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

For Valentine’s Day…Make Linked Möbius Hearts

The excellent Maths Careers site is managed and maintained by the Institute of Mathematics and its Applications. If your students wonder where Mathematics is used they will find plenty of answers here. See for example Who employs mathematicians?  We even have 9 Maths skills you need to win the Great British Bake Off!

careers-heartsFor Valentine’s Day, have a look at this post from Maths Careers with instructions on how to make this wonderful pair of linked Möbius hearts.

If you wish to get creative and try this I advise watching the Numberphile video carefully; following the instructions worked  as you can see from my creation here! I can verify that unless you follow the instruction to make sure the twist in each strip is in a different direction you will end up with a mess! Quite an interesting mess but certainly not two hearts!….
valentine-mobius-hearts
use-to-create-mobius-heartsNote the Desmos graphs on my strips. I created a file in Word valentine-mobius-hearts (or pdf: valentine-mobius-hearts) with Desmos images in a table. Adding dotted borders to the table gives guidelines for cutting. I began each cut by using the end of a paperclip to pierce the paper. See Valentine’s Day for the Desmos details.

To create my strips I printed the document and then printed again on the reverse. I then cut out and trimmed the strips so there was no white spce at the end – the picture here has been made using strips 10 cells long.

Valentine’s Day …

It’s that time of year again – save your money and send your loved ones (or anybody!) a math-o-gram!

..and see also “Make Linked Möbius  Hearts“.

Click on the image and move that slider!

Click on the image and move that slider!

Desmos have provided you with the means to send a math-o-gram to the mathematicians in your life!
valentine-design

Valentine’s Day seems an appropriate time to express love for Desmos!
Geeky people you could even use the Desmos API …

Remining with the loving theme you can also express your feelings for WolframAlpha!

I Love YOU

valentine-relay

Valentine Relay – Chris Smith


For some Valentine class activities, try the Valentine Relay from Chris Smith and note all the other relays in this excellent set of resources. You can find more excellent resources from Chris on TES and follow him on Twitter here.

Transum Valentine Puzzle
and here’s a Valentine logic starter from Transum.


 

Low Stakes Testing in the Mathematics Classroom

To download:
Low Stakes Testing in the Mathematics Classroom (PowerPoint File, takes a few moments to download)

Low Stakes Testing in the Mathematics Classroom (pdf)

Or use this shorter version to see the slides on the student survey on low stakes tests.
Mini Tests Colleen Young for PowerPoint
Mini Tests Colleen Young pdf version

Slides from the BERA Conference, Learning from the classroom – Practitioner research in mathematics education – July 2016

There are many hyperlinks provided in the presentation, for ease of access these are also provided below:

Further Reading

Solving Linear Equations

A collection of resources to use to demonstrate and practise solving equations.

Student Exercises
I find Owen Elton’s worksheet, Balancing Equations on TES Resources very useful when introducing equations, the diagrams emphasise that we must do the same to both sides.
(See Diagrams in Mathematics, for more on the use of diagrams to help understanding).

Balancing Equations

Balancing Equations – Owen Elton

Linear Equations - Don Steward

Linear Equations – Don Steward

As an alternative to the balance approach, consider this doing / undoing approach, described here by Don Steward, this is an approach I use for finding inverse functions. The exercises Don refers to are here.

A2 - Mostly Algebra

A2 – Mostly Algebra

One of the Standards Unit resources, A2 Creating and solving equations (in Mostly Algebra) uses this approach, students create an equation and  then undo it; this is a great exercise for demonstrating notation. Another resource with this approach comes from the Mathematics Assessment Project (the design and development was led by the MARS Shell Center team at the University of Nottingham) Building and Solving Linear Equations lesson

For a superb collection of ideas and student exercises for solving linear equations see all Don Steward’s posts tagged linear equations. Many of these outstanding resources use a very visual approach with very clear diagrams to help students’ understanding.

Diagnostic Questions

Diagnostic Questions

There are numerous questions on linear equations on Diagnostic Questions.


WolframAlpha solve equationA simple way to check a solution to an equation of any type is to simply enter your equation as a WolframAlpha query. Note that WolframAlpha includes a graphical illustration; it is so important for students to understand how equations may be solved graphically; I always illustrate graphical solutions when we are working with Algebra to help students make these links. Desmos of course, is ideal here.
Desmos & solving equations


Centre for Innovation in Mathematics Teaching
CIMT Interactive Resources
CIMT
have tutorials on equations: Linear Equations 1, Linear Equations 2, and Linear Equations with Brackets in their Interactive Resources


The following three resources work well for demonstrating the balance method of solving linear equations.

Duncan Keith’s Linear Equation Calculator is available on STEM Learning.


Choose the type of equation you require then the sequence of operations required to solve the equation.
Select Do it after each operation, for example -32 Do it were the keys selected to start the above problem.

The slideshow below shows how to use the calculator to solve equations where the unknown is on both sides.


Mathisfun

Mathisfun

Mathisfun

Mathisfun has this very clear and easy to use interactive illustrating the solution of linear equations.