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!
Our first look at the question involved some dice rolling and a computer simulation. Today we are going to look at an exact solution to the problem. That solution involves studying all of the different things that can happen when you roll 5 dice. It turns out that there are 7 different patterns that can happen, and these patterns related to the ways you can write 5 as the sum of positive integers.
(1) 5 different numbers, which I’ll represent as 1 + 1 + 1 + 1 + 1
(2) 3 different and 2 the same -> 1 + 1 + 1 + 2
(3) 2 different and 3 the same -> 1 + 1 + 3
(4) 1 different and 4 the same -> 1 + 4
(5) 1 different and 2 pairs -> 1 + 2 + 2
(6) 1 pair and 1 triple -> 2 + 3
(7) All numbers the same -> 5
For today’s project we’ll count the number of ways that each of these 7 patterns can occur. We know that the total number of arrangements is 7,776, so that’s going to help us make sure we have counted correctly.
Here’s the introduction to the problem and to the approach we are going to take today:
Now we began to count some of the arrangements. In this video we count the number of dice rolls in (1), (4), and (7) above:
Now we moved on to some of the more challenging arrangements. Here we looked at (6) and (2) above:
Now we looked at case (5). This case proved challenging because dealing with the 2 pairs caused a little confusion between over counting and under counting. But, after looking at the cases carefully we did manage to get to the answer.
At this point we had only one case left -> (3) from above. But, the counting practice that we’d had up to this point helped this case go pretty quickly.
Finally, we added up our numbers and checked that we’d found all 7,776 cases. We did!
The one thing left to do was to count the different numbers that we saw in each case and find the average. I’d done that ahead of time just to save a bit of time in the movie. Our final answer was (27,906) / (7,776) or about 3.588. The exact answer was (happily!) very close to the two estimates that we had found in our simulations yesterday.
I love Tanton’s problem. It is a great estimation problem as well as a great counting problem. We might do one more project tonight on yet a different way to solve the problem using Markov chains:
Looks like a fun idea – I’ll be thinking about how to talk through this approach with the kids during the day today.
So, I decided to make looking at this problem our weekend project. Today I wanted to try a few simulations. Tomorrow we’ll count the cases. That case work is pretty challenging, but I think we’ll be able to get through it.
Here’s the introduction to the problem:
Next I had the boys roll 5 dice 50 times and record the numbers they saw each time. Here are those results as well as our first estimate of the answer to Tanton’s question:
Finally we moved on to do a simulation. Normally I would have used Mathematica, but I’ve got a program related to one of our old Goldbach Conjecture videos running right now. This problem was easy enough to deal with on Excel, though, so I looked at a simulation there.
Here’s what they saw in the numbers:
Definitely a nice start to this project. Counting the different cases tomorrow will be a bit more difficult, but we’ll also get to talk about some pretty interesting mathematical ideas like partitions and choosing numbers. I’m excited to see if we are able to get all the way to the end.
Tonight I sat down with the boys to make sure they understood the problem. They noticed that half the numbers would have no powers of two – good start! After that observation they started down the path to solving the problem really quickly. In fact, my younger son thought that we might have a geometric series.
Since we covered a few ideas pretty quickly in the last video, so I stared this part of the project by asking them to give me a more detailed explanation for how they got the 1/2, 1/4, 1/8, . . . pattern in the last video. It turned out to be a little harder for them to give precise arguments, but they did manage to hit the main points which was nice.
At the end of this video my older son was able to write down the series that we needed to add up to solve the problem.
Now that we had the series, we had to figure out how to add it up. My guess was that they’d never seen a series like this, but my older son had a really cool idea almost immediately – rewrite the series!
The boys were able to sum the series in this new form – so yay!
At the end of of the last video my younger son said that he was surprised that the “expected value” wasn’t zero since zero was the most likely value. In this part of the video we talked a bit more about what “expected value” meant.
Once we had that I asked what I meant to be a quick question -> is the expected number of 3’s higher or lower. It turned out to be a longer conversation than I expected, though, because my older son was actually able to write down the answer!
Definitely a fun problem. I think it is fun for kids to see how to add up a series like the one in this project. I also think it is fun for kids to explore some of the basic ideas about primes that pop up in this problem.
As an aside, one other place where I’ve seen the series that came up here is in this post from Patrick Honner:
We’ve now got a few more years of 3d printing under our belts and a new program we are using – F3 by Reza Ali – is opening completely new 3d printing ideas for us.
Somewhat incredibly, F3 has a one line command that draws all of the points that are a fixed distance away from a cube. Here’s that beautiful shape:
Â
Seeing that command inspired me to revisit James Tanton’s old question. I wasn’t quite able to do it in one line (ha ha – my programming skills are measured in micro-Reza Alis . . . .), but I was still able to make the shape. Here’s how it looked on the screen:
After the boys got home from school we revisited the old project together and used both the old and the new 3d prints to help us describe the shape (sorry for the noise in the background – that’s a humidifier I forgot to turn off):
Maybe because it is one of our first projects ever(!), but I love this problem as an example of how 3d printing gives younger kids access to more complex problems.
I was flipping through James Tanton’s Solve This last night and found a project I thought would be fun. I had no idea!
One of the most exciting projects we've ever done coming later today – stupid slow internet 😦 . Thanks @jamestanton for the inspiration!! pic.twitter.com/9Rvxw991na
So, having see a few folding / cutting project for kids previously the projects about Möbius strip in Chapter 8 of Tanton’s book caught my eye. As I said above, though, I had no idea how cool this project would turnout to to be!
We started with the standard project of cutting a Möbius strip “in half”. What happens here??
Oh, and before getting to the move – most of the movies below have a lot of footage of us cutting out the shapes. I was originally going to fast forward through that, but changed my mind. The cutting part isn’t that interesting at all, but I left it in to make sure that anyone who wants to repeat this project knows that the cutting part (especially with kids) is a tiny bit tricky. You have to be careful!
Now, once we’d done the cut the boys were still a little confused about whether or not the result shape had one side or two. I thought it would be both important and fun to make sure we’d resolved that question before moving on:
With the Möbius strip cutting out of the way we moved on to what Tanton describes as “a diabolical Möbius construction”. All three shapes start as a thin cylinder with a long ellipse cut out. You then cut and twist the strips outside of the ellipse making a Möbius strip-like component of the new shape. Hopefully the starting shapes will be clear from the video.
Try to guess what the resulting cut out shape(s) will look like prior to them getting cut out 🙂
The first shape involves putting one half twist into one of the strips left over after cutting the ellipse out of the cylinder.
The second shape also starts as a cylinder with a long ellipse cut out. This time, though, we make put half twists (in the same direction) in both of the long strips that are left over after cutting out the ellipse.
Sorry about the camera being blocked by my son’s head a few times – oops!
The final shape for today’s project is similar to the second shape, but instead of two half twists going in the same direction, they go in opposite directions.
All I can say is wow – what an incredible project for kids. Thanks James Tanton!!
The boys knew from the video that the method could also be applied to sharing candy. Since we didn’t watch that part of the video I was wondering if the boys could figure out the connection on their own. Here’s the start:
Next we tried an example to see what would happen if our initial guess was a big over estimate of 1/3 of the Skittles:
Since we were struggling with our second time through the procedure in the last video, I thought it would be fun to try to be more precise in how we split the piles. That extra precision did lead to slightly better results.
So, a really nice math activity. It was really fun to see the procedure work when we couldn’t be totally sure we were actually dividing the piles in half. Such a great project for kids.
Saw a great new video from James Tanton today about folding a tie. The kids had spent yesterday hiking in New Hampshire and were a little tired, but Tanton’s project made for a perfect little afternoon project.
I’ll present the videos in the order that we did them, so Tanton’s video is the third one below. Showing his video later in the project will also give you a chance to think through the problem without spoilers.
Anyway, here’s how we started -> what do you have to do to fold a tie in half?
I was super happy with how the introductory problem went because at the end of the last video my older son said that he thought folding the tie into thirds would be hard. Well . . . that’s exactly what we are going to try to figure out!
Next we watched Tanton’s video. He talks about both folding ties and sharing candy, but for today at least we are just focused on the tie folding part:
Now we tried to replicate Tanton’s procedure. My 5th grader had a little bit harder of a time understanding the procedure than my 7th grader did, but they both eventually got it.
At the end we talked about why they thought the procedure worked.
So, a super fun project and a really easy one to implement, too. So many potential extensions, too – might be neat to see how kids approached folding into 5 parts after seeing Tanton’s video, for example.
A tweet from Tanton reminded me about his project earlier this week. I was excited to revisit it and got a double surprise when my older son told me that he actually did it in his 7th grade math class last week! It is nice – actually amazing – to see Tanton’s work showing up in my son’s math class!
An unfortunate common theme with some of our recent projects is that they aren’t going as well as I hoped they would. Still, though, this was fun and I’ll have to spend a bit more time thinking about the last bit – how to write 1/3 using base 3/2.
We started by reviewing base 2 and, in particular, how you can play around with binary using blocks.
Next we looked at base 3/2. I’m sorry that this video runs 10 min – I definitely should have broken it into 2 pieces.
Finally we accidentally walked into a black hole. I assumed that writing 1/3 in base 3/2 wouldn’t be that difficult and that an easy pattern would emerge quickly. Whoops.
Turns out that no pattern emerges quickly, and even playing around on Mathematica for a bit after we turned off the camera we couldn’t find the pattern. The discussion facilitated by the work on Mathematica was great – at least my kids learned that (i) there are multiple ways to write a number in base 3/2, and (ii) there are easy sounding project that I can’t figure out!
I hope to revisit this part after I understand it better myself. Any help in the comments would be appreciated.
I really like this project and am sad that a little bit of stumbling around by us might have obscured the beauty of Tanton’s idea. Hope we’ll be able to revisit it soon.
This is also a much more difficult question and one that I wouldn’t expect the kids to be able to handle on their own. However, they did walk almost all the way to the solution by themselves, which was pretty cool.
We started by talking about the problem and discussing a few possible approaches to solving it – the conversation about understanding the problem was fantastic.
I was super happy with my younger son’s idea for checking a 3-4-5 triangle against a 4-5-6 triangle.
In the next part of the talk we explored the area of the 4-5-6 triangle using Heron’s formula. As we’ll see in a few videos, it was really lucky that my older son remembered this formula.
Also – ha ha – “decimals and square roots don’t usually work out” 🙂
At the end of the last video the boys wondered if both triangles could be right triangles. It took a while for them to see how to use the Pythagorean Theorem to approach this question, but eventually we got going. At the start it seems that we have a really complicated algebraic expression, but, almost by miracle, a parity argument appears!
Having had the luck of stumbling on a parity argument in the last video, I showed them how a similar parity argument would apply with Heron’s formula to show that there were no solutions to Tanton’s question. This is the one step that I don’t think they could have stumbled on by themselves. Even so, it was really great that Heron’s formula and a parity argument came up naturally while we were discussing the problem.
Finally, we wrapped up by discussing our thoughts about the rectangle and triangle problems:
So, thanks to Josh for pointing out this problem that I’d missed, and thanks to James Tanton for inspiring another really fun Family Math project.
