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 🙂

Sharing Kelsey Houston-Edwards’s topology video with kids

Kelsey Houston-Edwards’s latest video is terrific:

This one is particularly easy to share with kids because there are several puzzles where she asks you to stop and think about the solution. I began the picture frame puzzle as the starting point for our project today.

The puzzle goes roughly like this:

A common way to hang a picture is to use two nails in a wall and run the wire around those two nails. Assuming the nails / wall are strong enough, if you remove one of the nails the picture will still hang. Is there a way to hang a picture with two nails so that if you remove either of the nails the picture will fall?

We took a shot at this puzzle using yarn and snap cubes. It was a good challenge for the boys:

In the last video we got the picture to fall once, but the boys weren’t quite clear what happened – but now they at least knew it was possible! Here we explored the idea more carefully:

Next we finished watching the video and then discussed what we saw (as I publish this post the video preview isn’t embedding properly, but is really just audio anyway):

Finally we looked at two sets of shapes that appeared in the video that we’ve looked at before. The first is a 3d print of Henry Segerman’s “Topology Joke” and the 2nd is a set of “rollers” that we’d made after seeing a Steven Strogatz tweet. The tweet and the roller project are here:

3d printing and rollers

Another fun project from Kelsey Houston-Edwards’s amazing math series. Sorry to be brief on this project, but I had to get this one out quick because of a bunch of activities going on today.

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!

Are the “juggling roots” related to Aztec Diamond tilings?

I was working over at MIT today and brought the print I made overnight so that I could sand it after it cured:

Occasionally there’s a grad student that I chat with and he walked by today and asked what the print was. I showed him the “juggling roots” from the John Baez tweet:

Here are the two recent projects that we’ve done after seeing that tweet.

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

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

Seeing the rotating roots, he said “Oh, that’s related to the Aztec diamond tilings.” Unfortunately he had to run to a meeting so I didn’t get to learn what the relationship was.

But . . . here’s a picture of the Aztec diamond:

Screen Shot 2016-03-02 at 7.07.20 PM

Here are a few of the projects that we’ve done on the Aztec diamond tilings:

The Arctic Circle Theorem

TA second example from tiling the Aztec diamond

It is funny the relationships you see when you know what you are looking at. I don’t see the connection, but I’m excited to learn what it is!

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!

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

We’ve done two projects so far with Grant Sanderson’s new calculus series:

Sharing Grant Sanderson’s Calculus ideas video with kids

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

I wanted to do a follow up to the 2nd project this morning to address one point that I sort of skipped over in that project. Unfortunately we got so caught up on what is the main point of the video that we didn’t get to it! Sort of good news / bad news, with the good news being that talking about graphs of distance and velocity with kids is amazingly fun.

Here’s the 2nd video in Sanderson’s series – the Derivative Paradox:

I got started this morning with a quick review of how to go from the graph of distance to the graph of velocity. The boys seemed to remember quite a bit from our last project, which was really nice:

Next I drew a graph that was similar to the one that gave the boys some trouble in our last project. I was hoping for this part to also be a review, but it gave them some trouble again. Part of the trouble was that I was a little careless in the presentation, unfortunately, but this was still the start of a very good conversation.

Drawing graphs of functions and derivatives was tricky for students that I was teaching in college. I’m happy to have these conversations with my kids, though, because I do think the ideas are accessible to them.

So, after stumbling through the conversation in the last video, I tried to take a different approach here. Almost starting over, actually.

One really interesting thing for me watching my older son talk about the pictures was that his intuition said that the velocity must be near 0 at time 0, but the graph was saying something different than that. He was really struggling to reconcile the two ideas.

Putting some numbers to the velocities helped straighten out that problem. By the end both kids had sort of an aha moment and realized that what we were talking about with these graphs

I can’t say enough good things about this new calculus series from Grant Sanderson. Not all of it is going to be accessible to kids, which is totally fine – he’s not pitching this series at kids. Some of it is, though, and his approach showed me how to make some of the ideas accessible. There’s probably at least 10 more projects to do that involve sharing his series with kids. I can’t wait!

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

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

Grant Sanderson’s new video series on calculus is incredible. We’ve done one project with the introductory video:

Sharing Grant Sanderson’s Calculus ideas video with kids

Today we looked at the second video in the series. This one introduces the idea of derivatives. The videos are not aimed at kids – not eve close. But watching it with them to explain (or just skip) some of the difficult parts was really fun.

The video below shows what they took away from the video when we talked about it roughly 2 hours after they watched it. I was pleasantly surprised by how much they remembered. They remembered the discussion about the moving car, they remembered the paradox of “instantaneous velocity”, and they remembered the rise over run idea of the derivative. Some of the main ideas stuck with them and those ideas definitely made them think!

Next we talked about the car and the graph of distance versus time. This discussion was really fun. The idea is not something that they’ve seen before but this short little discussion allowed us to explore the ideas in several different ways. It was a wonderful accidental moment when my younger son drew the graph heading back down to the x-axis – Sanderson does not touch on this idea in his movie!

I missed the chance to explore my younger son’s idea about the graph representing the velocity instead of distance – we’ll come back to that tomorrow.

Next we worked through one of the derivative calculations that Sanderson used to end his video. I picked the graph of y = x^2 just to keep the algebra on the easy side (he does y = x^3 in the video.

We didn’t have as much excitement here, but they were able to follow along pretty well. Most importantly, they were able to understand that “dx” was a variable rather than, say, “d” times “x” or something like that:

I’m not intending to teach my kids calculus right now. However, working through Grant Sanderson’s derivative video with them was really fun – and we have one more idea from this project still to study! I think I will go through some more of Sanderson’s videos with them – it is so great that he’s made something that makes some of the main ideas in calculus accessible to kids.