# What learning math sometimes looks like – rectangle edition

Recently my younger son and I have been study some introductory geometry in our Prealgebra book, and several of the questions in this chapter have sparked great conversations. Today I saw a fascinating question about a rectangle and couldn’t wait to hear his ideas. So, off we went.

We talk through the problem for a bit at the start – this is a challenging problem just to understand! Since he’s a little stuck, he decides to try out a few test cases and draws a 2×1 rectangle. This simple rectangle immediately gives him some new ideas – great start!

It is nice to hear all of his ideas, and especially “I think I see something . . . . ”

I broke the talk into two pieces when it seemed that his idea at the end of the last video didn’t quite work. The trouble he’s run into is that when the width increases by two units the area changes in a way that is different that when the length increased by one unit.

One bit of luck is that he’s working with some small numbers and that probably helped him notice that the change was twice what he was expecting. Upon noticing that fact, he’s able to write down the dimensions of the rectangle. The last part for him is just checking. At the end I show him a little bit of geometry that was hiding in the problem that helps us simplify the calculations a little.

When I think of what learning math looks like, a lot of the ideas in these two videos are what I think of. When I think of kids who are not enjoying learning math – or worse – I’d love to walk through a problem like this, or even just show them the ideas here so that they can see what learning math looks like. Questions, ideas, trying out a few things here and there, trying to find what works and what doesn’t, and maybe even finding the solution to a pretty challenging problem.

Fun morning!

# A project with 3D shapes based on a MoMath exhibit

I visited the Museum of Math earlier this week and shot these two videos in one of their exhibits. The videos show two different ways to slice a cylinder into smaller pieces (sorry for the vertical video – feel free to mock me!)

The reason I shot these two videos is that my older son and I have just started a chapter on three dimensional shapes. I wanted to use these two videos to show two different ways to calculate the volume of a sphere – one seemingly easy, and one seemingly pretty hard.

There is no intention here for my son to understand every step. What I do want to show, however, is two different ways of looking at a problem, the idea of building up a solution to a complicated problem in steps, and also the surprise that the extremely ugly looking sum of rectangles actually converges to the right answer. I think this is a fun example of the power of math, and, as always, it is really fun to hear how kids talk through advanced math ideas.

Here’s my son’s reaction to the first video showing how to chop up the cylinder into circles. Before seeing the second video with the rectangles, it is pretty hard to see the rectangles.

Here’s his reaction to seeing the rectangles in the second video:

So, now we take a deeper dive into the rectangles to see what we can say about them. This video and the next one are really just writing down a Riemann sum (informally). Again, my aim here is not for my son to understand every step, but simply for him to start to see the ideas of building up the solution to a complicated problem in a few steps. At the end of the second video it is nowhere close to obvious that the sum we’ve discussed will converge to the correct volume of the cylinder.

Since I have no interest in actually evaluating this sum, we go to Wolfram Alpha to do it. What we see there is that the sum does indeed converge to the value we thought it would – amazing!

So, a fun little project with a nice surprise at the end. The ideas of looking at a problem two different ways – in this case slicing into circles and slicing into rectangles – and the idea of building up a solution to a complicated problem in several (hopefully) easier steps are what I’m trying to show here. I was lucky to have seen the MoMath exhibit at the same time we started studying 3D geometry. It will be fun to return to this project when we study calculus – many, many years from now 🙂

# What learning math sometimes looks like: the triangle inequality

My younger son and I just started looking at some basic geometry in Art of Problem Solving’s Prealgebra book. Today I noticed a neat challenge problem at the end of the section about perimeter and it looked like it would be a fun problem to talk through. It turned out to be a great illustration of ideas that kids have when they are learning math.

In the first part I introduce the problem:

An isosceles triangle has integer side lengths and a perimeter equal to 25. What are the possible lengths of the sides?

We spend the first part making sure that my son understands the problem. He then dives in.

He doesn’t know what to do, so he decides to try some examples. The first example he tries is 1, 1, 23. When he draws a picture he notices that this combination of side lengths doesn’t work. From that he forms a conjecture and proceeds to write down some of the solutions.

We left off the last section with a list of possible solutions. I started off this part by asking him to describe the difference between the 1, 1, 23 case and the 12, 12, 1 case. They seem sort of similar when you look at the numbers, but one works and the other doesn’t. His thoughts on this point are really nice.

Next I ask him to go through some of the other cases he thought would not work based on his original conjecture. The first case is 8, 8, 9. Talking through this case leads him to a new idea, and this new idea is basically the triangle inequality! Again, his reasoning here is really great – kids have such interesting ideas about math 🙂

The rest of the video is just checking the other cases. It takes a little bit of time because of a little arithmetic mistake, but we make it to the end.

So, hopefully a nice example of what learning math can look like. Again, not a straight line to the end, but learning math seldom is. Although kids it would be silly to expect kids to formulate their ideas in super precise mathematical terms, their ideas and instincts about math are the key to helping them learn. It was so fun to hear my son essentially formulate the triangle inequality here:

Pursuing ideas like this one is what makes learning math fun.

# Going through Joel David Hamkins’s Graph Theory for Kids

Denise Gaskins published the latest issue of Math Teachers at Play today. As usual it is a wonderful compilation of math activities and one, in particular, caught my eye. Here’s the latest issue:

Math Teachers at Play issue 85

And here’s the activity from Joel David Hamkins that caught my eye:

Graph Theory for Kids

After downloading and flipping through Joel’s activity this afternoon, I thought that it would be really fun to run through it with the boys tonight. I’ll present the videos without much comment since they really speak for themselves. The short summary is that I absolutely love this activity and think that many kids will love it, too. My kids were totally engaged for the full hour that it took to work through it.

The only warning I’d give is that in the part where the kids draw their own graphs – part D in our set of videos – make sure they are careful when counting. My kids drew some pretty complicated graphs and we had to slow down and count really carefully.

So, here’s our tour through Joel David Hamkins’s Graph Theory for Kids:

Part A: getting into the activity and explaining a few of the terms that we’ll be using

Part B: Going through a few more examples and introducing the definition of Euler Characteristic

Part C: There were lots of examples to do in the last section, so I had the boys work through all of them off camera and then had them each explain their work on one of them. This section also introduces the idea of a “connected” graph:

Part D: At the end of last section the kids were asked to draw their own graphs. My kids drew some complicated graphs and had a hard time counting the various pieces. Here we are just going through and counting carefully.

Part E: The next section of the activity introduces the idea of a “planar” graph

Part F: Now we move on to the end of the activity and look at some calculations for 3-Dimensional shapes

Part G: I thought it would be fun to end with a shape that didn’t have an Euler Characteristic of 2, so we built a torus out of our Zometool set. We also looked up the Euler characteristic of a few other shapes off camera and my son mentions a Klein bottle at the end.

So, a really fantastic activity. As I said, my kids were really engaged for over an hour and seemed to find all parts of the activity to be really interesting. There’s a ton of fun math here and lots of ways for kids to get creative. Also lots of great opportunities to hear kids talking about their own mathematical ideas.

Thanks to Denise Gaskins for sharing this one and to Joel David Hamkins for putting this awesome activity together.

# Using a Richard Green google+ post to talk about geometry with my son

Patrick Honner shared an amazing post from Richard Green yesterday:

The post caught my attention for a couple of different reasons. First, the result is absolutely amazing, and following Green’s summary I clicked through and skimmed the original paper. It was really cool to see all 111 tilings. Second, the mention of the Paul Monsky result about triangles in a square was fascinating. Monsky was (and still is) a professor at Brandeis when I was in graduate school and he was always incredibly generous with his time and ideas. I found the Monsky paper with a quick google search and his proof is amazing (though pretty technical and not really something that you could share with kids).

Lastly, though, Green’s post intrigued me because I’m just finishing up a section about angles with my younger son and it sure seemed as though there was a project for kids hiding somewhere in this post. I tried one project idea with my younger son this morning.

The first thing we did was look at Green’s opening paragraph – “It is easy to cut an equilateral triangle into four smaller equilateral triangles . . . ” Perfect, let’s talk about that! Right away we get to have a fun little conversation about triangles and counting.

Following my son’s idea of how to chop up an equilateral triangle into smaller triangles, I had him build the object he described out of our Zometool set and I built an example that used a (slightly) different idea. He sees a pattern in the number of triangles that goes 1, 4, 16, 64, . . . . When you include the triangle that I showed him you get a different pattern 1, 4, 9, 16 . . . . So, we get a couple of nice patterns to talk through.

We also talk briefly about the Monsky result at the end of this video.

After that brief introduction, we moved to the end of Green’s post and I had my son talk about some of the shapes he saw. It is always fun to hear ideas that kids have about math, and these tilings are so cool that I’m sure that kids will have all sorts of really fun things to say about them.

Finally, let’s talk about some angles. We used the shape that caught his attention and then tried to calculate what some of the angles in the tiling. The first angles that he noticed were the right angles, and then the octagon at the center of the tiling caught his attention – what are the angles in that octagon?

After finding the angles in the central octagon, we went looking for one last set of angles to calculate, and my son chose the angles in one of the hexagons. This calculation is a tiny bit more difficult because not all of the angles are the same. I love hearing his ideas about how to find these angles, and also his surprise that two of the angles are actually right angles!

So, a super fun geometry project based on Richard Green’s post. It isn’t that often that you can use ideas from current math research in conversations with kids, but the ideas in Green’s post were just too good to pass up. Thanks to Patrick Honner for sharing the post yesterday and thanks to Richard Green for pointing out and explaining this amazing geometry paper.

# Nicholas Kristof’s “Are you smarter than an 8th grader?” article

Saw Nicholas Kristof’s article today via a Cristina Milos tweet:

The article links to a document showing a bunch of problems from the 2011 TIMSS exam. Three of those questions are included in the article. I thought it would be interesting to have my kids talk through those questions.

My younger son isn’t home right now, so I’ll go through the questions with him later (and update the post). The first three videos below show my older son working through the three problems. [now updated]

Just to be clear, I’m doing this exercise because I find it interesting to hear how kids think through problems, not because I think that answering a few multiple choice math questions makes you smarter than, well, fill in any end to that statement that you want.

Older son question 1:

Older son question 2:

Older son question 3:

Younger son question 1:

Younger son question 2:

Younger son question 3:

# The “rope around the Earth” problem

My son ran into a version of the classic “rope around the Earth” problem yesterday. The common version of the problem goes something like this:

You have a rope that goes all of the way around the Earth at the equator. If you wanted a second rope to also go all the way around the Earth but always be exactly 1 ft higher up than the first rope, how many feet longer than the first rope would the second rope need to be? (Assume that the equator is a perfect circle).

The version of the problem that my son saw was slightly different. I thought it would be fun to use that problem to motivate a short project about perimeter and area of some simple shapes:

The first step in our exploration was adding one blue strut to each side of the shapes. How did the area change? How did the perimeter change?

It was easier for my older son to see the change in area in his square than it was for my younger son to see the change in his triangle. However, after talking for a little bit, we were able to resolve the discrepancy.

The next step was looking at the pattern in the area and perimeter when we added one more blue strut to the sides of each shape. At this stage the boys seem to have a pretty good handle on how the area and perimeters are changing and even were able to make a conjecture about the next step.

In the last part of today’s project we take a look to see if the patterns that the boys thought would emerge on the last step do indeed emerge. We then turn back to the original “rope around the Earth” problem to see how to understand the different perimeters in play there.

So, a fun project motivated by a classic problem. Hopefully the talk allowed the boys to see that there more than just a clever little problem going on here.

# Two nice problems shared by David Coffey and Federico Chialvo

I saw two neat problems for kids on twitter this week and went through them with the boys this morning.

First up, the problem shared by David Coffey:

I couldn’t find any multicolored items to use, so we used snap cubes instead of cards. The first thing that we tried to do is see if we could solve the problem at all. The boys had a couple of ideas at the beginning, and it was interesting to hear the discussion. My younger son noticed that a procedure my older son was following was going in a circle, for example. My older son noticed that the last step would have to involve swapping out 3 orange cubes for blue cubes. That idea will help us find the minimum number of steps in the next video.

Eventually we found a way to swap the orange and blue cubes in 5 steps.

Next we tried to see if there was a way to swap the cubes in fewer than 5 steps. My younger son noticed that it would take at least 3 steps for a complete swap since you have to move 7 cubes, 3 at a time. He then noticed a way to make the complete swap in three steps.

At the end we compare both procedures that we found. This was a nice little activity.

The next problem we tackled this morning was from Federico Chialvo

The challenge on this problem is that my older son knows how to calculate each of these areas from studying geometry. I had him do that calculation at the end, but first I wanted to hear ideas that didn’t involve calculation. The “no calculations” requirement turned this into a fairly challenging task.

I split the 9 minute conversation into two pieces just to make it easier to watch. Their geometric instincts are to chop the hexagon into triangles and rectangles. Chopping up the hexagon in this way is interesting, but since you get 30-60-90 triangles my older son’s urge to calculate is hard to suppress 🙂

As we try to move away from calculating, they notice that the triangles they are looking at might actually form a square that is roughly equal in size to the original square. Eventually we see that the remaining rectangle has an area that is a little bit more than half the area of our original square. All of that information comes together to produce an estimate that the area of the hexagon is bit more than 2.5x the area of the square.

We finish up with a quick calculation of the area of the hexagon by my older son. The calculation shows that the hexagon is $6*\sqrt{3}$ times the area of the square. So, our original estimate of 2.5 times wasn’t that far off – yay!

# Three versions of the same problem

My younger son and I have been studying angles for the last week. I decided to run through a few clock problems for a review. The secret about this exercise is that all three problems are pretty much the same problem in a slightly different form.

Question 1: How many degrees does the hour hand move in 10 minutes. I love the way he talks through this problem.

Question 2: What acute angle do the hour and minute hand make at 2:10?

I really didn’t know how he’d approach this problem. He got a little confused in the beginning thinking that the hour hand would move 1/3 of an hour in 10 minutes. Eventually he found that it moved 1/6th of the way through and that observation got him to the answer. I was a little surprised that he didn’t notice a connection between this problem and the first problem.

Question 3: What acute angle do the hour and minute hand make at 9:50?

The interesting thing here is that he does pick up on the fact that this one is similar to the previous problem. That observation allows him to get to the answer quickly. It was nice to see him also notice at the end that all of the answers were the same. Fun little project.

# What learning math sometimes looks like – Geometry edition

I started making math videos with my kids about 4 years ago. One of hopes was that the videos would be useful for other kids because I wanted to show them what it looks like when kids do math. Not that it is necessarily super different than what it looks like when adults do math, but I suspect that most of the time kids see adults doing math they are seeing something that is a little more polished than average. I thought seeing kids doing math would be more motivational or instructive to kids.

Last night’s talk with my son is almost exactly what I had in mind. My intention was to review a topic that we’d covered in the morning, but right at the beginning he saw an alternate approach so we went in that direction. When we returned to the original approach – for what I thought was going to be a short review – what we covered in the morning had slipped out of his mind! It took 3 or 4 minutes (and a couple of false starts) for him to recall the prior approach.

But that’s always been the point – “doing math” doesn’t mean that you take a quick, straight line path to the answer. There’s usually lots of false starts – even with things you’ve seen before. Also, as the first minute or so of this video illustrates, sometimes you suddenly see something that you’ve been studying for a while in a totally different light. Those unexpected moments of inspiration are always really fun.

Here’s last night’s talk:

Here’s the talk from the morning that I was trying to review, too:

Power of a Point and some fun equalities