Using Matt Parker’s Platonic solid video with kids

Saw this neat video about Platonic solids from Matt Parker a few days ago:

and decided to try to use it with the boys. The obvious idea was to use our Zometool set to make a “hyper-diamond,” though actually making the shape proved to be tricky.

I used this video to get a better look at the shape:

and also looked here to see some other Zome versions of the 24-cell:

The 24-Cell on Dave Richter’s Zometool website

It was interesting to learn that there were multiple ways to make the 24-cell out of Zome struts. I chose the way that uses the green struts because that was the easiest shape to see in the videos. I built the shape myself because the connections were a little more delicate than I expected. Getting the green struts to connect without other struts disconnecting was particularly tricky (or maybe it was just that I was tired and have a cold . . . ). Anyway, here’s the shape:

And here’s what the boys had to say about the shape after watching the videos. It is always really interesting to hear kids talk about the 4th dimension:

Older son first:

Younger son second (and I love how he describes the rotating shape in the video):

So, a fun project giving kids an interesting peek at a 4 dimensional shape. Thanks to Matt Parker for inspiring this project.

A challenging algebra problem from the 2011 AMC 10b

This problem gave my son some trouble this morning:

Problem #19 from the 2011 AMC 10b

The problem asks you to find the product of the roots of the equation:

\sqrt{ 5|x| + 8} = \sqrt{x^2 - 16}

A lot of ideas from Algebra come into play on this problem and we had a good conversation about it tonight.

We started by talking through his ideas about the problem and found a couple of interesting misconceptions about algebra and equality right at the beginning:

Next we went back to look if the 4 solutions that we found to the equation were actually solutions:

Finally, I went back through the problem to show him how the 3 and -3 showed up even though they are not solutions.

The emergence of these spurious solutions makes this a really instructive problem.