Tag algebra

Steve Phelp’s 3d pentagon

Sorry that this post is written in a bit of a rush . . . .

I saw a neat tweet from Steve Phelps earlier in the week:

The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.

So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.

Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:

Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.

I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!

This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.

A problem about cones for kids courtesy of Dan Anderson

Saw a fun tweet from Dan Anderson when I got up this morning:

Here’s a direct link to the CNN article:

The artificial glacier growing in the desert

The article is interesting all by itself, and the mathematical question Dan is asking was the subject of our project this morning.

First I asked the boys to read the article – here’s what they thought:

I was happy that the idea about the cone having the least surface area for a given volume came up when the boys were summarizing the article. We now moved on to investigating that question.

We first looked at a cube:

The calculations for the cube were pretty easy. Now we moved on to a slightly more complicated shape -> half of a sphere.

Working through the various volume and surface area formulas is a nice introductory algebra exercise for kids:

Now we moved on to looking at cones. Looking carefully at cones is quite a bit more complicated than looking at cubes or spheres. So, first we played with the formulas and reduced the surface area formula to one variable. We got that formula at the end of this movie:

The formula we found in the last video was a bit complicated, so we moved to Mathematica for a bit of help. The graph of the surface area for different values of radius of the cone is a shape that the boys haven’t seen before.

It was fun to talk about how this shape could be helpful in studying the question that Dan asked in the tweet.

It was also fun for me to hear how they thought about ways to zoom in on the minimum.

Definitely a fun project – would be especially good for a calculus class, I think.

Some beautiful geometry in a challenge problem from Alexander Bogonolny

I did the project below with the boys on Sunday before they went off to camp for a week. The idea wasn’t to get into heavy math, but rather just a relaxed walk through some fun shapes. We got one detail wrong in the 4th video which I was sort of kicking myself for, but then I saw a tweet from Nassim Taleb showing some of the geometry in a different problem that Alexander Bogonolny had posted and it made me realize the connection between the algebra and geometry in our problem was still fun to show:

So, despite the error I thought I would publish the project anyway.

Here’s the original problem:

Below are the videos showing our walk through the geometry. First, though, here’s the quick introduction to the problem:

After that intro we looked at the region described by the constraint in the problem. We have to thicken up the region a little bit using the absolute value function in order to see it, so the Mathematica code looks a bit more complicated than in the problem, but that extra complexity is just to make the picture easier to see.

One cool thing about our discussion here is that my younger son thought there should be 3 fold symmetry in the shape because there was 3 fold symmetry in the equation 🙂

Now we looked at the situation in which the surface achieves the maximum value subject to the constraint in the problem. My younger son made the nice observation that the two surfaces appeared to be “blending together” at certain points. That “blending” is an important idea in Lagrange Multipliers – though, don’t worry, we aren’t going down that path today.

Next we looked at the minimum value of the surface subject to the constraint in the problem. The error I made here was accidentally reversing the two surfaces. The fixed surface – the one describing the constraint – is now on the outside rather than the inside.

Finally, I asked the kids to pick a value smaller than 45/4 for the curve so that we could see what happened. Unfortunately they picked 7 which is too small – there’s no surface! – so they chose 10 and that allowed us to see that the shrinking surface inside of the original shape. Also we can see fairly clearly (after some rotation) that the two shapes do not intersect.

Definitely a fun project showing the boys a beautiful side of a really challenging problem.

Exploring a neat problem from James Tanton

I didn’t have an specific project planned for today and was lucky enough to see a really neat problem posted by James Tanton:

I didn’t show the tweet to the boys because I thought finding the patterns would be a good exercise for kids. We started with the k = 0 case. This case is also good for making sure that kids understand the basics of functions required to explore this problem:

Next we looked at the k = 1 case.

Next we looked at the k = 2 case and then my younger son made a really fun little conjecture 🙂

At the end of the last video my younger son thought that the k = 3 case might produce the pentagonal numbers. I had to look up those numbers ( 🙂 ) while the camera was off, but I found them and we checked:

We ended by looking at Tanton’s challenge problem -> what happens when k = -1? I had the boys take a guess and then we looked at the first few terms and the boys were, indeed, able to solve the problem!

The boys had a lot of fun playing around with this problem and I was really excited they found a different pattern than the one Tanton was asking for!

Sharing a Craig Kaplan post with kids part 2

Yesterday we used a recent post from Craig Kaplan as a way to talk a little bit about algebra and geometry:

Here’s that project:

Sharing a Craig Kaplan post with kids

After the project I printed 12 of the pentagons and had the kids play with them today. See Kaplan’s post for some historical notes about the pentagon. The historical importance is probably too advanced for kids to appreciate, but what they can appreciate is that this pentagon can be surrounded in multiple ways. I had the boys play around to see what they could find.

Here’s what my younger son found:

Here’s what my older son found:

This project was really neat. I think making shapes like the one in Kaplan’s post is a great way for kids to review (or even get introduced to!) both equations of lines and some elementary geometry. Also, as always, it is extremely fun for kids to explore ideas that are interesting to professional mathematicians 🙂

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Sharing a Craig Kaplan post with kids

I saw the latest post from Craig Kaplan via a tweet from Patrick Honner:

The picture in the middle part of the post looked like something that kids could understand:

Screen Shot 2017-06-27 at 10.17.26 AM.png

For our project today I thought it would be fun to talk about how to make the polygon tile in the above picture. After we understand how to describe that polygon, we can 3d print a bunch of the tiles and talk more about the idea of “surrounding a polygon” with these tiles tomorrow.

This project is a fun introduction to 2d geometry (and especially coordinate geometry) for kids. We also use the slope / intercept form of a line when we make the shape.

We got started by looking at Kaplan’s post:

Next we began to talk about how to make the shape – the main idea here involves basic properties of 30-60-90 triangles. My older son was familiar with those ideas but they were new to my younger son.

We also talk a little bit about coordinate geometry. The boys spend a lot of time discussing which point they should select to be the origin.

In the last video we found the coordinates of 3 of the points. Now we began the search for the coordinates of the other two. We mainly use the ideas of 30-60-90 triangles to find the coordinates of the first point.

The 2nd point was a bit challenging, though:

The next part of the project was spent searching for the coordinates of the last point. The main idea here was from coordinate geometry -> finding the coordinates of the middle of the square. The coordinate geometry concepts here were difficult for my younger son but we eventually were able to write down the coordinates of the final point:

We were running a little long in the last video, so I broke the video into two pieces. The last step of the calculation is here:

After finding all of the coordinates we went upstairs to make the shape on Mathematica. We used the function “RegionPlot3D” that allows us to define a region bordered by a bunch of lines. Below is a recap of the process we went through to make the shape and a quick look at the shapes in the 3d printing software:

This isn’t our first 3d printing / tiling project. Some prior ones are linked in a project we did last month after seeing an incredible article by Evelyn Lamb:

Evelyn Lamb’s pentagons are everything

I’m excited with the boys to play with the tiles from Kaplan’s post tomorrow.

 

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!