Revisiting Equilibrio

Over the summer we did a project on a game called Equilibrio by Fox Mind Games:

A review of Equilibrio by Fox Mind Games

I originally learned about this game from this Tracy Johnston Zager tweet:

Since doing that original project we’ve lent the game to several family’s. It just returned to us yesterday so I decided to have the boys take a fresh look at the game this morning for today’s math project. The boys have a few different things going on today, though, so it made sense to have them work one at a time.

My younger son went first – here’s his thoughts on the game and the first puzzle he was planning to build:


Here are his thoughts after finishing that puzzle.


He wanted to build one more shape and picked a pretty challenging one:


Next up was my older son. Here are his thoughts about the game:


and here’s what he had to say after completing of the first shape:


Finally, he also chose a more challenging shape. The funny thing is that almost all of the shapes look pretty much impossible on paper, but the kids figure them out eventually (though we’ve not completed the book).


I really like this game and think it is an absolutely wonderful activity for kids. There are lots of great lessons about problem solving and perserverance the game. Can’t recommend Equilibrio enough.

Justin Solonynka’s Train counting project

I few weeks (months?) ago I learned about Justin Solonynka’s train counting project from a Tracy Johnson Zager tweet:


It looked like a super fun project and we finally got around to trying it ourselves last night. This is one that I’d like to have a 2nd crack at because the first time around didn’t go quite as well as I was hoping – but that’s life I suppose.

Still . . . I was happy about the counting ideas we did get to discuss in the project, and I was happy that we got to the end of a pretty difficult counting problem. Hopefully Justin’s incredible video all by itself is enough proof of how great a project this is, but here’s how ours went:

I started by having them open the puzzle box and ask some mathematical questions:


So, the boys decided to count the total number of trains you could make, and we started counting. The first path we went down didn’t quite capture all of the arrangements, but the mistake in counting was actually pretty instructive.


At the end of the last video the kids thought there would be only two trains with one train car, but at the beginning of this video they notice that there are actually 20. This observation means that we’ll have to correct the counting from the last video and they work through this revised counting problem here.

There’s a little bit of confusion about how to revise our counting procedure, but working through that confusion is a really important part of learning to count.


Finally, I wanted to show them a slightly different way to count the trains by keeping the factorials from the second video around. I didn’t do nearly as good a job here as I’d have liked to do, but eventually they found the idea of choosing numbers.

So, we got a nice contrast from the last video where we just looked at permutations. For three train cars, for example, the contrast is that 10*9*8 arrangements is the same number of arrangements as choosing 3 cars out of 10 and looking at the 3! ways you can arrange those cars.


So, a nice counting project even if our walk through it wasn’t so great. I’m happy that Tracy shared Justin’s video – it is a great way for kids to learn about counting.

Tracy Johnston Zager’s latest blog post makes me happy and sad

Tracy Johnston Zager has an excellent blog post summarizing her experience at a recent conference. The post is a must read:

Disrupting the Usual Rhythm

Her post made me happy because everyone should experience math the way Zager describes in her post. In fact, everyone should experience math in this way all time time!

Oddly, though, the post left me a little sad, and I couldn’t really put my finger on why. One reason came to mind as I was thinking about it tonight, though.

I remember this post from David Coffey last year:

Whose Fault is it that you aren’t good at math?

One line, in particular, came to mind tonight (and I’d encourage you to read the original blog post to understand the full context):

“I’ll grant you that it might take a gift to be great at math, but if you’re not good at math, it’s no because of your genes. It’s because of your experiences.”

So, I guess I’m a little sad because the great time that Tracy wrote about in her blog seems to be quite the exception rather than the rule when it comes to experiencing math.

My response to Coffey’s post is here (and seems to have acquired a little formatting problem that I’ll try to fix tomorrow):

Responding to David Coffey’s Challenge

Zager’s post is probably a better response, though, because it directly shows the power of that great experiences in math can have. It certainly gave me a goal to have everyone walking away from my math projects at least as excited as Zager was after this conference!

A question from Tracy Johnston Zager that caught my eye

I saw an interesting question from Tracy Johnston Zager on twitter tonight. Unfortunately twitter isn’t cooperating with me right this second and I’m having trouble linking to all of the tweets (ARRGH!!!!), so I’ll have to summarize because I’m trying to get to be early tonight.

The conversation went along these lines:

Tracy was in a classroom working on a game similar to Nim. On each turn of the game a player can add either 1, 2, or 3 blocks to a pile. The player who adds the 10th block loses (or maybe wins, I forgot this detail, but luckily it doesn’t matter for purposes of this post).

The one tweet I can get to link are some of the “notice and wonder” questions that the students had:

The question that caught my eye was about the students’ questions about the actual pieces used to play the game. I’m paraphrasing, but the question from Tracy as I understood it was essentially – there’s not really that much math behind the questions about the game pieces, so is it productive to talk about those questions in a math class?

I thought that addressing those questions might be interesting, but I’m also a terrible judge as to what will be interesting, so I decided to talk about those ideas with my younger son (a 4th grader).

Here’s how the conversation went.

First I introduced the game and we played a few rounds (just 2 min here – the rounds go quick!):


Second – I asked him if he thought the game would change if we played with yellow blocks instead of orange blocks. He thinks that the game will not really change, but importantly for this blog post he does not appear to think this is a silly question:


Finally, I asked him what would change if we played with Lego mini-figures rather than blocks. Again, he thinks the game will not change.

Then I asked him what *would* change the game and something really cool happened!


It was an incredibly lucky break that the conversation about what would change the game led to my son figuring out how to solve a new game. But, even without that fun ending, I think talking about what would change the game and what wouldn’t change it was productive.

I can definitely believe that a kid would have questions about the game pieces and wonder if changing the pieces would change the game. In my mind it is similar to a kid learning algebra wondering if the way you solve an equation like 3x + 1 = 5 is the same way you solve an equation like \pi x + 1 = 5.

Anyway, I’m glad I tried out the question with my son – it was interesting to hear his thoughts 🙂

A neat game shared by Tracy Johnston Zager

Saw this tweet from Tracy Johnston Zager earlier today:

The game looked fun, so I thought I’d turn it into a project with my younger son.

Also, as a note, I haven’t really studied much about games, but this game looks a lot like the game of Nim. It may even be exactly on online version of the game of Nim, I’m not sure. I didn’t look into Nim, though, rather I just watch my son play with this game.

I had him read through the rules while I set up the camera. Here are his initial attempts to play the game and his thoughts as he played. He begins to notice some losing positions and explores a few strategies to try to win.


In the second bit of the project I asked him to think about how to analyze this game. His first thought is to work backwards because he know the scenario with two piles of just one drop each is a losing situation if it is your turn.

He also thinks that watching how the computer plays would be good way to learn. The computer does seem to win a lot!

Then we got lucky – I was searching for a game with just two columns of drips and in the set up we found, he saw an easy way to win! It turns out the computer doesn’t win every time! Winning this game creates a few other ideas for how to win the two column game.


For the last part of the project we tried a few three column games. He tries out a few ideas at the beginning that turn out not to work. However, after a bit of experimenting he’s able to get a win!


So, a really fun game to play and a great game to get kids talking about strategy. I liked simplifying the game to two and three columns to narrow the potential strategies, but I’m sure that playing whatever board showed up on the screen would be fun, too.

A neat student conjecture shared by Tracy Johnston Zager

Saw this interesting tweet from Tracy Johnston Zager today:

I decided to ask my kids what they thought about the conjecture. Each kid took the question in a direction I wasn’t expecting:

4th grader:

6th grader:

We’ve actually had quite a bit of fun talking about area and perimeter over the years. Here are a few sample projects – including projects with computers, Zometool, and manipulatives – that I think kids would find fun in investigating area and perimeter:

(1) Area Perimeter and Fence Posts

(2) What learning math sometimes looks like: the triangle inequality

(3) The rope around the Earth problem

(4) An interesting Robert Kaplinsky post shared by Tracy Johnston Zager

(5) A Zometool Koch Snowflake

(6) Using the Koch Snowflake to introduce fractals to kids

(7) Using manipulatives for a fun twist on Dan Meyer’s geometry problem

(8) Some fun with Archimedes and Pi

(9) The area inside of a parabola

(10) A fun fractal project – exploring the Gosper curve

Fun with abundant numbers

A few weeks ago Tracy Johnston Zager asked about this problem on twitter (and these embedded tweets should be amusing during Halloween week!):

Her post actually played a role in two blog entries so far:

A Neat number theory problem for kids from Tracy Johnston Zager


A neat number theory problem from David Radcliffe

Today’s post was inspired by one of the responses to her question which sort of stuck in the back of my mind:

That response moved from the back of my mind to the front when when my younger son and I got to section 6.4 of Art of Problem Solving’s Introduction to Number Theory book:  Perfect, Abundant, and Deficient Numbers.  Ha – maybe 10 days apart, but a funny coincidence nonetheless.  Time to see if there is indeed fun to be had with these numbers.

We ended up spending two days in this section because there was so much to cover.  As I’ve written a few times recently, finding ways to build up number sense has been on my mind and playing around with factoring (and then adding up the factors) seems like a perfectly fine way to spend time playing around with numbers.   So building number sense was definitely of the goals here.  One other thing that I thought would be fun in this section was talking about why people are interested in these properties of numbers in the first place.

As an aside, I was happy to see this post from Justin Lanier on twitter the other day discussing a time when he and his class worked through one amazingly complicated equation from a theoretical math paper about the Goldbach conjecture:

I think it is fun to try to figure out ways for kids see examples from current math research and I am going to try to put together a special project about prime numbers this weekend after seeing Justin’s post.

Anyway, back to abundant numbers.  To show my son one example where abundant numbers appear in number theory we looked at the Wikipedia page about super abundant number:

Superabundant Numbers

The first equation on that page looks pretty intimidating, but explaining that equation turned out to be a great way to talk about abundant and super abundant numbers with a kid.  Of course, in the background we got lots of good practice with numbers – finding factors, dividing, fractions, and sums.  Yes!!:

Since my son really seemed to enjoy talking about super abundant numbers the next day we went through a similar computation to see if we could find all of the numbers from 1 to 20 that were abundant.  I’m not aware of any really quick way to do this exercise other than checking most of the numbers individually (as my son points out in the video, you don’t have to check the primes).  Although this might be a dull exercise in isolation, he seems to be pretty excited about finding the abundant numbers and he doesn’t appear to find all the arithmetic work here to be dull.

Conclusion:  Twitter was right –  there was indeed some fun to be had with abundant numbers.  Of course, no one would think that a deep dive into the number theory would be appropriate for elementary school kids, but a deep dive isn’t necessary.  As I said, there’s a nice opportunity for building up number sense here and even a lucky opportunity to mention some advanced math.  All in all a neat topic and a fun couple of days!

A neat number theory problem from David Radcliffe

[sorry if this isn’t edited well, limited editing time due to lots of kid activities today]

Saw this post on twitter earlier in the week:

It made for a great follow up to a previous Family Math which also happened to arise from a problem we saw on twitter:

The first thing that we did this morning was talk through the problem to be clear about a couple of math terms – proper divisors, for example.   We also looked at a few other integers to help make sure that we really did have our arms around the problem.   One neat thing is that the kids were able to see that when we listed the factors of a number, the solution to Dave’s problem required exactly four factors (or exactly two rows of divisors the way were were listing them).

The next step was to try to understand how you could sort out which numbers had exactly two rows of divisors.  My older son noticed that numbers that were the product of two primes (to the first power) would have exactly four divisors.  We worked through a couple of examples of integers of this form and found that they did indeed have that property.   My younger son was able to explain why this was the case – thanks to Art of Problem Solving’s Introduction to Number Theory book!  The last thing we talked about in this part was whether or not numbers of this form comprised all of the solutions to Dave’s problem:

Next up we started searching for other potential solutions.  We started off down the path of looking at powers of primes – it turns out this was a lucky road to head down.  We looked at squares and fourth powers, but neither of these types of numbers seemed to solve the problem.  It did give us the idea to look at cubes of primes, though, and that showed us one more set of solutions:

The last task was to see why the two types of numbers that we’d found – products of exactly two primes, and a single prime cubed – formed a complete set of solutions to the problem.     This part of the talk has a little more theory in it, but I think the lesson here is important – how do we know that our solution is complete?   We talk about how you count the number of divisors, and then how that counting process could arrive at exactly 4 divisors.   Funny enough, the divisors of 4 play an important role in answering that question.   The fact that there are only two ways to multiply integers together to arrive at 4 (2×2 and 4×1) tells us that we have a complete solution to the problem.  Yay!

So, in a couple of weeks we went from a neat problem shared by Tracy Johnston Zager about the sum of the divisors to a neat problem shared by David Radcliffe about products of proper divisors of integers.  Of course this journey was helped by the fact that I’m going through a number theory book with my younger son (and have gone through the same book with my older son previously).  These are challenging problems for kids to think through for sure, but I think that kids will enjoy the challenge.  These problems also do a great job of building up number sense because as you work through them you are constantly thinking about factoring, multiplying, and integers that share certain types of properties.    I forget the exact phrase, but in my mind problems like these definitely belong in the “low entry point / high exit point” problems that the people who study math education seem to really like.

As always, I’m glad that people are sharing problems like these ones on Twitter!

A neat number theory problem for kids from Tracy Johnston Zager

I been traveling the past couple of days but did notice this post on twitter a few days ago:

There was some debate about whether or not this is a good problem for kids (4th graders, in particular), but I thought it had some fun possibilities and looked forward to going through it with the kids when I got home.   By coincidence I’m in the middle of a Art of Problem Solving’s “Introduction to Number Theory” book with my younger son right now,  so I’m probably a little more primed than usual to be interested in this type of problem.   Part 3 of the post below – the video FamilyMath185c – is one of the best math talks that I’ve ever had with the boys, so I’m sort of double happy that we discussed this problem today.   Sorry this post is written up so quickly – I’m tired from the trip, but really excited about part 3.  Probably not the best combination for good writing 🙂

I started by simply introducing the problem and then asking each of the kids to come up with a new (but similar) problem that they would also be interested in solving.   It turned out that they both came up with a fairly neat twist, so we had a nice little math talk set up just from their ideas.

Next up the revised problem proposed by my younger son – what if instead of of adding up the factors, we alternate adding and subtracting.   I certainly wasn’t expecting this idea and didn’t really know what to do with it, so the direction I went was mainly arithmetic practice.  If there’s a easy to understand number theory ideas here, I’d love to hear about them.

My older son picked an alternate problem that does turn out to have some really interesting math hiding in it.   What made this part of our talk more fun than usual was that both kids picked up on the interesting math reasonably quickly – even noticing that the answer would be different if you were dealing with a perfect square!!  Yay!  This part of the discussion shows both why I think kids will find problems like the original one to be interesting and also why I love having the opportunity to talk about math ideas with my kids.  So much fun:

Finally, the extension of the problem I chose was counting the total number of divisors.  This is a topic that both kids have seen before, so I expected them both to remember at least some of the ideas.  I wanted to cover this angle of the problem originally more as a reminder of something that we’ve discussed before but when my older son wanted to look at the product of divisors, it turned out that this topic fit in pretty naturally.

I’m really happy to have seen the original problem on twitter.   Part of the attraction of problems like these is the opportunity to get in some arithmetic practice in the same way that you can with “number talks.”   But, in my mind this type of problem brings a little more than the basic number talks do because there is also some pretty interesting math hiding in and around the problem.  I have always believed that many kids will find the math in problems like this to be really interesting and the fun we had today – particularly with the product of the divisors part – makes me believe in this type of problem even more.

I’m not advocating for a book like Art of Problem Solving’s “Introduction to Number Theory” to become a part of a standard elementary school’s math curriculum, but I would advocate for kids to see problems like this one occasionally.  You never know what sort of surprising fun you’ll have talking through it!