access the notebook.

For reference materials see the various resources on the Reference page which includes links to online dictionaries.  


If you like to watch videos to help you learn then you may find some useful resources on the Videos page. Though of course you need to actually do lots of questions!


underground-mathematicsThe best way to learn Mathematics is of course to do Mathematics and there are some excellent sources of problems for students of all ages to try.

Underground Mathematics has an extensive collection of questions to get you really thinking about your Mathematics. Suggestions and full solutions are provided but as always make sure you really do everything you can first with the question.

There are several sites with questions and examples for students of all ages. See more posts with many more resources in the Questions Category.


exponentialsFor revision you can use questions and examples already mentioned, Underground Mathematics includes examination questions for students age 16+. Note the above question comes from an Oxford University Mathematics aptitude test; it is one of the many Review Questions.

Diagnostic Questions

Diagnostic Questions

The Revision pages include questions from UK 15-18 Mathematics Examinations. These all include very challenging questions as well as more routine practice.

For 17-18 Year olds, MadAsMaths includes some very challenging questions for those aiming at the top grades. My student who recommended the site went on to achieve an A* grade!

If your teacher is not using Diagnostic Questions shown here, you can sign yourself up as a student. Note the many Collections of Questions as well as questions by topic. (You need to be signed in for the link to work).


brilliant-levels

Brilliant – Level 5 example

On the Challenges page you can see resources such as the UK Maths Challenges, Nrich, Underground Maths and Brilliant. Signing up to Brilliant (including an easy option for sign in for Facebook users) will allow you to join an international community and get free weekly, personalised problems. Questions at various levels are available. Follow Brilliant on Facebook.


think-outside-the-flockWe all like to play Games, many games are available to help you practise Mathematics, you can see a whole collection on Mathematics Games.

Try Bart Bonte’s logic game ‘Think Outside the Flock’. The game is on Math Playground which I know has many games students enjoy. For more logic games and puzzles try the Logic page.


maths careers

Make a badge!

Are you wandering what mathematicians do or are thinking about a career in Mathematics? Aimed at anyone from age 11 to adult the Maths Careers site will answer your questions.

The Maths Careers site offers you many articles to read, for further reading materials try Plus Magazine from The Millennium Mathematics Project – University of Cambridge or perhaps Math in the News from the Mathematical Association of America or Mathematical Moments from the American Mathematical Society.


The 11 Commandments of Mathematics is perhaps a good way to end this post – and note 10 on Learning Mathematics – think about how you learn not just what you learn. Have a look at these Study Strategies from The Learning Scientists.

For valuable resources to support the techniques described here see the excellent downloadable materials on study strategies. Note how each strategy is backed up by research.


Obviously all these sites are those that I think are particularly good, I do know that many of my students use a lot of the sites I have mentioned here. You will find more recommendations on the Useful links pages. Students do let me know your own particular favourites.

Desmos Apps

This slideshow requires JavaScript.

The best graphing calculator around is free and available on Android, get it on Google play here and is also available for Apple devices – iOS apps.

Desmos on Android has had a major upgrade and you can now access your account with all your saved graphs and also create new graphs. Press the three lines in the top left corner to sign in, sync up, and take all of your graphs on the go.
Android App

Tons of examples

Tons of examples

Also included in this update:

* implicit inequalities
* draggable table columns
* a ton of new example graphs
* bugfixes and performance improvements
…..
Maths students everywhere get this app! It’s great for younger students as it is easy to use for learning about graphs but has the sophisticated features that older students need.
From simple linear graphs to polar or parametric curves – Desmos does it all!
……

There are several examples in the slideshows here to show you the syntax you need on Desmos if you are unfamiliar with it and a series of pages giving you more information.

Box Plots

Box&Whisker - WolframAlpha

Box&Whisker – WolframAlpha

Box Plots show summary information for data – the box shows the median and quartiles and the whiskers are extended to all points which are not outliers.

A very clear description for younger students can be found on BBC Bitesize. For A Level Students the Centre for Innovation in Mathematics Teaching includes Box Plots in Chapter 3 of their Statistics text (page 26 of the pdf file, which is page 76 of the text).

Box Plots are particularly useful when comparing one or more distributions;

CIMT notes

CIMT Statistics Text

To help older students think further about about outliers and skew, have a look at this excellent illustration of Nobel Prize data from Plotly. Who is that outlier for the Nobel Peace Prize?!

Plotly - Nobel Prize winners by field

Plotly – Nobel Prize winners by field

Note that it is possible to display points with Box Plots in Plotly, the following data and charts demonstrates skew well:

Data

Box Plots & Skew

You can learn more about Plotly for graphing and sharing your data here: Learn and explore Plotly 

Desmos on Android

This slideshow requires JavaScript.

The best graphing calculator around is now available on Android (free).  Get it on Google play here. (Desmos iOS apps have been available for some time).

Maths students everywhere get this app! It’s great for younger students as it is easy to use for learning about graphs but has the sophisticated features that older students need. From simple linear graphs to polar or parametric curves – Desmos does it all!

There are several examples in the slideshows here to show you the syntax you need on Desmos if you are unfamiliar with it.

Matrices Resources

A new slideshow on matrices has been added to the WolframAlpha series.
Matrices 

See the spreadsheets by Mike Hadden MatrixDeterimant and MatrixInverse, these enable you to check answers and see working out as well.

If you are looking for notes and examples on Matrices then the following are useful sources and appropriate for Advanced and degree level students.

Notes and examples
Chapter 9 on Matrices and Transformations from the CIMT Further Pure Mathematics A Level material,
Just the Maths,
The Math Centre
and The HELM Project. If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include clear worked examples. You can see on the list that a very small number of titles are ‘not ready yet’; for the sake of completeness the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

Calculators & Tools

Calculating FinancesA whole new section.

I thought I would gather together useful sites/tools for checking your work.

This will be kept updated with any new tools I come across that I think will be useful. I will also add relevant notes and examples.

Do explore the various pages;

I have checked all links are working and made some happy discoveries along the way where tools have been improved. There are many calculators here from basic to rather more advanced. I will keep these pages regularly updated.

On Decision Mathematics for example, check the excellent Linear Programming grapher from zweigmedia.com:

Linear Programming Grapher - zweigmedia.com

Linear Programming Grapher – zweigmedia.com

And look at the calculator from zweigmedia for normal distribution probabilities on the Statistics 16+ page, not only are probabilities calculated but a very clear diagram illustrating those probabilities is also provided.

Normal Distribution Calculator - Random Science Tools and Calculators

Normal Distribution Calculator – Random Science Tools and Calculators

The Graphs section has also had a major update with all the Desmos resources gathered together

Statistics – WolframAlpha

A new slideshow has been added to the page demonstrating WolframAlpha syntax. WolframAlpha can be very useful for checking for example normal probabilities. Each query as you will see in the slides is illustrated with a diagram. It is always useful to sketch a diagram when solving any normal distribution problems.

Integration – applets

If you are learning about integration, there are some excellent online resources to help you. You can of course check your work with WolframAlpha and note that if you ask for a definite integral WolframAlpha will also return a visual representation of the integral, illustrating the area found. This is particularly helpful where parts of the curve are above the x axis and parts below. (You can see further calculus examples as part of Slideshow 4 on the WolframAlpha page).


Using the Integral Machine from The Center for technology and Teacher Education, University of Virginia, you can study approximate methods of integration such as the Trapezium Rule, try increasing the number of strips used and note how the area found using the Trapezium Rule (called Trapezoid here) gets closer to the actual area. You can also see from the shape of the curve why the trapezium rule can give an over or underestimate. Experiment with the four functions given.

Numerical Integration – Zweigmedia

If you use the Numerical integration utility and grapher from zweigmedia’s Finite mathematics and Applied Calculus you can choose your own function, then compare the actual value of the integral with that found by various approximate methods.

For some notes and examples on Integration, you could use resources such as The Math Centre or Just The Maths (starting at Unit 12.1).

WolframAlpha Examples

Can you use WolframAlpha to answer the following questions?
If you need to check the syntax then check the slideshows which illustrate many examples.
Use this file if you want to print out the questions: WolframAlpha Questions

Slideshow 1

  1. -8 + -2
  2. -7 × -6
  3. 6/7 − 3/5
  4. Change 67% to a decimal and a fraction
  5. Find all the factors of 24
  6. Find the highest common factor of 18 and 24
  7. Find the lowest common multiple of 12 and 16
  8. Find the mode of this data set: 6, 7, 8, 8, 9, 9, 8, 7, 2, 3, 6
  9. Find the median of the data set: 6, 7, 8, 8, 9, 9, 8, 7, 2, 3, 6
  10. Find the mean of the data set: 6, 7, 8, 8, 9, 9, 8, 7, 2, 3, 6
  11. Simplify 6a − 3b + a − 2b
  12. Expand 5(2x − 8y)
  13. Solve the equation 6x − 4 = 57
  14. Find the nth term and the next few terms of this sequence: 7, 11, 15, 19, 23, 27 …..  (enter the sequence and scroll down)
  15. Plot the graph of y = 5x − 1
  16. Plot the graph of y = 5x −  1 for values of x from -5 to 5
  17. Reflect the point (2,3) in the line y = 5
  18. Rotate the point  (1,2) 90 degrees about the point (0,0)
  19. Find the surface area of a cuboid with dimensions 4cm by 5cm by 3cm
  20. Find the volume of the same cuboid
Slideshow 2
  1. Solve the equation 5x−1=3x+2
  2. Solve the inequality 6x−1 > 7
  3. Solve the inequality 3x + 7 ≤ 8
  4. x = w+2y, make y the subject of this formula
  5. A = πr2, make r the subject of this formula (you can enter pi for π)
  6. Find the value of  b2+c2  when b = -1 and c = 2
  7. Solve the simultaneous equations 3x+2y=10 and 2x-1=4
  8. Factorise 4ab2+8a2b 
  9. Factorise x2 – 7x + 12
  10. Find the interior angle of a regular octagon
  11. Find the area of a circle of  radius 7cm
  12. Change 42000000 to standard form
  13. Change 5cm to m
  14. Change 4m2 to cm2
  15. Say ‘thank you’ to WolframAlpha! (see Slideshow 7)

Slideshow 3

  1. Find the intercept for the line y = x+3
  2. Find the intercept for the line 3x+4y=24
  3. Find the gradient (slope) of the line y = 2x–6
  4. Find the gradient of the line 3y+5x=10
  5. Find the midpoint of (1,2) and 5,8)
  6. Plot the graph of the cubic function y = x3+2x2–8x+1
  7. Plot the graph of the reciprocal function y = 5/x
  8. Plot the graph of the power function y = 3x
  9. Factorise 2x2 +3x+1
  10. Complete the square for the expression x2–8x–3
  11. Find the minimum value of the expression x2–8x–3
  12. Solve the quadratic equation 2x2+3x+1=0
  13. Solve the simultaneous equations y=2x2+3x+1 and x+y = 5
  14. Simplify x/2 + y/3
  15. Simplify ab2/a2b

A Level students (age 16-18) should study slideshow 4 and experiment with the examples given.
Slideshow 5 is on Differential Equations, Slideshow 6 is on Set Theory and Logic

…..and Slideshow 7 – a few of the sillier questions you can ask WolframAlpha!