Friday, 4 October 2013

My favourite technology-based maths problem

This is my favourite technology-based maths problem, and possibly my favourite maths problem.  It needs to be attempted in software that has linked geometrical and algebraic views (and preferably some others too).  GeoGebra or TI-Nspire are ideal for this.

Start with a square of variable side with one vertex on the origin and one on the positive x-axis:


The problem is to create a rectangle with the same area as the square whose sides are in the ratio 2:1.

Creating the initial square in GeoGebra


  1. Add a point A at the origin.
  2. Add a point B on the x-axis.
  3. Use the Regular Polygon tool selecting vertices at A and B and set the number of vertices to 4.

Creating the initial square in TI-Nspire


  1. Add a new Graphs page.
  2. Add a point at the origin by finding the Intersection Point of both axes .
  3. Add a new Point On the x-axis.
  4. Display the Coordinates of this point.
  5. Use Measurement Transfer to transfer the value of the x-coordinate to the y-axis.
  6. Add a line Perpendicular to the axis through the point on x-axis.
  7. Add a line Perpendicular to the axis through the point on y-axis.
  8. Find the Intersection Point of the two lines.
  9. Add a Polygon through the four points.  NB it is important to use the Polygon tool and not the Rectangle tool.
  10. Measure the Area of the square.

Solutions

If you have a correct solution the area of the square and the rectangle should remain equal as you drag the point on the the positive x-axis.  I know 8½ distinct methods of constructing a solution using GeoGebra and 6½ using Nspire.  How many can you find?

Thursday, 14 February 2013

Modelling mechanics in software


Mechanics is more than just an application of algebra and geometry, it is the primary application of geometry and algebra.  Newton’s development of calculus was to answer problems in what we would now call mechanics and much of his work was geometrical in nature.  However, students study less geometry than they use to and this, along with difficulties in algebra, can result in issues when they start studying mechanics.

As a remedy to this software that links geometry and algebra is perfect tool in which students can practise modelling situations in mechanics.  This can complement their studies in mechanics and enhance their skills.

In particular two main methods that lend themselves well to being represented in software are interactive force diagrams and animations of position.

Interactive force diagrams

Interactive force diagrams can be created where the vectors for the forces acting on an object are represented dynamically.

The diagram below shows the forces acting on a block when there is a pulling force at some angle to the upwards vertical.


Animations of position

Many other situations in mechanics are about describing how objects move and this movement is often defined as function as time.  This may be something as straightforward as the position being given as a linear function of the time (in the case of constant velocity) or more completed examples where the velocity, acceleration or force are given as functions of the time.

Most graph-plotters and dynamic geometry software allow for a function, or the coordinates of a point, to be defined in terms of a parameter, t.  This means that if you know the x and y-position of a particle as a function of the time these can be entered to give the path of the particle.

To further enhance this a slider can be added for t and the both the position of the particle and vectors for velocity (and acceleration) can be shown.  In Geogebra once a slider has been added this can be animated. 

The example below shows the position, velocity and speed for a particle where these are functions of time.


Further examples

A full A level Mechanics 1 paper converted to dynamic Geogebra files can be found at:  http://www.mei.org.uk/?section=resources&page=ict#geogebra

Tuesday, 22 January 2013

Could the drive for using CAS come from students?


In a previous post I wrote about how Computer Algebra Systems (CAS) are really useful tools for teaching when writing questions: http://digitalmathematics.blogspot.co.uk/2011/08/using-cas-for-writing-questions.html  This focussed on teachers using them for preparation but not necessarily exploiting them in the classroom.

Many teachers are reluctant to use them in lessons when they aren’t allowed in examinations.   There is also the view that the purpose of mathematics lessons is for students to learn processes and therefore a tool that can perform those processes will hamper students’ progress.

CAS is out there

There are free tools available online that have CAS built into them such as websites like Wolfram Alpha, and software such as Geogebra and Microsoft Mathematics.  If students find these they can use them when set basic algebraic tasks.  A series of questions on factorising quadratic equations can be performed very quickly on Wolfram Alpha!


Once students are aware of these then the standard tasks set to them may appear pointless.  They may also question why the school subject of mathematics is not using a widely available tool that everyone has access to.  This could be the driver that leads teachers to think about how CAS can be used effectively in the teaching of mathematics, as opposed to ignoring it or banning it.

When to use CAS


CAS is very powerful and used appropriately can give students access to results very quickly that they would not necessarily be able to find accurately without it, at least not without taking a lot of time.  This gives the scope for relationships to be investigated without the limiting factor of always need to perform processes by hand.

A parallel can be drawn with four-function calculators and arithmetic.  If a problem required students to divide two numbers I wouldn’t always want them to demonstrate that they could divide these numbers with a pencil and paper method, especially if the numbers were large or given to a lot of decimal places.  I would often be looking for them to know that division was need and that they could give the answer to a suitable degree of accuracy when found using a calculator.  That is not to say there is or there isn’t a place for students learning a pencil and paper method for division but that is a debate for another time.

If the same logic is applied to students who have access to a CAS calculator this would mean that problems could be set where I would be interested in whether students could select the correct process and give a suitable answer.  For example, in a question that required a derivative I would want the students to know that a derivative was needed and to be able to work with the expression for the derivative obtained using CAS but not demonstrate that they could find this by pen-and-paper (such as using the quotient rule).  Again, this is not to say that there isn’t a place for learning the quotient rule.


A lot of the decision as to whether to use CAS comes down to an individual’s perception of what mathematics is.  For those who perceive maths as a series of processes where the mechanics of them are important then CAS can appear like cheating; for those who see it as the relationship between objects then it can be a useful tool for freeing-up students to investigate these relationships.