Playing with shapes of constant width

Tonight’s project was just for fun – they boys both had long days of school / music.

So, last night I downloaded and printed some shapes of constant width from Thingiverse:

Anenome’s Object of Constant Width on Thingiverse

With 4 of them on the table I asked each of the boys what they thought the shapes were and then let them play around with them. After they played for a bit I put a book on the shapes and asked them how they thought the book would move as the shapes rolled.

Here’s what my younger son thought:

Here’s what my older son thought:

I always find it fun to hear what kids think about complicated shapes. Lots of neat ideas and then a good “wow” when you learn the secret property!

A neat counting problem I saw on Twitter today

Saw a fun problem in this sequence of tweets today:

Seemed like a great problem to cover with the boys tonight. They had some pretty good ideas right from the start:

Once they had the idea for the first 100 numbers down we moved on to trying to find the 1000th number with no 5’s or 7’s. Their approach was a little different from how I would have proceeded, but it was nice to see their idea. They also noticed and avoided a little trap right at the end 🙂

Really happy I saw this problem today. Excited to get the book when it comes out.

A challenging but super instructive inclusion / exclusion example

My son had a really interesting problem as part of the homework for an enrichment math program he’s in. I’m writing this post from the road so I don’t have the exact statement of the problem in front of me, but it went something like this:

You are going to make 7 digit numbers using the digits 1, 2, 3, . . . , 7 exactly once. How many of these numbers have no consecutive digits with common divisors?

So, for example 1,234,567 is a perfectly fine number, but 2,413,567 doesn’t work.

My son’s solution was nice, but complicated. He found the number of ways to separate the even numbers (there are 10) and then found the ways to fill in the odd numbers in each of those cases.

I couldn’t find an easier solution and wondered on Twitter if there was one. One response I got pointed me to a similar problem that was discussed on the Art of Problem Solving problem forum:

Looking through the thread I stumbled on a really clever inclusion / exclusion solution. Since we’ve been taking a closer look at inclusion / exclusion ideas I thought it would be fun to step through this solution with the boys. I think this a really instructive inclusion / exclusion example. One thing that was a little tough for the boys to understand was that the elements we were “excluding” were pairs of integers.

Also, just to be clear, I’m not expecting the boys to have a complete understanding of this solution. Rather, I just wanted to show them an inclusion / exclusion example that had some interesting twists.

So, we started by introducing the problem because my younger son hadn’t seen it before:

Next we dove in to the inclusion / exclusion solution. The “no restrictions” case is easy! Seeing the way to express the restrictions is pretty challenging. Once we understood that case we looked at subtracting away the cases with 1 restriction.

Next we looked at the 2 restriction case. Now things get really tricky – the fact that we have now have pairs of pairs of numbers is one bit of confusion. Another bit of confusion comes because one pair of pairs is not like the others.

Finally we looked at the case with 3 restrictions. This part, I think anyway, is really cool. The surprise is that several of the cases are impossible!

Despite being a very challenging problem, I love this problem as an inclusion / exclusion example for kids. No individual piece is beyond their reach and if you walk through the problem slowly everything is accessible to them.

Extending our arithmetic / geometry connection project to calculus

Yesterday we did a fun project connecting arithmetic and geometry:

Connecting Arithmetic and Geometry

While we were talking about the shapes my older son commented that one of the shapes looked like a pyramid. I thought it would be fun to make the shapes look even more like a pyramid and see what the kids thought.

We started by just talking about the shapes – the most interesting thing to me here was how challenging it was for them to compare the volumes of the shapes:

Because they were having a little bit of difficulty with the volumes I spent a little extra time on the idea. Things seemed to clear up a little bit, luckily:

Finally, I thought it would be interesting for the boys to see some of the math I used to create these shapes. Although this section goes on a little longer than I would have liked, I think this is a fun little introduction to functions and scaling even if we don’t define those ideas explicitly:

A fun little project. I think that some of the broad ideas from calculus are within the grasp of kids even if the underlying calculations probably aren’t. It was fun for me that a question from my older son led from us jumping from arithmetic to geometry to calculus 🙂

A few fun Perfect Bracket stats questions for students

ESPN had 18.8 million entries in their bracket challenge for the NCAA men’s basketball tournament. There were also several other bracket contests going, too. Below are a couple of fun bracket-related questions for students learning about statistics:

(1) Perfect Brackets after the first round:

The ESPN contest went from 18.8 million entries to 164 perfect brackets at the end of the first round of games:

So . . .

(i) If you were running a contest that had 100,000 entries instead of 18.8 million, how many perfect brackets would you expect to have in your contest?

(ii) What do you think the probability of having 0 in your contest would be?

(2) Perfect brackets after Michigan St. beat Miami

The number of perfect brackets in the ESPN contest fell from 952 to 513

Prior to the Michigan St win you had 5 perfect brackets left in your contest. Given what happened in the ESPN contest how many do you think you’ll have after the Michigan St. win?

What do you think the probability is that you will have 0?

(3) The USC vs SMU game was the 22nd game of the tournament

You had 241 perfect brackets going in and 22 after USC won.

In the ESPN contest 81.6% of the entries picked SMU to win and 18.4% picked USC.

(i) Suppose you have a coin that flips heads 18.4% of the time. If you flip it 241 times what is the probability that you will have 22 or fewer heads? (probably best to use a computer for this one . . . )

(ii) Do you think having 22 brackets left rather than 44 (which would roughly be 18.1%) was random chance or was there another factor in the reduction?

(4)  Expected upsets

I’ll make up the numbers for purposes of this problem, but you can get the actual numbers here if you want:

Some statistics for the ESPN bracket tournament

Suppose that the 18.8 million entries have selected the winners of the games this way:

Teams 1 – 5:  95% of the entrants guessed these teams would win their 1st round game

Teams 6 – 10: 90% of the entrants guessed these teams would win their 1st round game

Teams 11 – 15:  85% of the entrants guessed these teams would win their 1st round game

Teams 16 – 20:  80% of the entrants guessed these teams would win their 1st round game

Teams 21 – 25:  70% of the entrants guessed these teams would win their 1st round game

How many games out of these 25 do you expect the team that was not favored by the ESPN contestants to win?  Why?

Connecting arithmetic and geometry

I’ve been kicking around a few ideas about connecting arithmetic and geometry. My first thoughts were revisiting an idea we’ve played with a few times before:

So, today I decided to look at these two sums with the boys:

$1 + 2 + 3 + 4 + \ldots + n$, and

$1^2 + 2^2 + 3^3 + \ldots + n^2$.

We got off to a slower than expected start because my younger son didn’t remember the formula for the first sum correctly. I’m trying hard to break through the idea of relying on remembering formulas, so I was actually happy to review where the formula came from, though.

After the introduction we moved on to studying the first series using snap cubes. What is the geometry hiding behind the formula.

This part of the project to a totally unexpected turn, though:

I decided to keep going with the new sum that added up to $n^2$ to see if we could make another connection. The boys did remember that the sum of odd integers connects to perfect squares, so I challenged them to find the connection between that formula and the new one they just stumbled on.

Finally we moved on to the sum of squares formula. Lots of fun questions from the boys here, including if the idea extended to 4 dimensions!

The shape here is more difficult to build that it initially seems, but they got through it and now hopefully have a better idea of where the formula comes from.

We wrapped up by looking very briefly at pyramids.

I’d like to do more projects like this one and develop a bunch of different ways to share connections between arithmetic and geometry with kids.

An introduction to the “inclusion / exclusion” principle

My older son is enrolled in a math enrichment program that is currently assigning some great counting problems. The program has a 3 week break now so I thought I’d continue on a little bit with some counting ideas and study the “inclusion / exclusion” principle.

We started off last night with the classic case of “derangements” after getting loads of good suggestions on twitter thanks to Patrick Honner, Justin Lanier, Bowen Kerins, and David Butler.

The initial question went like this: If you permute 4 numbers (or snap cubes) how many of those permutations leave no number in its original position?

First we just studied the problem to make sure that we understood it:

Next we moved on to see if we could count all of the cases. Unfortunately our camera ran out of memory in the middle of this video, so sorry for the jump. We did manage to count all of the cases, though:

Having written down all of the cases, we now tried to see if we could count them. The fun thing is that my older son’s idea led us down the “inclusion / exclusion” path:

Finally, I worked through an example with 5 blocks instead of 4. The purpose of this example wasn’t to lay out all of the theory but rather just to reinforce the ideas from the original example of 4 blocks. We’ll build from here in the next project:

A good introductory counting problem for kids – counting rectangles

My older son had a neat problem about counting rectangles on his homework (from an enrichment math program). He and I talked about these types of problems for a little bit during the afternoon, but they are such good examples that I thought it would be fun to do a short project on them.

We started with an introduction to the problem and a simple case – how many rectangles (with integer coordinates) are there in a 1×6 grid?

The neat thing here is that there was an accidental connection to Kelsey Houston-Edwards’s latest “Proof” video that we did a project on a few days ago:

Kelsey Houston-Edwards’s “Proof” video is incredible

Next we moved on to a more complicated problem – how many rectangles are in a 4×3 grid? Can we use the choosing number idea from the last video to count all of the rectangles here, too?

Finally, I wanted to show a counting procedure that was a little different than the choosing number approach in the last video. Here we defined a rectangle by the corners rather than the rows and columns.

This is a more complicated approach, but I thought it would be useful to show that two seemingly different counting strategies led to the same answer.

Revisiting Stephen Wolfram’s MoMath talk

Last week Stephen Wolfram posted an incredible summary of his talk at the Museum of Math:

We did a project using some of the code here:

Sharing Stephen Wolfram’s MoMath talk with kids

I think the ideas from the talk can provide kids with a really wonderful opportunity to explore math. We’ll hopefully revisit the ideas many times!

Today’s exploration follows the same line of ideas that we followed in the first project. The procedure we are looking at goes like this:

(2) Whatever number you get here, cycle the digits to the left -> so, 123 becomes 231, 1045 becomes 0451 (so just 451 for computations), 110110 becomes 101101, and etc . . .

(3) Now multiply the number from step 2 by a fixed number N and add 1.

We look at the sequence of outputs from this procedure in base 2, 3, 4, and 5 today. Quite amazingly, Stephen Wolfram showed that this entire procedure could be done with some very short code in Mathematica. Here’s a pic of the short code and also patterns we see in the digits when we multiply by 1, 2, 3, 4, 5, 6, and 7 at each step when we reun the procedure above in base 4.

If this seems way too complicated I’m not explaining the procedure well enough – go back to our first post on the subject or to Wolfram’s blog. I promise you’ll see that the explorations are totally accessible to kids.

We started our project today by revisiting the results in base 2 and looking for strange or unusual or really anything that caught our eye in the digit patterns.

Also, I’m sorry that the zoomed in shots are so fuzzy (so, the first minute here and basically all of the 4th video). I didn’t realize how bad the footage was until it was published. Even with the fuzziness, though, you can still hear how engaged this kids are and how interesting it was for them to explore all of the strange patterns:

For the 2nd part of the project we looked at the patters of the digits in base 3:

Then we looked at base 4 and immediately saw something that we’d not seen before:

So, having explored bases 2, 3, and 4 we went back to some of the patterns we’d seen and got a nice surprise – we were able to find structure in some of those patterns. This video is the exploration that led to us finding the pattern in base 2.

Again, I’m sorry this video is so fuzzy – wish I would have caught that when we were filming 😦

Now we moved on to exploring some of the patterns that we’d seen in base 3 and base 4 – that exploration allowed us to predict a pattern in base 5 even though we’d not yet looked at any of the digit patterns in base 5!

I can’t wait to play with Wolfram’s ideas a bit more. The ideas are such a great way to expose kids to exploration in math!

A math exploration via an interesting integral from Nassim Taleb

Yesterday Nassim Taleb posted an interesting integral yesterday:

Here’s the integral:

Taleb frequently posts difficult problems and I have a thought about maybe 1 in 10 of them. This one caught my eye, though, because I’ve been reviewing basic complex analysis trying to understand more about recent work by Laura DeMarco’s and Kathryn Lindsey:

3-D Fractals Offer Clues to Complex Systems

Convex shapes and harmonic caps on arXiv.org

Anyway, I had 90 min this morning while my older son is at an archery class, so I thought it would be fun to write up the little exploration I had yesterday with this integral.

In Taleb’s integral, the substitution $u = e^z$ transform the integral (after a little algebra and ignoring some constants) to one in the form:

Other than the square term in the exponential (which will become a big deal shortly!) this integral looked a lot like one of the contour integral examples in my old complex analysis book:

Playing around a bit on Mathematica encouraged me that integrals in the form above did have closed form solutions:

Also encouraging was this paper I found online showing how to evaluate the usual Gaussian integral via countour integration (see example 9):

“The Gaussian Integral” by Keith Conrad

I was surprised that the countour integration solution was only recently (well, recently for math) discovered, but the really amazing coincidence is that the form of the integral used to evaluate the Gaussian integral via countour integration is nearly the same form as the integral we are trying to evaluate here. So, I dove in to the calculations but unfortunately didn’t have much luck finding a contour that worked.

Then this paper was posted on Math Stackexcange and on twitter:

The form of the integral had been studied before and, unfortunateley, the general solution was unknown. However, lots of interesting stuff was known including a recurrence relation that gives the values that Mathematica had found plus a few more. It is a surprise that you can know the values of these integrals when $k$ is an integer, but not in other situations. Another surprise was the odd form of periodicity discussed in the paper.

Along the way I also found this surprise on Mathematica, which probably has an easy analysis-related explanation but isn’t obvious to me geometrically:

So, despite not getting to a complete solution, this problem Taleb posted was a really fun one to study. It never ceases to amaze me how much fun math gets shared on Twitter.