A Fawn-tastic start to the school year

Just like last year, I made it exactly two days into the school year before working through an activity that Fawn Nguyen shared with the boys. Guess the over / under for nest year should be pretty easy to set!

Today Fawn tweeted about her new project:

The 2nd lesson – Algebraic Thinking – caught my eye and I thought I’d have each of the boys work through 2 of the 8 problems.

Here’s my younger son’s work on problem #1:

and on problem #5 – I love the language that he uses when he solves this problem:

Here is my older son working through problem #3 – his language solving this problem is also great:

and here’s his work on #6:

So, another great start to the year thanks to Fawn Nguyen. Sharing her project with the boys always makes for a fun evening.

A challenging AMC 10 problem involving some basic statistics

I was traveling for work the last two days and asked my son to work on some old AMC 10 problems rather than working in his geometry book. When I got back home tonight I asked him to pick one of the problems that gave him trouble for us to work through together. He picked #19 from the 2005 AMC 10 b:

The 2005 AMC 10 b

Here’s the problem:

On a certain math exam, 10\% of the students got 70 points, 25\% got 80 points, 20\% got 85 points, 15\% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?

This choice proved to be lucky since I’ve just started the section on basic statistics in Art of Problem Solving’s Prealgebra book with my younger son. It was a nice problem to work through with both of them.

We started by reading through the problem carefully and making sure that both boys understood what mean and median meant. The boys decided to solve the problem by assuming that 100 students took the exam. We solved for the median first:

Next we found the median by working through a long arithmetic calculation. Since the calculation itself isn’t really that interesting, I tried to focus more on building up number sense, and, in particular, on ways to make the calculation easier. My older son is definitely more comfortable working through calculations like this one, but I think my younger son was able to see a few good math ideas in action:

So, a nice problem giving some good practice in basic statistics as well as some good arithmetic review. It is also a nice illustration of why I like the problems from the old AMC 10 tests. This is #19 out of 25 problems on the AMC 10, meaning it is one of the more difficult problems. Not so difficult that the kids aren’t able to understand the solution, though. The AMC folks do a great job producing problems that strike that balance, which is something that I’d probably struggle mightily to do on my own.

Modular arithmetic, Khan Academy, and a NY Times puzzle

I’ve had a busy week at work and haven’t been able to put as much time in with the boys as I normally do. Yesterday I spent most of the time with my younger son catching up on some of the problems in our Introduction to Number theory book and ended up with nothing obvious for him to work on for homework. I ended up telling him just to play around on Khan Academy.

This morning I got a bit of a surprise when he asked me what modular arithmetic has to do with cryptography. I was sort of caught off guard by the question – we are on the modular arithmetic section in our number theory book, but I don’t remember the book mentioning anything about cryptography, and I certainly haven’t.

His reason behind the question was interesting – he saw the connection on Khan Academy. Apparently some of the feedback he’d gotten on the modular arithmetic section was that he had made 5% progress towards a cryptography badge (or some other achievement, I don’t really know). It is always surprising what catches a kid’s attention.

This question about applications of modular arithmetic made me switch up what I was planning on talking about with him today. I saw this puzzle in the NY Times posted by Stephen Strogatz yesterday:

If you don’t have time to click the link, here’s the puzzle:

“A chess master is preparing for a tournament. She plays at least one game a day, but no more than 12 games over any seven-day stretch. Can you show that, if she keeps practicing like this for a long time, there will be a series of consecutive days in which she plays exactly 20 games?”

This is a modular arithmetic question in disguise, so I figured that he might like working through this question even though it is certainly not something that I would expect him to be able to solve on his own. I hoped that the surprise connection to modular arithmetic would be interesting to him.

Before diving into the videos, I need to say that this is tough question and I did no prep work at all for this project. I mean none at all – he asked me the question about cryptography as we were walking into the study. Partially as a result of the lack of prep, the conversation is far from polished and it takes us a while to get to the punch line. Still, a couple of little smiles at the end showed me that he found it to be a really interesting puzzle. With a little more work, I think there’s a really fun project for kids hiding in this project somewhere.

We started with a quick review of what it means for two numbers to be equal mod m. The idea here was to remind him of the congruence definition in the book – two numbers are equal mod m if their difference is divisible by m.

 

With our short review out of the way, we started in on the puzzle. The first step was to write down a sequence of games played on each day that meets the conditions of the problem. For a young kid, just writing down this numbers is an interesting problem.

 

Now that we had a sequence of games per day written down, we added up the total number of games played from the start. Luckily we didn’t hit 20 exactly, though, trust me, that was just luck! The next step was to look and see if there was ever a sequence of days were she played 20 games exactly. Noticing where the 20 games were hiding was a tough question for him, but not totally inaccessible.

 

Finally the connection to modular arithmetic. He notices the connection mod 20 (after noticing a connection in mod 10 – ha!). The reason that our list would definitely have a repeat mod 20 took a while to understand (in fact, the bulk of the video is about why a repeat happens), but at least I think he understood the explanation.

I’m really happy to see how excited he was at the end of this little project 🙂

 

So, a surprise connection from a little work on Khan Academy combined with a neat NY Times puzzle leads to a fun day. Watching this again just now I’d like a 2nd shot at this project, but I’m still really glad we did it. It is neat to see the math you are studying show up in a some surprising places!

The Joy of teaching my kids

A few weeks ago for our weekend Family Math project we talked about fractions and decimals in binary.    That blog post is here:

https://mikesmathpage.wordpress.com/2014/04/05/fractions-and-decimals-in-binary/

These family math project are just for fun.  These projects tend to cover either fun math we find around the house – see the paper folding example from all the way back in Family Math 1:

or, if not stuff from around the house, they are intended to be a fun overview of some advanced math.  The overview of fractions and decimals in binary was supposed to be in the second category, but it led to a really great surprise this morning.

Today with my younger son we moved on to a new chapter in our book – repeating decimals.   A few days ago we had started off talking about decimals and fractions by reviewing why .9999…. = 1, so I was hoping to play off of that to show why 1/3 = 0.33333….  However, when I sat down and asked my son what he thought the decimal expansion for 1/3 would be I got a little surprise:

“I don’t know, but I know what it is in binary.”

So fun that he remembered this talk from the Family Math project from a few weeks ago:

With that, we started down a totally new path – how does knowing what 1/3 is in binary help you understand the decimal expansion?

Such a fun morning!!