Lessons taught; Lessons learnt

I run a weekly maths club for a group of primary children in the area, and part of this involves creating or altering resources so that they can be picked up and understood by ten year olds. The first resource from the club that I'm contributing to this blog is my template and instructions for creating a trihexaflexagon.

While there are no shortage of flexagon sites, descriptions and videos around (and I particularly like the Murderous Maths one, as I've seen Kjartan perform, and he was very entertaining), there does seem to be a gap in the market for a simple blank template, together with a simple one-page instruction guide to putting them together.

Source: Inkscape SVG and PDF.

Source: Inkscape SVG and PDF.

I'm happy to work on producing similar instructions for the more complicated flexagons -- leave a comment if you'd be interested.

(Source: conic_family_plot.ggb.)

Questions and suggestions

Press the play button, and the red curve will animate, showing you some of the possible curves you can get by changing a, b, c and d.

Click 'show controls' and you can alter a, b, c and d yourself. You can also click the checkboxes next to a, b, c or d to see a 'family plot':

These family plots will also animate (and look quite pretty when they do!).

• What are the values of a, b, c and d in the red curve?

  [Answer: at the start, they are all equal to 1.]

• What types of curves would you get for other values of the constants?

  [Answer: The curves you get from these equations are ellipses and hyperbolas.]

The equation is a special case of the equation , which generates all conics.

As well as being a pretty thing to have in my room as a class enters, it could also serve as the basis of investigations into conics (either in the special case shown in the applet, or in general). For example:

• Can we tell by looking at an equation whether it will be an ellipse or a hyperbola?
• Can we tell by looking at the equation whether any points will appear at all, or whether the equation has no solutions?
• Can we classify all the equations which go through one/two/three/more specified points?
• Can we go backwards, from a diagram we want to the equation?
• Why do the family plots look like they do?

Inspired by this excellent post on the 'Point of Inflection' blog, here's an interactive cylinder, which will let students explore the relationship between radius, height, and volume. The post linked gives some great throughts on the benefits of using interactive examples like this in classes.

Incidentally, I've just noticed while creating this post that the new version of Geogebra allows you to embed a simple Geogebra applet completely in HTML, without having to upload a separate .ggb file. A wonderful advance, but it does make it impossible to save the applet to your own computer. I would love to link to the .ggb file here, but the new version of Wordpress seems to have implemented some odd 'security guidelines' for uploads that I need to hunt down and disable!

The new version of Geogebra has several nice new features which make it much more useful for a range of statistical uses. Firstly, it comes with a simple spreadsheet-style view, which allows you to enter and manipulate data in a grid of cells, similar to a spreadsheet. Secondly, it has a number of new statistical functions, covering a range of data creation, summation, and visualisation options.

Here is an applet which demonstrates a couple of these new features. It takes 50 random values, generated to fit a Normal(8,4) distribution, and plots a box-and-whisker plot and a histogram.

All this just uses what are now built-in features of Geogebra. My contribution is to make the histogram interactive: move the blue points on the x-axis around to alter the class boundaries. This lets you explore the ways that small changes to the class intervals can sometimes have large effects on the histogram.

Enable Java to see this Geogebra applet.

(Source: adjustable_histogram.ggb.)

Double click on a cell in the spreadsheet view to change its value. Also, just as in Excel, press F9 while in the spreadsheet view to regenerate all the random numbers. You may see the purple x-axis points move: they have been constrained to always be below the minimum and above the maximum value.

This term my school is running some maths competitions involving timed rounds. The questions are in Word, and I've been getting quite frustrated with the countdown timer built into Activstudio -- in particular, it makes resetting the count very fiddly. There are a lot of countdown/timer applications around, but the ones that do what I want seem to be either stupidly overcomplicated, not free, or to crash all the time.

To get around my frustrations I've cobbled together the following simple countdown timer, written in Javascript:

Source: countdowntimer.html.

Not particularly flashy, but it gets the job done!

You can start/stop/reset the countdown, and alter the starting count in 30s increments, just by pressing buttons, which makes it easy to use on an Interactive Whiteboard. In addition, if you click on the ^^ link (at the bottom right) the timer will open in a new window, with as few toolbars as your security settings allow.