# Playing with Three Sticks

I saw this tweet from Justin Aion at the end of July and immediately ordered the game:

When I returned from a trip to Scotland with some college friends the game was on the dining room table – yes!! Today we played.

In this blog post I’ll show how the game ships and two rounds of play (and we might not be playing exactly right) to show how fun and accessible this game is for kids.

First, the unboxing. The game comes out of the box nearly ready to play.

Here’s our first round of game play. I think we misunderstood one of the rules here, but you’ll still see that the game is pretty easy to play:

Here’s the 2nd round of play. I think we understood the rules better this time, which is good. You’ll also see how this game gets kids talking about both numbers and geometry:

Finally, here’s what the boys thought about the game:

I’m really happy that I saw Justin Aion’s tweet and now have this game in our collection. It is a great game for kids!

# Taking about Kate Nowak’s shape

Saw this neat drawing from Kate Nowak the other night:

I was interested to see if we could make the shape from our Zometool set, and . . . .

The boys really enjoyed making the shape last night and both also made several comments about how interesting it was. This morning we talked about it a bit. Both kids focused on symmetry. I spent a bit more time with my older son exploring the different kinds of symmetry, but it was great to hear what both kids had to say. It really is an amazing shape!

Younger son first:

Older son next:

This was a really fun project. The shape didn’t take that long to build, which was lucky. It is always fun to be able to pull out the Zome set to explore something that we saw on Twitter ðŸ™‚

# An interest rate question extending Kate Nowak’s rate post

Earlier in the week Kate Nowak wrote a neat post about rates. The perspective in the post (in my words) is coming from writing curriculum materials for 6th grade math:

Here’s an alternate perspective on the same (or at least similar) issue that I encountered at work this week.

Suppose I ask you to play the following game:

(1) You pay me \$2 today.
(2) I’ll then select an integer from 1 to 10 at random (uniformly)
(3) At the end of year 1 you pay me \$1, and if my random number was 1 I’ll pay you \$10 and the game stops. If my number wasn’t 1 we’ll meet again next year.
(4) In general, at the end of year n, you’ll pay me \$1 and if the random number I picked was n, the game stops.

The interest rate question relating to this games is this: What is your expected rate of return for playing my little game?

Here are two different ways to think about it:

(1) Internal rate of return

You’ll see an expected set of cash flows that look something like this:

The “internal rate of return” on those cash flows is about 12%, so you might say (and I think that many people would be quite comfortable saying) that your expected rate of return playing my game is about 12%.

(2) Accounting for the costs and the investment returns differently

One possible objection to the internal rate of return calculation is that your cash outflows are really part of your investment in the game and so are quite different than the investment return. In fact, to play the game all the way through, in addition to the \$2, you need to be sure that you have access to \$10 over time to play.

So, you might prefer to discount your cash outflows at a less risky rate – I’ve picked 4% just for example purposes – and discount the inflows (the investment returns) at a risky rate to measure your return. That calculation looks something like this:

Using this method the expected investment return you’ll get for paying \$2 to play my games is more like 8% per annum.

So, what is the correct way to think about the rate of return for playing my game?

I think the rate of return question here is pretty interesting to think about and gives a real life example of the things that Nowak is thinking about writing 6th grade curriculum.

# Using Kate Nowak’s rotated parabola with kids

Saw a Kate Nowak exercise via a Dan Anderson tweet earlier in the week:

and wrote about an estimation problem coming from that picture that I thought was fun:

A fun estimate question inspired by Kate Nowak

I thought it would be fun to see what the boys thought about the rotated parabola, so this morning I showed them a few rotated parabolas and asked them what they thought:

My younger son was interested some of the pieces of area that were cut out by the rotated parabola. Funny enough, my older son was interested by a similar question a few years ago:

The area inside of a parabola

It was fun to explore their ideas about the different areas of the graph. We had a neat detour when my older son wondered what the graph of $y = x^2$ would look like of we were really zoomed out.

Finally, my older son was interested in what the parabola would look like under a variety of different rotations. The discussion here ended up being a neat surprise as what grabbed the boys’ attention was how to change the x- and y- coordinates following a rotation so that the graph would display correctly. I wouldn’t have thought to talk about that, but they were pretty interested in understanding how the coordinates changed. The funny thing is that walking down this path gets you really close to talking about trig functions.

So, a fun morning project even if math need to compute or calculate the rotation of the graph is obviously way, way over their heads. But, since the picture is actually pretty simple, there’s still plenty of interesting things to talk about with kids, AND plenty of stuff that kids might be interested in!

# In Defense of Short Stories

Saw this tweet fromÂ  Kate Nowak presentation today:

When I saw the tweet I thought that it would be fun to take the other side this argument.Â  As the day went on, though, that thought changed to I want to take the other side of that argument.

I’ll begin with one of my favorite – and short! – math stories:

Julie Rehmeyer’s “Inspired by Math” interview

What’s always stuck with me from this interview is the story that begins around 31:30 and in particular the part beginning around 34:40 about proving that 0 + 0 = 0.

What I find so wonderful about Juile’s story is that you just never know what’s going to stick out from your own math education, and you certainly never know what is going to suddenly inspire a student at the time.

For me, the main source of math inspiration was Mr. Waterman’s Enrichment math class at my high school. This class was nothing but short stories for 3 years. For the most part you had no idea whatsoever ahead of time what we’d be talking about during that class. It sure taught me to love math, though.

A book I received as a prize in that class – essentially a collection of short stories – was my first introduction to math beyond school and contest math. I couldn’t wait to learn more about subject like abstract algebra and game theory after reading it:

My teaching style was heavily influenced by how Mr. Waterman taught, and though I’ve never really thought of it in these terms before that style is surely more “short story” than “coherent novel.”

Even with the stuff I do with my kids now, I’m happy to use, and super duper happy to have the opportunity to be inspired by, material shared by:

Fawn Nguyen

Patrick Honner

Dan Anderson

Kate Nowak

Stephen Strogatz

Numberphile

and on and on and on . . . .

I don’t view these ideas as great for my kids because they are coordinated, or connected in any special way – other than that they allow me to show really beautiful math to the kids.

One short story that I really look forward to sharing with them is Chapter 8 from Art of Problem Solving’s Precalculus book.

That chapter covers the geometry of complex numbers in a way that is so beautiful that I actually wrote to the author to thank him for writing that chapter. Just a beautiful little short story ðŸ™‚

So, don’t sell the short story short! You never know what’s going to grab a kid’s attention ðŸ™‚

# Patrick Honner’s construction challenge shows the beauty of math

Last week Patrick Honner posed this clever geometry question on twitter:

There were many wonderful solutions posted on Twitter and on Google+, and I wanted to highlight 3 of them as wonderful examples of mathematical reasoning. These are the kinds of fantastic ideas that really highlight the beauty in math to me.

So, in the order that I saw them:

(1) Kate Nowak’s solution:

This solution uses two (related) ideas from geometry: similar triangles and ratios. There are two reasons that this solution was appealing to me. First, it is a fantastic use of abstraction – the point X that is part of the similar triangles and also part of the original triangle is not on the page, but we can still use properties involving these triangles in the solution.

Second, one of my great memories from learning geometry in Mrs. Whitney’s class (in, gulp, 1984 . . . ) was learning how to use ideas from geometric constructions to divide. Who knows why, but that idea just blew me away. In this solution, Nowak uses the ideas that Mrs. Whitney showed my class to divide a segment into two pieces of just the right proportion. The especially cool thing about Nowak’s construction is that we don’t know what the value of the ratio, only that the two line segments are divided into two pieces that are in exactly the same proportion. What a great, and super instructive, solution.

(2) Alexander Bogomolny’s construction:

I liked this solution because the approach is a clever twist on the ideas that Nowak’s used. Instead of dividing segments into equal, but unknown, proportions, Bogomolny divides two specific line segments in half. To achieve this goal he rotates a line about a point.

This gives rise to a nice question for a geometry student -> how do you do that with a compass and a straight edge?

Next he uses a neat property of trapezoids – the line connecting the midpoints of the two bases also passes through (i) the intersection of the diagonals, and (ii) the intersection of the two legs (when extended).

That give rise to a second great question for geometry students – prove that this statement about trapezoids (with non-parallel legs) is true!

So, I love the way that this proof not only answers the question posed by Patrick Honner, but also can be used to expose geometry students to a few other ideas and challenges.

(3) Patrick Honner’s solution:

Finally, the solution that Honner provided to his own problem is terrific:

The surprising (to me) idea Honner uses in this solution is that the altitudes of a triangle intersect at a single point. At first glance it is not at all clear how you might use this idea to solve the problem, but Honner shows how to find a triangle with X as vertex whose altitudes intersect at the point P. This solution shows a beautiful way in which two seemingly unrelated ideas in geometry are connected.

As I mentioned above, there were many solutions given on Twitter to Honner’s problem, but these three really stood out for me. I love that this problem illustrated how different people can approach a problem in different ways, and all of the different ways can be instructive.

Also, in a week when the internet spent way too much time discussing the differences between 5 + 5 + 5 and 3 + 3 + 3 + 3 + 3, it was nice to get this wonderful reminder from Nowak, Bogomolny, and Honner that math is about beautiful ideas and not just about getting an answer.

# A really intersting twitter conversation sparked by Kate Nowak

This tweet from Kate Nowak has been on my mind all day:

This part of the conversation on Twitter really got me thinking – in response to the question from Nowak “what is a hook to you?”:

That bit of the conversation caught my attention because I really liked Meyer’s definition, but at the same time I share Nowak’s concern about what happens if kids don’t have the tools to resolve the questions in their mind. However, despite sharing that concern, I occasionally show my kids ideas that are probably way over their head that I think they will find interesting.

My all time favorite example is the Numberphile -1/12 video.

/

I’ll never forget my younger son (who was in 2nd grade at the time) screaming at the computer screen “no no no no no.” I loved that something about the video bothered him – deeply and almost physically – even though it was hard for him to identify what had gone wrong.

You can see how much it bothers him in this video (at 6:50) from the project below:

/

After I read Jordan Ellenberg’s idea of “algebraic intimidation” in How not to be Wrong we talked about the series in this project:

Jordan Ellemberg’s “Algebraic Intimidation”

Ellenberg’s idea was a great way to explain to my younger son that it was ok not to believe the calculations he saw in the video.

Another fun project where the explanation was over the head of both kids was our look at the Chaos Game.

Computer Math and the Chaos Game

Around 2:17 in the video below we start down the path to an amazing result. That result – around 3:05 – was something that I hoped would plant the idea that computer math can be really fun even if the math was a little over their head.

/

More recently a Dan Anderson blog post got me thinking about how to use a little computer math and 3d printing to illustrate some math ideas usually reserved for advanced courses:

Those shapes allowed us to have a neat discussion about when some ideas in math work (finding the area of a triangle by approximating it with rectangles), when they don’t (finding the length of the hypotenuse using the same approximations), and how our old friend the -1/12 series actually played a role in the first part of the project ðŸ™‚

As with the first two projects, I’m happy to show them a strange result just to show that something interesting is happening. Even if they don’t yet have the tools to really understand what’s going on, I think it gets them thinking. As my older son says around 2:04 in the video below – “things get weird when you go to infinity.”

/

Here’s that project:

3d printing and Calculus concepts for kids

To circle back to Kate Nowak’s idea, I’m also a little dubious of trying to lead with ideas when kids don’t have the tools to fully understand them. Probably 95% of the math talks that I’ve had with my kids are not at all like the 3 projects I shared above. Every now and then, though, there’s been a more advanced idea that seemed like something they might find interesting. Sort of a hook to show that math has some amazing ideas that can be both fun and surprising.

# Patty Paper Geometry

Saw this interesting recommendation from Kate Nowak on Twitter last week:

The book (and some extra patty paper) arrived today and it looks like it has some great activities. I found one that seemed fairly easy to implement and gave it a shot with both kids. The project is a little lesson about the perpendicular bisectors in a triangle.

I tried it out first with my older son. We’ve just finished studying Art of Problem Solving’s Introduction to Geometry book which means that he’s seen a little bit about perpendicular bisectors before. He recognizes pretty quickly what he needs to do to make the project work, so we moved fairly quickly. Based on how this project went, I’m pretty excited about reviewing some more geometry concepts with him using this book.

My younger son has not had much geometry and I’m pretty sure has not heard the term “perpendicular bisector” before. I thought this project would still be interesting for him, though, because of the surprising result that the three perpendicular bisectors intersect in a single point. We went through the project a little slower than my older son’s pace, but he seemed to really enjoy it. He even wanted to try out a second triangle.

I was glad that he wanted to look at another triangle because that gave us a chance to try a triangle that wasn’t a right triangle. His interest and engagement in the overall project left me pretty excited to try teaching him some new geometry using the approach in the book.

So, I’m really happy with how this project went with both kids. Looks like this book gives you a great opportunity to review and to learn new geometry. Can’t wait to try out a few more of these explorations. Thanks to Kate Nowak for this awesome recommendation!

# The insane speed of the Mathcounts final

[note: published w/o much of any editing because we were heading out for a Memorial day hike]

Last week Kate Nowak tweeted about one of Art of Problem Solving’s contest prep programs:

This program from AoPS is timed, another one called Alcumus is not.

My kids practice with some of the AMC 8 and 10 exams and have also participated in the MOEMS competitions, too, but we haven’t really done anything with the compeitions requiring insane speed yet. I’m not sure if that type of competition is going to interest them, but if it does then I’m sure Art of Problem Solving’s training is going to be something that we use.

We had some people over for dinner last night and the conversation turned to some of these speed compeititons because one of the kids had just been part of a science bowl team. I don’t actually know anything about the science bowl stuff, but since she mentioned the speed I thought it would be fun to show everyone a video from an old Mathcounts national final because the speed is just unbelievable. One of the competitors here – Bobby Shen – went on to be a 2 time gold medalist at the IMO and also was one of the winners in the Putnam exam last year. Just watch the first couple of questions to see what I mean about the speed:

For our Family math project today I though it would be fun to go through some of these questions with the boys. I had two goals. The first was simply to think through the problems, and fortunately the boys found all of the problems to be pretty engaging. The second was to try to understand how anyone could solve these problems so quickly.

So, following the problem sequence in the video above, here’s the first problem:

The graph of 16x – 2y = 48 intersects the y-axis at (a,b). What is a + b. Mathcounts solution time – 2 seconds, maybe.

Question 2: A rhombus had sides 10 inches. The lengths of the diagonals differ by 4 inches. What is the area of the rhombus? Mathcounts answer time – 1 second or so.

Question 3: When dribbling a basketball up the court Gloria dribbles at a rate of two dribbles for every 3 steps she takes. How many dribbles does she take in her 51 steps up the court? Mathcounts answer time – about 1 second.

Question 4: This a geometry problem, so probably easier to listen to it. This one actually gave the competitors a bit of trouble, too, so it is interesting to see.

Question 5: When you multiply $(x^7 - 2x^4 + 5x^3 + x - 9) * (-3x^6 - 3x^4 + 4x^3 - 5x^2 + 1)$ what is the coefficient of $x^4$? Mathcounts answer time: about 2 seconds.

Question 6: The average age of 3 members of a quartet is 57 years. The average age of the whole quartet is 62. What is the age of the 4th member? Mathcounts answer time – about 2 seconds.

Question 7: This was the first question for the final 2 competitors.

What is the area in square units of a parallelogram having diagonals 8 and 5 that make a 45 degree angle with each other.

This question was pretty hard for my kids, but we got there in about 10 minutes which was nice.

So, a fun morning with some problems from an old Mathcounts exam. Amazing to see the speed of the fastest kids in the country, but also really fun to work through these problems with the boys.

# Going through a Kate Nowak exercise

Tonight on twitter Kate Nowak published an exercise she was planning to use in her geometry class tomorrow. I may need to, um, borrow this exercise for my son tomorrow since I may need to run off to work early. I decided to go through it carefully just in case I get any questions at work. That short exercise helped me see the problems much better than just reading the list. Here’s are my thoughts, but first the twitter post:

(a)

For this one I used the ruler to draw the 3 inch side and the compass to draw the range of points 5 inches away from one of the endpoints of that side. Lots of possible triangles and hopefully a good start to the lesson.

(b)

Here I started with the 3 inch side at the bottom of the page. That was lucky because 5 inches and 6 inches are longer on the paper that I expected. If you put the 3 inch side in the middle of the paper, you’ll not have room (on 8 x 11 paper anyway!). I used my compass to draw a 5 inch circle centered at one of the endpoints of the 3 inch side and then used it again to draw a 6 inch radius circle centered at the other endpoint. One of the intersection points for these two circles was on my paper and allowed me to draw the triangle.

I used the compass again only for problems (d) and (i).

(c)

Had to use the protractor for the first time on this one. Lots of possible choices of triangles here.

(d)

I like this one. After drawing the 120 degree angle it took me a bit to figure out where to put the 5 inch side. Then I remembered the compass and picked a random point on one of the sides and drew a 5 inch radius circle. Just reading this one I missed that it would be more challenging than some of the other problems.

(e)

Having measured the 120 degree angles in the last problems in the right hand side of the protractor, I did the same thing here with the 30 degree angle at the start and accidentally began by nearly drawing a 150 degree angle. Oops! I’ll be interested to see if the students notice the similar triangles from the different groups on this one.

(f)

Now that I was a pro at drawing acute angles, this one went pretty smoothly ðŸ™‚

(g)

This one also went pretty smoothly.

(h)

I found it easier on this one to draw the 100 degree angle and then the 30 degree angle on the other side of the 6 inch side. I sort of feel like I cheated, but I didn’t see where the intersection point with the 50 degree angle would be right away so I solved a different problem.

(i)

Hey, two possible solutions!! I’d like to hear what the kids have to say about this one, too!

So, fun little exercise. If I do have to run off to work early tomorrow, I’ll definitely give the worksheet to my son and have him attempt to create all of these various triangles.