Seeing ideas about substitution for the first time

My son had an interesting problem on his enrichment math homework this week, and it gave him a lot of trouble this morning:

Tonight I thought it would be good to talk through the problem since I think the main idea he needed to solve it was new to him.

Here’s the introduction and some of the ideas he tried this morning:

Next we took a look at the equations on the computer and talked about some of the ideas we saw:

After looking at the graphs of the equations on the computer we came back to the whiteboard to talk about substitution.

Finally, having worked through the introductory part of u-substitution in the last video, I let him finish off the project on his own.

I can’t remember talking through this topic previously, but it was fun. It is always neat to be there when a kid is seeing a math topic for the first time.

Today I got one step closer to a long-term goal

One of the math mountains that I’ve always wanted to try to climb is to find a way to explain to kids why 5th degree polynomials can’t be solved in general.

The “one step closer” came from a comment by Allen Knutson on one of our projects on John Baez’s “juggling roots” tweet. Here’s the tweet:

Here are the two recent projects that we’ve done after seeing that tweet. Knutson’s comment is at the end of the first post:

Sharing John Baez’s “juggling roots” tweet with kids

Sharing John Baez’s “juggling roots” post with kids part 2

The comment pointed me to a video that shows how the “juggling roots” approached can be used to show that there is no general formula for finding the roots of a 5th degree equation:

The neat thing about the combination of this video and Baez’s post is that you can see some of the ideas from the video in the “juggling roots” gifs in the post.

Tonight I used some of the 3d prints of the juggling roots that I’ve made in the last few days to talk about the ideas a bit more and then we watched just a few minutes of the video.

We started with with a print that I accidentally made twice – but luckily the two prints give us a way to view the juggling roots through two cycles:

Next we looked at a different print to see a different juggling roots pattern. Here I was trying to set up the idea that the roots can move around in different ways. The way those different movements interact is the key idea in the video that Allen Knutson shared.

Finally, we went upstairs to watch a little bit of the video. Sorry for the sound issues, I don’t know why I left the sound on in the video. I mainly wanted the boys to see a different view of the juggling roots and I told them that the video gave the explanation for why 5th degree polynomials can’t be solved in general:

So, although I don’t quite have a full explanation of 5th degree polynomials for kids – I feel like I took a giant step towards getting to that explanation today. It is an extra happy surprise that 3d printing is going to come into play for that explanation!

Sharing John Baez’s “juggling roots” post with kids part 2

Yesterday I saw this incredible tweet from John Baez:

We did one project with some of the shapes this morning:

Sharing John Baez’s “juggling roots” tweet with kids

The tweet links to a couple of blog posts which I’ll link to directly here for ease:

John Baez’s “Juggling Roots” Google+ post

Curiosa Mathematica’s ‘Animation by Two Cubes” post on Tumblr

The Original set of animations by twocubes on Tumblr

Reading a bit in the comment on Baez’s google+ post I saw a reference to the 3d shapes you could make by considering the frames in the various animations to be slices of a 3d shape. I thought it would be fun to show some of those shapes to the boys tonight and see if they could identify which animated gif generated the 3d shape.

This was an incredibly fun project – it is amazing to hear what kids have to say about these complicated (and beautiful) shapes. It is also very fun to hear them reason their way to figuring out which 3d shape corresponds to each gif.

Here are the conversations:

(1)

(2)

(3)

(4)

(5)

(6) As a lucky bonus, the 3d print finished up just as we finished the last video. I thought it would be fun for them to see and talk about that print even though (i) it broke a little bit while it was printing, and (ii) it was fresh out of the printer and still dripping plastic 🙂

The conversations that we’ve had around Baez’s post has been some of the most enjoyable conversations that I’ve had sharing really advanced math – math that is interesting to research mathematicians – with kids. o

Sharing John Baez’s “juggling roots” tweet with kids

I saw an incredible tweet from John Baez last night:

The tweet links to a couple of blog posts which I’ll link to directly here for ease:

John Baez’s “Juggling Roots” Google+ post

Curiosa Mathematica’s ‘Animation by Two Cubes” post on Tumblr

The Original set of animations by twocubes on Tumblr

So, I think the path that the animation took to our eyes was from twocubes to curiosamathematica to John Baez to us. Sorry if I do not have the sources and credit correct, but I will make corrections if someone alerts me to an error.

I’d never made any sort of animation before, but since the pictures looked like they came from Mathematica I started to play around a little bit last night to see what I could do. In doing so I learned about Mathematica’s “Animate” and “Manipulate” functions and made some progress, though the animations that I made were not nearly as good as the ones from the above posts. This Stackexchange post was helpful to me in improving the quality of my animations, but still mine aren’t in the same league as the original ones:

Why is my animation so slow?

Anyway, with that introduction, I thought it would be really fun to share these animations with kids and do a tiny bit of background explanation. I stared this morning by just showing the boys some of the pictures and asking them to describe what they were seeing:

Next I showed them one of the animations that I made and asked them to see if they could see some similarities with any of the previous animations:

Next we went down to the living room to talk about roots of equations. My older son knows a little bit about quadratic equations, but only a little bit. I didn’t want this part of the conversation to be the main point, but I did want them to get a tiny peek at the math behind the animations we were looking at today:

Finally, we went back up to the computer to look at some of the animations for quadratic and cubic equations. My maybe too open-ended task for them here was to compare the animations of the roots of quadratic and cubic equations to the animations of the roots of the quintic equations.

I’ve always wanted to be able to share some of the basic ideas from Galois theory with kids. I’ve never seen anything like these animations previously. They make for a neat starting point, I think, since kids are able to talk about the pictures. I would **love** to know what a research mathematician sees in the pictures. In particular, is there something in the pictures that gives a clue about why the roots of 5th degree polynomials are going to be more difficult to study than 2nd, 3rd, or 4th degree ones?

What a kid learning math can look like – understand equations

This morning a problem from Art of Problem Solving’s Algebra book gave my son  some trouble.  I wouldn’t have picked it as a tough one ahead of time, but he was stuck and we decided to talk through it.

Here’s the problem and his initial thoughts – the idea of turning the words into an equation is what was giving him trouble:

 

Next we moved on to the 2nd part of the problem -> writing a new equation once we’ve removed 84 pennies. This step goes relatively smoothly after having worked through the problem in the first step:

 

The last step was to solve the equations. Ahead of time I would have guessed that this would have been the most difficult step, but it turned out that the mechanical steps required here went pretty well:

 

So, an interesting problem and also interesting to see where the difficulty was. I sometimes (actually quite often) do a pretty bad job of guessing where the difficulty in problems will be. This problem was a nice reminder that turning words into math is something that takes a little practice and a little getting used to.