# What a kid learning math can look like: struggling through a great geometry problem

Saw an incredible problem posted on twitter today:

Here’s the picture in case it isn’t clear from the tweet:

When my son got home from school I asked him to give it a shot.  The video is over 8 min long, but it shows what a kid struggling through a challenging problem can look like.  It is important – especially in middle school, I think – that kids know that math isn’t just about speed.

Also, though, give the problem a shot before checking out the video – it is wonderful and involves no calculating that can’t be done in your head (i.e. this isn’t a trig problem):

# A nice series problem for kids from Five Triangles

Back in 2013 we did a neat problem on Numberphile’s “Pebbling the Chessboard” video:

That video also reminded me of a neat “proof without words” that Patrick Honner had written about:

Our project is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

and Patrick Honner’s blog post is here:

Proof Without Words: Two Dimensional Geometric Series

Tonight I saw a neat tweet from Five Triangles that reminded me of the prior project:

I thought it would be a fun one to try out with my older son, though I didn’t quite know how to introduce the problem. I started with a slightly easier series as a trial: 1/2 + 2/4 + 3/8 + 4 / 16 + . . .

Since things seemed to go pretty well with the first problem I decided to go ahead and try out the series posted by Five Triangles:

So, a neat problem for kids building off of a the “simple” infinite series 1 + 1/2 + 1/4 + . . . . As our project from 2013 shows, the more complicated versions can have interesting geometric interpretations, but I’ll leave those for another time. Tonight it was just fun to see some neat arithmetic with infinite series.

# A neat counting problem shared by Five Triangles

Saw this tweet from Five Triangles earlier in the week:

Because of some unlucky travel issues we didn’t get a chance to talk about it until today. Our original talk through the problem happened this morning and the boys had a tough time with it. The whole conversation this morning probably took close to 45 minutes.

To reinforce some of the ideas from this morning I thought it would be useful to go through the problem again with them tonight. This is a tough problem, but it gives plenty of great opportunities to talk about both counting ideas and geometric ideas with kids.

We started the project tonight by just talking about the problem again. This morning the boys eventually settled on an approach that involves looking at lines on the surface of the shape, and we took that approach again tonight with snap cubes:

In the first part the boys counted 12 cubes that were “cut” by lines passing through them. Now we counted other cubes that were cut by the plane. I’m not sure we got the count right the first time (though I didn’t quite get what they got wrong, I just felt that something had gone astray), so we actually went through the whole counting process twice. There’s a lot to keep track of in this problem!

Next I showed them the short program in Mathematica that I wrote during the day. Having a 3D picture with piece sliced out but with grid lines makes counting the sliced cubes much easier. Having this picture of the sliced box on the screen gave me the idea to have them count the sliced cubes in a different way. This led to my older son noticing a few other geometric ideas in the picture:

Next, I showed them the bit that was sliced off. Despite one little inaccuracy in the way the picture was drawn (which may not even show up in the video) we were able to see the 18 sliced cubes again.

To wrap up I showed them that I’d 3D printed these two pieces and left those pieces for them to play around with.

So, a tough project but a fun project. I was surprised how difficult counting these sliced cubes was, but despite the difficult I’m glad we went through this problem because it has so many great mathematical ideas hiding in it.

# A neat problem from 5 Triangles and Dave Radcliffe

Saw a neat exchange on twitter last night:

Though my kids are far too young (4th and 6th grade) to find this solution on their own, I thought that going through this solution with them would be a useful and fun exercise. Each step in the solution is something that they can understand and Dave’s approach is also a great lesson in problem solving.

So, instead of our typical morning projects this morning we talked through this problem.

Here’s the introduction to the problem and a few initial thoughts. Right off the bat the kids have some nice thoughts about prime numbers:

The boys had some good thoughts about simplifying the problem in the last section. We looked at a few other simple examples – 3 consecutive integers in which two are prime (this led to a nice discussion about twin primes).

Next we moved to the computer to take a look at Dave Radcliffe’s idea. Luckily Mathematica has a function – PrimePi[n] – that counts primes less than or equal to n. We wrote a little program to count the number of primes occurring in a list of 2015 consecutive integers.

In this part of the project we began to use this program to explore the number of primes in various lists of 2015 consecutive integers.

At the end of the last section my younger son noticed the point that Dave Radcliffe had made last night -> as you go down the list the number of primes changes by +1, 0, or -1, but no other number.

In this part of the project we discussed why the changes were never greater than 1 and also how this property might help us solve the original question.

Finally we discussed how we could find a long list of consecutive integers with no primes. Unfortunately we were running a little longer than I expected so this part was a little more rushed than I would have liked. Still, though, they seemed to understand the idea.

So, I think walking through this problem with kids is a fantastic exercise. There are lots of interesting mathematical ideas from arithmetic and from number theory kids might find fascinating. Also, the idea that the proof shows a list of 2015 integers with exactly 15 primes exists without actually finding it is also an amazing idea (and likely one that is brand new to kids).

Finally, Dave Radcliffe’s idea to look at lists of 2015 consecutive integers to see how many primes they have is a fantastic problem solving idea. Seeing how a simple idea like Dave’s changes an almost unapproachable problem into a one that is now much easier to understand is an important example for kids to see.

Definitely a fun project.

# Fun trapezoid coincidence on twitter tonight

Earlier tonight I saw this great question on twitter from Wendy Menard:

The discussion following Dave’s tweet was about why the triangles ACD and BCD in his picture have the same height (their areas are the same because the have the same base and the same height, but as Dave says in the discussion, the fact that the heights are the same isn’t obvious).

Well . . . funny enough before I saw Wendy Menard’s post tonight my son and I were talking through problem 20 from the 2008 AMC 10a:

Problem #20 from the 2008 AMC 10 A

Here’s the problem: Trapezoid ABCD has bases $\overline{AB}$ and $\overline{CD}$ and diagonals intersecting at K. Suppose that AB = 9, DC = 12, and the area of $\triangle AKD$ is 24. What is the area of trapezoid ABCD?

Working through the AMC 10 problem you’ll see that triangles AKD and BKC have the same area. That fact gives a little insight into why the two shaded triangles in Dave’s picture have the same area (and why the heights of triangles ABD and BCD are the same) -> ABCD in Dave’s picture is a trapezoid because angle DAB is 72 degrees!

How fun that two problems brought to our attention almost randomly have really similar ideas that help you get to the solution!

# What learning math sometimes looks like part 2

Yesterday we had a really good talk about Pythagorean triples.  My son struggled a little bit with the algebra, but I was happy to see that struggle since I think it represents what learning math really looks like.  That post is here:

What Learning Math Often Looks Like

Today we talked a little more about Pythagorean triples during our normal school time and it seemed like it was time to move on to another topic.  Luckily for me Kate Nowak had shared her thoughts on a great Five Triangles geometry problem yesterday evening:

Because of her post I didn’t have to think too hard about what problem to talk about next.  My son worked for a little while and came up with a solution to the problem that had a small error.  We discussed the error and I wanted to see if he would be able to talk about the error and how to modify his solution on camera.

I was happy to see him talk through that solution and followed up with a new question.  That question gave him a little trouble, but just like yesterday that trouble turned out to be a good example of thinking about math.  As I said in the first blog post, learning math isn’t always a straight line.   I’m really happy to see (and share) these examples because I really think this is what learning and thinking about math looks like:

# Soccer Ball math

Five Triangles posted a neat picture of a standard soccer ball earlier this week:

Seemed like building the shape out of our Zometool set would be a fun exercise after a week of camping, so we gave it a shot this afternoon:

After introducing the problem we started building (off camera).  We’ve done a few other fun exercises with our Zometool set and actually just bought George Hart  and Henri Picciotto’s “Zome Geometry” so the kids are pretty familiar with building structures out of the Zome pieces.  The only trick for this little exercise is that you want to start with edges that are three times the normal length to make the truncation easier.   Also, once you have the icosahedron, it isn’t so obvious where the soccer ball is hiding:

Next we move to the truncation.  Since we started with side lengths that could be easily divided into three parts, truncating the icosahedron isn’t that hard.  It is, however, incredibly interesting to see the “soccer ball” shape emerge from the icosahedron.  The kids were surprised to see that “the pentagons made the hexagons.”  Here’s a peek from about half way through:

After we finished building we did a quick wrap up and talked about a few other questions that we could ask about our new shape – things like the number of edges, or number of pentagons.  I also asked them if they thought the Zometool shape was actually the same shape as the soccer ball and was surprised to hear that they thought it wasn’t.  We talked about that for a bit, too:

All in all a fun little geometry exercise.   Didn’t want to go into too much depth here since they just got back from a week of camping, but even without the depth they seemed to find all of the building to be really engaging.  Thanks to Five Triangles for the inspiration.

# A Fawn Nguyen inspired geometry problem

Last week Fawn Nguyen posted that she was going to so a fun Five Triangles problem with her class:

I typically love the problems posted by Five Triangles and their geometry problems, in particular, are consistently outstanding.  Too bad I’ve not really covered any geometry with either kid yet 😦

But, I have been working on fractions and decimals with my younger son and this problem had a really interesting infinite series hiding in it, so I though it would be fun to talk through with the boys even if I would have to skip over the interesting geometry.

I spent the first 5 minutes just introducing some basic concepts about triangles so that they could understand the problem.  The fact that we are dealing with an equilateral triangle here significantly simplifies the explanation because we can work with the medians rather than the angle bisectors.  Also, though we didn’t dwell on it, with an equilateral triangle it isn’t hard to believe that the medians intersect in a single point.

With that basic introduction out of the way, now we could spend a little time talking through the problem.  The first challenge is to find the radius of the second circle.  My older son had a one geometric idea that was going to be a little more difficult to work through than I was hoping for, but then my younger son noticed that we could draw in a new line segment that would make finding the radius of our new circle pretty easy.  From there we moved on to talking about the infinite series that is hiding in this problem:

From our picture in the last video we were able to see that 2/3 + 2/9 + 2/27 + . . .  = 1.  Now we try to see if there’s a way to sum up that series without appealing to the geometry.  This particular problem is pretty similar to converting repeating decimals to fractions which is what my younger son and I have been talking about for the last week.   I really loved the various ideas that the kids threw out here:

Finally, we wrap up by showing how to sum up the above series by using base 3.  We start by talking about why .9999… = 1 in base 10 and move on to show how the same argument shows that .22222…. = 1 in base 3.  Luckily the series we are looking at is easy to write in base 3.  Fun!!

So, yet another thanks to Fawn Nguyen for alerting me to a really great problem.  Though not quite the point of the problem as originally posed, I love the connection between arithmetic and geometry hiding in this problem.  It was really fun to talk through with the boys.