Tag algebra

Looking at the complex map z -> z^2 with kids

Yesterday we did a fun project using Kelsey Houston-Edwards’s compex number video:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

The boys wanted to do a bit more work with complex numbers today, so I thought it would be fun to explore the map Z \rightarrow Z^2. The computations for this mapping aren’t too difficult, so the kids can begin to see what’s going on with complex maps.

We started by looking at some of the simple properties. The kids had some good questions right from the start.

By the end of this video we’ve understood a bit about what happens to the real line.

After looking at the real line in the last video, we moved on to the imaginary axis in this video. The arithmetic was a little tricky for my younger son, so we worked slowly. By the end of this video we had a pretty good understand of what happens to the imaginary axis under the map Z \rightarrow Z^2.

At the end of this video my younger son noted that we hadn’t found anything that goes to the imaginary axis. My older son had a neat idea after that!

Next we looked at (1 + i)^2. We found that it did go to the imaginary axis and then we found two nice generalizations that should a bunch of numbers that map to the imaginary axis.

Finally, we went to Mathematica to look at what happens to other lines. I fear that my attempts to make this part look better on camera may have actually made it look worse! But, at least the graphs show up reasonably well.

It was fun to hear what the boys thought they’d see here versus their surprise at what the actually saw 🙂

I think this is a pretty fun project for kids. There are lots of different directions we could go. They also get some good algebra / arithmetic practice working through the ideas.

A fun calculus problem for kids – playing with derivatives and absolute value

I’ve been doing a few “calculus for kids” projects after seeing Grant Sanderson’s essence of calculus series. The series made me see that some of the high level ideas are completely accessible to kids and it has been fun to explore some of those concepts.

Today I thought it would be fun to see what they thought the derivative of absolute value would look like – they had some neat ideas:

Next I thought I would turn the problem around – what if absolute value was the derivative! What would the function look like. This problem was much more challenging. In the first video they spent most of the time just getting their head around the problem:

So, now that they had the ideas in place to solve the problem, they started drawing pictures. The process of getting to the correct graph was really interesting to watch:

The more I think about this calculus project, the more fun I think it is going to be. Many of the ideas in Sanderson’s series will be out of their reach, but some of the high level concepts are incredibly fun to share with kids.

More calculus ideas for kids inspired by Grant Sanderson

I’m enjoying thinking about how to share Grant Sanderon’s latest calculus video series with kids. My goal is not remotely to develop a calculus course, but just to give kids an opportunity to see and explore some of the basic ideas that Sanderson shares in his video series. At a high level, things like slope of the graph of a function are easily accessible to kids even if the calculations required to make the ideas precise might be beyond them. Our projects so far are here:

Sharing Grant Sanderson’s Calculus ideas video with kids

Sharing Grant Sanderson’s “derivative paradox” video with kids is really fun

Sharing Grant Sanderson’s derivative paradox video with kids part 2

Sharing Grant Sanderson’s “derivatives through geometry” video with kids

So, walking the dog tonight I came up with two ideas for discussion:

(i) How does the length of the hypotenuse of a right triangle change as the length of one of the sides changes?

(ii) If a function has the property that the slope of the tangent line is the same as the value of the function, what would that function look like?

We began with a quick review / discussion of slope in the context of a curve. This concept is still new to the boys and I wanted to have one quick review before we dove into the main project:

Next we moved on to the right triangle problem – how would the length of the hypotenuse change when one of the side lengths changed? The boys were able to grasp some basic ideas around when the changing side was short (near zero length) and very long (near infinite length), and we were able to make a sketch of what the derivative might look like just from these basic observations:

The next project was a basic (the most basic?) differential equation -> a function has the property that the derivative at a point is equal to the value of the function at that point. The value of the function at 0 is 1. What does this function look like?

Finally, we went up to the computer to use Mathematica to explore our two questions. For purposes of this higher level conceptual overview, it is nice that Mathematica’s built in functions allow us to study these two questions without having to do the calculations ourselves:

The more of these project I do, the more I’m convinced that this is a useful exercise for kids. For now at least, I can’t think of any reason why learning about these basic ideas at the same time you are learning about functions will cause problems.

My week with “juggling roots”

A tweet last week from John Baez made for a really fun week of playing around. I’ve written several blog posts about it already. Here’s the summary to date, I guess:

(1) The original tweet:

(2) The blog posts:

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

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

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

(3) A video from a comment on one of the posts from Allen Knutson that helped me understand what was going on a bit better:

So, with that as background, what follows are some final (for now at least) thoughts on what I learned this week. One thing for sure is that I got to see some absolutely beautiful math:

Dan Anderson made some pretty neat 3d prints, too:

For this blog post I’m going to focus on the 5th degree polynomial x^5 - 16x + 2. I picked this polynomial because it is an example (from Mike Artin’s Algebra book) of a polynomial with roots that cannot be solved.

So, what do all these posts about “juggling roots” mean anyway?

Hopefully a picture will be worth 1,000 words:

What we are going to do with our polynomial x^5 - 16x + 2 is vary the coefficients and see how the roots change. In particular, all of my examples below vary one coefficient in a circle in the complex plane. So, as the picture above indicates, we’ll look at all of the polynomials x^5 - 16x + A where A moves around a circle with radius 8 centered at 10 + 0 I in the complex plane. So, one of our polynomials will be x^5 - 16x + 2, another will be x^5 - 16x + (10 - 8i), another will be x^5 - 16x + 18, and so on.

The question is this -> how do the roots of these polynomials change as we move around the circle? You would certainly expect that you’ll get the same roots at the start of the trip around the circle and at the end – after all, you’ve got the same polynomial! There’s a fun little surprise, though. Here’s the video for this specific example showing two loops around the circle:

The surprise is that even though you get the same roots by looping around the circle, with only one loop around the circle two of the roots seem to have switched places!

Here’s another example I found yesterday and used for a 3d print. Again for this one I’m varying the “2” coefficient. In this case the circle has a radius of 102:

When I viewed this video today, I realized that it wasn’t clear if 3 or 4 roots were changing places in one loop around the circle. It is 4 – here is a zoom in on the part that is tricky to see:

Next up is changing the “-16” in the x coefficient in our polynomial. Here the loop in the complex plane is a circle of radius 26:

Finally, there’s nothing special about the coefficients that are 0, so I decided to see what would happen when I vary the coefficient of the x^2 term that is initially 0. In this case I’m looping around a circle in the complex plane with radius 20 and passing through the point 0 + 0i:

So – some things I learned over this week:

(1) That the roots of a polynomial can somehow switch places with each other as you vary the values of the coefficients in a loop is incredible to me.

(2) The idea of thinking of these pictures as slices of a 3-dimensional space (which I saw on John Baez’s blog) led to some of the most visually striking 3d prints that I’ve ever made. The math here is truly beautiful.

(3) I finally have a way to give high school students a peek at a quite surprising fact in math -> 5th degree polynomials have no general solution.

What a fun week this has been!

Sharing Grant Sanderson’s “derivatives through geometry” video with kids

We’ve done a few projects inspired by Grant Sanderson’s incredible new calculus series:

Sharing Grant Sanderson’s Calculus ideas video with kids

Sharing Grant Sanderson’s “derivative paradox” video with kids is really fun

Sharing Grant Sanderson’s derivative paradox video with kids part 2

Today we returned to Sanderson’s series to look at his “derivatives through geometry video”:

We watched the video last night. To get started today I reminded the boys about the concepts in Sanderson’s video. The specific example we looked at was how the area of a square with side length’s X changes as the side length’s change:

Next we moved on to the first of two challenges in Sanderson’s video. In this video we tackle the function y = 1/x. How does this function change as the value of x changes?

The second challenge problem involved the function y = \sqrt{x}. The ideas here are slightly more complicated than in the prior video and my younger son wanted a more detailed explanation. I’m glad he did, though, because going though this example a little slower I think helped the general ideas sink in.

I didn’t want to have the project end with all of the algebra in the last video, so I decided to return to the two challenge functions and look at their graphs again. Did the answers we found for the derivatives match up with what we were seeing with the slope of the function in the pictures?

This new calculus series from Grant Sanderson is one of the best “math for the masses” projects that I’ve seen. He is not pitching the series at kids, but I think there are many ideas throughout the series that are accessible to kids. I have no intention of trying to teach my kids a full course in calculus, but I do think that they will find exploring a few ideas here and there to be really fun. After we finished today my younger son’s first comment was that he wanted to do more projects like this one – yay 🙂

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?

Sharing Kelsey Houston-Edwards’s bridge video with kids

Kelsey Houston-Edwards published another fantastic video this week:

By coincidence my kids had been making domino runs this week and I was already planning on doing this bridge activity with the boys – perfect timing! Even better, Houston-Edwards’s video shows that this activity is a great way to introduce kids to basic mathematical modelling. That topic has been on my mind quite a bit this week too because of the new Grant Sanderson calculus series.

So, we watched the video on Friday night and talked about some of the ideas this morning.

Here’s a short introduction to the problem and a bit about what the boys remembered from the video:

Next we moved to the problem of trying to use some basic math to describe what’s going on with these bridges. Although they’d seen the math modelling in Houston-Edwards’s video previously, the modelling ideas were not the first ideas that came to their mind. Instead they were able to solve the first step in the bridge problem. Instead they were able to just see that solution.

With the algebraic solution not being quite the first thing that came to their mind, I decided to dive into that solution for the bridge with 3 bricks. The nice thing about the 3 brick bridge is that the numbers are still not that complicated.

Once we had some equations written down we talked about various different approaches to solving them. My younger son found a pretty clever way to solve these equations without too much algebraic effort.

Finally, I had the boys make some bridges on the table rather than the slightly cheating way that we did in the first video. We had a bit of debate off camera about whether or not their top bricks were fully off the table, but they were certainly very close 🙂

Definitely a fun project. I’m going to try to do more of these modelling tasks in the next month or so. Right now I’m not completely sure where to find good introductory modelling tasks for kids, but hopefully solving that challenge will be a fun project for me!

A challenging counting problem for kids learning algebra

My son is in a weekend enrichment math program and that program has been great for him. It comes to an end this week. The last problem on this week’s homework assignment gave him some trouble, so I thought it would be fun to see if we could work through it together.

I was a little worried because I’d not seen the problem until just before the project, but luckily things went ok.

Here’s the problem:

a, b, c, and d are positive integers less than 10. How many solutions are there to the equation a + bcd = ab + cd?

[post publication note: Originally the text presented the problem incorrectly. It is correct in the videos. Karen Carlson pointed out the typo to me. Sorry about that.]

Here’s how we got started – my son had found several cases, but not quite all of them:

After the introduction to the problem and my son’s work so far, we moved on to try to find more solutions. The main idea I gave my son involved writing the equation in a slightly different form:

Now that we had a plan, we moved on to counting the rest of the cases that we found in the last video:

Finally, we went to Mathematica to write a little program to count the solutions for us. This part of our project turned out to be more interesting than I was expecting. It was interesting to compare the brute force solution of the computer to the case by case counting technique that we’d just gone through.

So, a fun problem that definitely made my son think this week. It is