Inverses of 3x3 matrices calculator applet

Inverses of 3x3 matrices using cofactors appletMy new applet on inverses of 3x3 matrices

I’m pleased with this applet because I’ve been think­ing of doing this one for years. To get what I’ve wanted I’ve had to do some hard think­ing about the pro­gram­ming, and I now have some tech­niques  I can reuse time and time again.

Why a cal­cu­la­tor? The applet can be used to enter any reasonably-sized matrix and cal­cu­late its inverse if it exists, so you could call it a cal­cu­la­tor. I expect some peo­ple will use it as such. But it’s real pur­pose, as is usual with Wal­do­maths applets, is to demon­strate the method of cal­cu­la­tion and give insights into the math­e­mat­i­cal behav­iour of matrices.

Already I’ve played around with it myself for hours. It’s great not hav­ing to do end­less repet­i­tive cal­cu­la­tion myself to see results.

I’m already plan­ning an applet of find­ing inverses using row reduction.

 

Lengths and Mid-points of Line Segments applet

My new applet on Coor­di­nate Geom­e­try con­cen­trates on two impor­tant skills for post-16 maths, espe­cially for UK AS lev­els Core 1 and 2.

Coordinate Geometry 1 applet

Lengths of Line Segments

This topic often seems to be the first use­ful math­e­mat­i­cal appli­ca­tion of the Pythago­ras Rule (Pythagorean The­o­rem) that kids come across. Ear­lier the focus is on con­trived right-angled tri­an­gles. How many of them are going to find the height or the guy-length of a flagpole?!

d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2}+{{\left( {{y_2} - {y_1}} \right)}^2}}

It also encour­ages stu­dents to cal­cu­late by using a for­mula rather than a graph — you don’t need to draw or even imag­ine the triangle.

Mid-points of Line Segments

Once again a sim­ple idea, but one which allows stu­dents quickly to ditch the dia­grams in favour of an easily-applied formula.

P = \left( {\frac{{{x_2} + {x_1}}}{2},\frac{{{y_2} + {y_1}}}{2}} \right)

 Pretty soon the idea that the coor­di­nates are just the means of the x- and y–val­ues and it is this rather than the dia­grams which sticks in the stu­dents’ minds.

So why ..

So why haven’t I given the for­mu­lae in the applet? This is because of a basic premise of Wal­do­maths — allow stu­dents to see things for them­selves. This way the for­mula seems almost obvi­ous and hence much eas­ier to remem­ber. Stu­dents also learn the tech­niques their own way and apply them with con­fi­dence, rather than the all-too-common rote learn­ing of for­mu­lae which leads to stress and errors.
Some­times it’s depress­ing to think that an applet which has taken me hours of work can achieve its aim within a few sec­onds of a stu­dents time. But then again that’s what Wal­do­maths is all about.

So why do I show the work­ing? Surely they need to able to work it out for them­selves? Well, you can turn the work­ing off!

Quadratic discriminants applet

I’ve just fin­ished updat­ing my qua­dratic dis­crim­i­nants applet. It is larger, more infor­ma­tive, and is in the two usual for­mats, dark on light back­ground or light on dark back­ground. It draws the graph of the func­tion on coor­di­nate axes, gives solu­tions,  in surds (rad­i­cals) if nec­es­sary, of the equations.

Quadratics - discriminants and solutions of equations applet

It also uses my recently-developed drag-boxes to change the val­ues of the para­me­ters. These are sim­ply a box on screen with a num­ber inside: if you click the num­ber it increases as you drag right or decreases as you drag left. I hope that once you’re used to it works much more quickly than the old but­tons. I’m quite proud of these boxes!

Quadratics - discriminants and complex conjugate rootsThe applet can also be used to demon­strate the com­plex con­ju­gate roots that equa­tions have when the dis­crim­i­nant is neg­a­tive. This is use­ful for more advanced stu­dents, study­ing (UK) Fur­ther Math­e­mat­ics (FP1) or (US) 10th grade+ (Num­ber) Mathematics.

Enjoy play­ing around with this new ver­sion, and let me know what you think.

 

 

Extended Factors applet

Gallery

This gallery contains 1 photo.

I’ve extended my Fac­tors applet to include num­bers up to 9000. This is an exten­sion into 4-digit num­bers, and the list now con­tains the first four per­fect num­bers. Num­ber of fac­tors applet I’ve also included a prime fac­tor­iza­tion for each … Con­tinue read­ing

Welcome to my new blog

Hi and wel­come to the new Wal­do­maths blog.

I aim to announce all new stuff and updates on this blog, and to keep my vis­i­tors up to date with what I’m try­ing to do. Please let me have any ideas that you feel like sharing.

I also intend to write about my thoughts on Life, the Uni­verse and Every­thing, with a light-hearted math­e­mat­i­cal slant.