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.

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!