Lessons taught; Lessons learnt

Several people have emailed me recently, asking me how I include the Geogebra applets (like this one, showing the nine-point circle) in my posts. First, note that I’m using Wordpress, rather than any of the millions of other blog systems. Secondly, and more importantly, I’m self hosting.

Wordpress.com

As far as I can tell, it’s impossible to include Geogebra files in wordpress.com blogs. Wordpress severely restrict the content you can upload to wordpress.com, both for security reasons, and for financial reasons (you can pay them extra to be allowed to embed video/audio). When I tried, it just stripped out the applet information, leaving the ‘please install Java’ text behind.

If you have a wordpress.com blog, then, it looks like you are restricted to hosting your geogebra files on an external system, like Geogebra.org’s own Geogebra upload manager.

Self Hosting Wordpress

Self-hosting means that I’ve got my own Wordpress installation on this site, rather than using wordpress.com, or one of the other free blogging services.

While you might expect that all the wordpress.com restrictions would be lifted on a self-hosted site, this isn’t necessarily true. While there is no issue with using <applet> or <iframe> tags, I recently upgraded this site to Wordpress 2.9 (from the 2.8 version it was running previously), and found that the new version was much more picky about the file-types it allows everyone, even the site admin, to upload. While this is for good security reasons, it was still a little annoying!

There were several ways to deal with this increased security:

1. I could start hosting the Geogebra files on an external site, as above.
2. I could use the new fileless embedding feature which has recently appeared in Geogebra. This lets you embed small Geogebra creations without having to upload a separate .ggb file. Just make sure the ‘.ggb file and .jar files’ option in the Export dialogue is unticked, and the HTML it generates has the file embedding inside the <applet> tag (encoded using Base 64, from the looks of it).
3. I could add .ggb as an allowed filetype to the Wordpress system.

The fileless embedding is useful, but is only sensible for small files — I wouldn’t want to embed a 500kb geogebra file directly into my HTML! For large files I do still want to upload the .ggb files. Rather than hand-editing the Wordpress source, I installed the PJW Mime Config plugin, which lets you edit the allowed filetypes from the standard admin section of the blog.

Other blogs

Another option would be to investigate switching to another blogging platform. For example, it looks like Blogger/Blogspot will let you embed applets in their free service, without too many problems.

One of the teachers in my school regularly provides a ‘problem of the day’ to his students. This one, from last week, caught my eye, as containing some ‘hidden’ maths that would make a good lower-secondary investigation:

The average of a set of 64 numbers is 64.
The average of the first 36 numbers is 36.
What is the average of the last 28 numbers?

Solution

The total of all 64 numbers is .
The total of the first 36 numbers is .
The total of the last 28 is therefore ,
which makes their average .

Comments

In its ‘raw’ form this is just a basic test that someone understands how to calculate means. It’s nice that the final number is an integer, though… and is it a coincidence that ?

Let’s try the same problem with different numbers:

The average of a set of 51 numbers is 51.
The average of the first 17 numbers is 17.
What is the average of the last 34 numbers?

The average of the remainder is .

Not only do we get a whole number, but the answer is .

Can we conjecture that the answer will always be the sum of the two set sizes? Well, it’s happened twice, so it must be true (!). Looking at the general problem:

The average of a set of  numbers is .
The average of the first  numbers is .
What is the average of the last  numbers?

The average of the remainder is .

Here we might get stuck, unless we remember (or discover, or are taught) at this point the difference of two squares identity: .

This makes our average , and our conjecture is proven.

Extensions

Does a similarly nice thing happen if we specify extra conditions? For example, we could say that there are 5 numbers with an average of 5, and another 9 numbers with an average of 9.

Can we say anything about the median in these situations?

(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!

1% of a day plus 1% of an hour is exactly 15 minutes.

Check with Google if you don’t believe me!