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Month January 2015

Going through “Count like an Egyptian” with the boys

Earlier in the week I read Evelyn Lamb’s review of David Reimer’s Count like an Egyptian and thought it would have many fun projects for the boys. Lamb’s review is here:

Evelyn Lamb’s review of Count like an Egyptian

Link to the book is here:

Count Like and Egyptian by David Reimer

The book arrived a few days ago, and I was super excited to do our first project today!

The first chapter is about multiplication and division. The book explains a procedure for multiplication which is (i) so interesting and (ii) so simple that I was stunned that I’d never seen it before.

We started with a simple example -> 9 x 7

 

Next we more on to see how this simple procedure could be applied to a slightly more complicated multiplication problem -> 34 x 51. After we calculated this product two different ways, I showed the boys how this Egyptian multiplication process was connected to representing the numbers in binary.

 

With these two examples out of the way, I wanted to see if the boys understood the procedure, so I had each of them work through an example on their own. First up, my younger son working through 13 x 19:

 

and the my older son working through 27 x 36:

 

I was happy to see that they understood the procedure and were able to work through examples on their own after seeing only two examples from me. That sort of gets back to my surprise at never having seen this idea previously. It really is a nice way to multiply, but also a nice way to sneak in a little arithmetic practice while learning some new math. Seems like a great project for kids.

I didn’t want to go too long this morning but I did want to show them that the procedure was also pretty easy to reverse. We’ll probably revisit the division algorithm next weekend, but here’s a quick look:

 

So, a fun project from the ideas in the first chapter of Count like an Egyptian. Happy to have seen Evelyn Lamb’s review of the book and really looking forward to several more projects based on the ideas in this book.

I love opportunites to share math like this with my kids

My younger son and I are going through the section in Art of Problem Solving’s Introduction to Number theory book about linear congruences. This is a pretty advanced topic. I actually skipped it with my older son, but my younger son absolutely loves it. He’s totally captivated and I love hearing his ideas about how to solve the problems.

Today we were discussing how you solve linear equations using modular arithmetic. In particular, which equations have solutions and which ones don’t. Again, a pretty advanced topic for kids – and, to be clear, not one that I plan on covering in a lot of depth – but talking through this with him was so fun. It is also an area of math where knowing the definition of division as multiplication by the reciprocal is useful.

We started off talking about an equation that has a solution. I really enjoyed hearing him reason his way through this problem – especially the part where he pauses to make sure that 3 has an inverse.

 

Next we talked through an example where there is no solution. One interesting part of this discussion for me is his body language. You can see that the equation makes him uncomfortable – there’s no solution – but he struggles for a bit to figure out how to put this idea into words. Then he just crosses out the whole equation – ha!

 

Finally an example where the solution is slightly harder to get to than in the first example. Here we have a common factor between three parts of the equation. That fact allows us to modify the equation to find a solution. A complete understanding of the ideas here is likely a little bit beyond his understanding right now, and definitely beyond what I’m trying to teach him. It is still pretty interesting to me to hear him work through the problem here.

 

So, a tough, but fun topic. I’m thankful that I have the chance to spend a little extra time with my son working through this section. It makes me really happy to see him having so much fun with the math here.

Talking through a tough geometry problem

We probably spent 30 minutes this morning talking through a pretty tough example problem in Art of Problem Solving’s Geometry book. For our movie project this morning I changed up the numbers and had him work through the problem again. It still gave him a little trouble, but he was about to get through it in about 10 minutes.

In the first piece we introduce the problem and he finds the relationships between the two sets of similar triangles in the problem. One of the big difficulties from this morning was dividing up the bottom part of the diagram into “y” and “100 – y”. As I wrote in a recent post, I’m really struggling to help him see how to label geometry problems in a way that helps get to the solution.

A concept I’m struggling to communicate

 

Next we moved on to solving the equations for x and y. Solving these two equations wasn’t really the main focus of the day, but intstead of just jumping right to the solution, I thought it would be good to talk about easy and hard ways to solve these equations.

 

I wish that I had a better sense of how to help my son get through some of these struggles. Going to keep trying lots of different ideas, though it feels a bit like I’m just throwing a bunch of spaghetti at the wall and hoping that something sticks . . . .

Going through Christopher Danielson’s new book “Which One Doesn’t Belong?”

Recently Christopher Danielson published an interesting shapes book online – “Which one doesn’t belong?”

As of January 26th, the 3rd version of it can be found via this tweet:

I decided to spend some time today having the kids talk through 5 of the 11 pages. It was an interesting and enjoyable exercise where I purposely played virtually no role at all. If an explanation contained an incorrect statement (about a right angle, say) I did not correct it. The goal was simply to hear their explanations.

I had my older son pick 5 numbers from 1 to 11 at random, and then I went through those slides in the book with each kid individually. Those 10 conversations are presented below. For each slide, the first video is my younger son and the second is my older son.

I think the videos speak for themselves, so I won’t add much comment except to say that both kids found talking about slide 8 to be the most difficult.

Overall my impression is that this was a good exercise to go through with both kids. Both of them stayed engaged for all five of the exercises and found several of the explanations of what didn’t belong to be pretty challenging. For my younger son (who is in 3rd grade), I think one important challenge was translating the mathematical ideas of similarity / difference he saw into words. For my older son (5th grade) the challenge was to provide explanations that didn’t rely on computation.

I would happily recommend spending time talking through the pages of this book to anyone working with elementary school kids. It would probably be interesting for older kids, too, though that’s a little outside of my experience just now.

Here are the conversations for slide 1 (plus the introduction to the exercise with each kid):

 

 

slide 3:

 

 

slide 7:

 

 

slide 8:

 

 

and finally slide 11:

 

 

A concept I’m struggling to communicate

As I’ve referenced many times, I’ve not taught any of this elementary material to kids before and there are lots of struggles learning (and teaching) this material that are totally new to me. Often concepts that I think will be easy are difficult, and just as often concepts that I think will be difficult seem to be easy.

One thing that I’m really struggling to communicate to my older son right now is the importance of properly labeling diagrams. Maybe this is something that just about every student struggles with, I don’t know, but for sure I am struggling to communicate how important it is.

The struggle with labeling is illustrated well watching my son work through a problem about similar triangles from this morning. He does a good job identifying the geometric ideas in the problem, but has a hard time connecting the dots because nothing is labelled. It was really hard for me to sit on my hands and not help him through this particular difficulty today:

 

Once he does label one side of the square as “x”, the solution to the problem comes really quickly. I wish could find the right way to emphasize the importance of labeling diagrams properly – hopefully struggling through problems like this one will help get that lesson sink in.

 

Some fun geometry and a challenging number theory topic

Some days teaching the boys seem to go super well – could be some interesting ideas from them, or just general enthusiasm, but days like this make me really happy that I have the opportunity to teach them math.

Yesterday my son struggled with this AMC 10 problem:

Problem #14 from the 2012 AMC 10 B

It is a pretty challenging problem. We were all pulled in slightly different directions today and didn’t get a chance to do any regular school work until I got home from work tonight. I thought it might be interesting to revisit a different version of this problem and hear him talk through it.

Here’s the problem I came up with and his initial thoughts:

 

I was interested to see that his initial approach was to try to compute, and the desire to compute was driven by remembering the formula for the area of a rhombus. Drawing the long diagonal of the rhombus gives him a different geometric idea, though. I been trying to emphasize geometric ideas over computation, so I was happy to see the change in approach.

The new approach led us down the path of congruent triangles and then principles of counting. Fun!

So, I stopped the last movie after about 5 minutes. The final part of his solution is here. At the end I ask him to re-explain the geometric idea he’s using ( over counting, I guess), and then showed him an alternate geometric solution.

 

So that was fun. I love how the geometric ideas pulled him away from the straight computation.

Next up was a new section in our Number Theory book – linear congruences. This is (obviously) a pretty advanced topic, and I actually skipped it with my older son when we went through the same book a few years ago. But, my younger son has found this introduction to number theory to be really interesting, so what the heck.

I just love his enthusiasm when he sees that 2x = 3 (mod 4) has no solutions.

 

Happy to head into the blizzard on this math high note tonight 🙂

Volume, Scaling, and a surprising relationship to the Fibonacci numbers.

We used our Zome Geometry book for another Family Math project today. The project we stumbled on used the blue struts to talk about volume and scaling. It starts by asking a seemingly simple question about the volume of two boxes you can make with the blue struts. Here’s the problem:

 

and the constructed boxes – I liked hearing their ideas about the volume of the larger box:

 

Next they re-built the “golden box” using only medium and short blue struts. Building the box this way gave them a little bit of extra insight into the volume. Their first guess about the volume was that it would be the same as a cube with side length equal to 2x the small blue struts.

 

We took a few minutes to build the cube they wondered about in the last video and looked to see if the volume of that cube matched the volume of the golden box:

 

Once we saw that the 2x2x2 cube was too large, the kids thought to build a cube out of the medium struts. It seems possible that these two structures have the same volume – but we need to find a relationship between the mediums and the small blue struts in order to calculate the volume.

 

While we had the camera off, the boys played with the medium and small struts to find an aproximate relationship. They used this approximation to estimate the volume of the golden box. It was really interesting to me to hear and see their approach here:

 

After we finished the project, my younger son was playing around with the struts a little more. I thought it would be fun to see if they could see the interesting relationship between the medium and small blue struts. We did a little postscript. Here they see the numbers 5, 8, 13 for the long struts and 3, 5, 8 for the short ones and guess that the next numbers will be 13 and 20.

 

We checked if 20 shorts matched 13 mediums and found a fun surprise!

 

So, a fun project with a surprise postscript. Who would have thought you’d see the Fibonacci numbers pop up just by building Zometool boxes 🙂

A little Zome geometry in our new house

I was playing around in our Zome Geometry book looking for a project today. Stumbled on chapter 12 in the book about Archimedean solids. Unfortunately as we are currently transitioning between houses, I didn’t have any green zome struts. Luckily we found a few shapes that we could make using only blues.

The goals for today were:

(1) Construct some shapes from seeing their pictures on the computer, and
(2) Have them explain the things that they noticed about with these new shapes.

That noticing and wondering led to the creation of this awesome “snowman” at the end of the project:

We started with the Icosidodcahedron – seen here on Wikipedia:

The Icosidodecahedron on Wikipedia

First what they thought of the shape:

 

Here are their thoughts after building it:

 

Next up was the Truncated Icosidodecahedron seen here in Wikipedia:

The Truncated Icosidodecahedron on Wikipedia

Here is their initial reaction to the shape:

 

and their thoughts after building it – relating these first two shapes led to an interesting conversation.

 

 

The final shape we looked at was the Rhombicosidodecahedron, seen here on Wikipedia:

The Rhombicosidodecahedron on Wikipedia

Here’s their initial reaction to the shape:

 

and their thoughts after building it:

 

Finally – just for fun – here’s the snowman 🙂

 

Introductory geometry proofs

Michael Pershan has been writing about kids and proofs lately. That writing has had me thinking a lot about how my kids approach proofs. Here’s one of Pershan’s posts:

Maybe it’s ok to prove obvious things

With the ideas about kids how kids learn proofs in my mind, I’ve decided to give a little bit extra focus on proofs while I review similar triangles with my older son. The problem we looked at last night was pretty challenging – and also pretty abstract. I was interested to see how he’d approach it:

 

The move from general to specific didn’t surprise me too much. If fact, I’m probably even encouraging this approach inadvertently with all of the math contest problems we look at. Rather than asking him to go back and work through the proof in general, I thought that I’d show him what the general approach would look like. Next week I’ll try to have him try some of the general proofs on his own.

 

Fun probability talk – Bertrand’s Paradox

This morning my son had an interesting question on a probability problem from an old AMC10:

Problem #13 from the 2011 AMC 10b

Here’s the problem:

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?

Until today we’d mainly discussed probability in settings where there are a finite number of outcomes. In those cases I loosely define probability to be the number of “good” outcomes divided by the number of total outcomes. By “good” I mean the outcomes that you are looking for.

The difficulty in this problem is that there are an infinite number of outcomes, so you are dividing by infinity to calculate the probability if you use my loose definition. I’m really glad he asked me about this problem.

 

At the end of the last video I said that we’d now move on to Bertrand’s Paradox, but I put that idea aside for a second after I turned off the camera. There were still a couple of loose ends in my explanation of the prior problem and now that we’d be spending the whole morning on this problem, I thought it would be a good idea to at least mention some of these loose ends.

In particular, I wanted to show him one strange idea. We just showed that probability that the product is positive is 5/9. A similar argument will show that the probability that the product is negative is 4/9. The positive and negative probabilities add up to 1, but that doesn’t make sense, really, since there’s another case – the product can be zero!

 

Now we get to Bertrand’s Paradox. The question itself is surprisingly simple. Take a circle and inscribe an equilateral triangle in the circle. Now, randomly select a chord in the circle. What is the probability that this chord is longer than the side of the equilateral triangle?

Part 1: The probability is 1/4

 

Part 2: The probability is 1/2

 

Part 3: The probability is 1/3

 

So, geometric probability is strange 🙂 Really strange. Pretty much summed up by my son’s reaction around 1:50 in the last video – “This makes NO sense.” Indeed!