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Month December 2014

Why I don’t like the math shaming of the NYT

Saw several different people pointing out a math-realted idea from a NYT story about The Interview. One example out of the many is a blog article here:

The New York Times Doesn’t Know any Math Teachers

with follow up on twitter here:

It all leaves me a bit uncomfortable. Here’s are some reasons why:

(1) What is the message being conveyed?

If the message is something other than “ha ha, the reporter couldn’t do some fairly basic math” the message is lost on me.

I say “fairly basic” because analogous problems appear as early as elementary school – and certainly before algebra. After seeing some of the posts about the NYT article yesterday, I looked through some of the MOEMS books and found two similar problems in the first few pages of exams. For example:

(A) At a special sale, all pens are sold at one price and all pencils at another price. If 3 pens and 2 pencils are sold for 47 cents, while 2 pens and 3 pencils are sold for 38 cents, what is the cost of a set of one pen and one pencil, in cents?

(B) Admission to a local movie theater is $3 for each child and $7 for each adult. A group of 12 people pay $64 admission. How many children are in this group?

These questions are from the set of exams for kids in 3rd through 5th grade. Aside from having smaller numbers, they more or less exactly the same as the question answered in the blog article above.

Since kids can understand the problem as well as the criticism, I worry about what they will think about the criticism if they themselves can’t solve the problem.

To me the it has the familiar ring of what Jordan Ellenberg calls “algebraic intimidation” in How Not to be Wrong. In the section “Evanescent Increments and Unnecessary Perplexities” there is this passage (page 43 of my copy). The topic in discussion is a common proof of why 0.99999…. = 1:

“These arguments are often enough to win people over. But let’s be honest: they lack something. They didn’t really address the anxious uncertainty induced by the claim 0.9999… = 1; instead, they represent a kind of algebraic intimidation. “You believe that 1/3 is 0.3 repeating – don’t you? Don’t you?

I see a pretty similar thing going on here: you believe these two equations – don’t you? Don’t you?

Which is an easy transition to the next point

(2) What about the uncertainty

Oh, there’s lots of uncertainty in this problem and the equations in the blog article ignore all of it. Put yourself in the position of a NYT editor (or fact checker). Which would you prefer:

(a) The claims in the article: “Sony did not say how much of that total represented $6 digital rentals versus $15 sales. The studio said there were about two million transactions over all.”

(b) The math from the blog: “According to our analysis of the information provided by Sony, there were 1,666,666 $6 rentals and 333,333 $15 sales.”

[post publication edit. Dan Meyer contacted me on twitter and expressed concern about part (b) above. He thought that the quotation marks made it appear that part (b) is a direct quote from somewhere else. I did not notice the potential for that confusion when I wrote this post and certainly didn’t intend for there to be any confusion on that point. To be clear, part (b) is a hypothetical submission to the editor / fact checker designed to report the solution to the math problem that led to the various posts.]

Some commentators have pointed out the error of ignoring the uncertainty in the numbers. Here’s a good graph, for example:

Jamie Cleveland’s graph in the comments to Dan Meyer’s blog article

But there are other problems too. For example – How do you know Sony’s numbers are correct? How do you know that all of the sales and rentals took place at $15 and $6 and not some other price(s)?

Blindly relying on the math to get to answers can and has lead to all sorts of problems in the business world. A great starting point for understanding some of the problems in finance is Roger Lowenstein’s book about the rise and fall of the hedge fund Long Term Capital Management, When Genius Failed. Bethany McLean’s book about the 2008 financial crisis – All the Devils are Here – is another good one. Or, if you think these books won’t interest you, just follow Nassim Taleb on twitter to learn daily that blindly relying on equations to solve problems that arise in business can lead you to answers that turn out to be drastically wrong.

Finally . . . .

(3) Is this a good example of why you should learn algebra?

For me – no. As I said above, pretty similar problems on elementary school math contests. No algebra is necessary to solve them.

I also do not think that many people would find the question of trying to back into Sony’s rental and sales numbers to be all that interesting. I’d be curious if anyone has ever tried to do a similar calculation for any other movie ever.

Some of the math that comes into play behind the scenes in When Genius Failed and All the Devils are Here may actually be a good example. An excellent starting point is Zvi Bodie’s paper:

On the Risk of Stocks in the Long Run

If you think the math behind $15 million in movie sales would interest kids, just think about the math behind trillions and trillions of dollars in derivatives!! (to be clear: snark)

So, the whole shaming of the NYT for this bit of math leaves me uncomfortable. I think the implicit message it sends about the “we know math” crowd vs the “you don’t know math” crowd is terrible. I also think that the calculation is much more subtle than is being let on and ignoring those subtleties is something that’s caused lots of problems in the business world over time. Thinking that these sorts of problems can be solved by simply plugging into formulas is thinking that is mistaken. Finally, even ignoring the first two points, I do not see the problem itself as that compelling, and especially not compelling as a reason to learn algebra.

A follow up to our Tina Cardone geometry project

Had a ton of fun turning a problem that Tina Cardone shared in to a 3D printing project earlier this week:

A Cool Geometry Problem Shared by Tina Cardone

So when I saw this problem in my son’s Geometry book this morning, I couldn’t resist trying a similar project today:

Geometry Problem

The problem is fairly straightforward to state: You go around a square connecting the midpoints of each side to an opposite corner (see the above picture). What is the ratio of the area of the middle square (shaded in the picture) to the area of the original square?

We started our project this morning prior to the videos talking through the problem and actually got all the way to the end. For our project we revisited the problem and then talked about how we could write down the equations of the lines. We need those equations to print the shapes. Our talk this morning lasted about 8 minutes and I broke it up into two pieces here. The first piece is us talking about some basic coordinate geometry and the equations of the horizontal and vertical lines:

 

The second part of our talk this morning involved writing down the equations of the lines cutting across the square. Turned out to be a nice review of lines, which was one of the things I was hoping to get out of this project:

 

When I returned from work tonight we revisited the equations of some of the criss-cross lines and eventually wrote down the equations for all four of them. I’m happy for the review of lines that we got in this project – it has been a while since we touched on that subject and a problem about lines from an old math contest gave him a bit of trouble last week, too. Getting all four equations in this part of the project enabled us to tell Mathematica how to draw the shapes we needed to print:

 

Next up was our little session in Mathematica. Not the best talk we’ve ever done, but we eventually get all of the shapes. As raw as this video is, I think it is a pretty honest description of my nearly non-existent programming skills . . . .

 

While my son was at his evening karate class I printed the 4 triangles, the 4 quadrilaterals, and the one small square that make up the large square. Here’s a quick shot of the printer finishing up one of the quadrilaterals:

 

When he got home from karate we explored the problem using the newly printed shapes. I had him go back to our study first to review the original shape. Having been out of the house I didn’t want him trying to reconstruct the original shape wasn’t fresh in his mind. After he built the large square we looked at a different way to see that the small square is 1/5 th of the larger square. In a way, that’s one of the important lessons here – if there’s a geometric result that seems simple, there’s often a fairly simple geometric explanation, too!

 

So, a super fun project building off of the earlier project from Tina Cardone’s problem. I’m really excited to see ways that our 3D printer can help us with plane geometry – honestly this is an application of 3D printing that I wasn’t really expecting.

Blogging year in review

Saw a link yesterday to blog year in review and it made me want to summarize my year in blogging. The other blog listed their top 10 most read posts and top 10 favorites to write in 2014. I’ll do top 5 for each:

First, my most read piece – by a mile – was not a math piece but one about ultimate frisbee. Skyd magazine wrote about 21 influential people in ultimate and included hardly any women. My response was here:

21 people in and around Women’s ultimate you should meet

Someday I hope my math posts can be as popular as my rare ultimate frisbee posts. Not in 2014, though . . .

My top 5 most read math posts were:

1: Dan Meyer’s Geometry Problem

This was a project inspired by a neat geometry problem that Dan Meyer posted on his blog. One fun part about the project was talking about the “other” solution to the problem.

2: 10 Fun math things I saw in 2014

This post was a look back at some of the fun math ideas that I saw in 2014. Part of the inspiration for this post was David Coffee’s post asking the question Whose Fault is it that you aren’t good at math? His interesting response to the question is that a large part of the reason that you “aren’t good” at math is that you didn’t have the right mathematical experiences. I wanted to show that that there are so many fun math experiences being shared on line these days, that the fun math experiences are much easier to find than they’ve ever been.

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

This project was inspired by a surprising connection between a Numberphile video and an old blog post of Patrick Honner. The Numberphile video shows a really fun game, and Patrick Honner’s proof without words in his post is amazing.

4: A small mistake in Numberphile’s video about Graham’s Number

Numberphile published several neat videos about Graham’s number. I’d previously talked through Graham’s number with the boys An Attempt to Explain Graham’s Number to Kids. The prep work for that original project had left me with enough knowledge about Graham’s number to notice a small mistake in a new Graham’s Number video Numberphile released this year.

5: Michael Pershan’s Exponential Post (2/3)

This is an odd one because the high number of views isn’t due to the content of the post – which was just a short reaction to a Michael Pershan post on twitter – but to Cliff Pickover linking to the picture at the beginning of the post:

Formula

6: A 3D Geometry Project for Kids and Adults Inspired by Kip Thorne

Since #5 wasn’t really due to the content of the post, I’ll include the 6th most popular – a project inspired by an interview I saw with Kip Thorne about his work on the movie Interstellar. This was a super fun project and is a nice way to transition to the posts I enjoyed writing the most . . . .

And the 5 project that I enjoyed the most:

1: The Kip Thorne project was part of a larger group of projects I did with the boys where I used talks or writing by mathematicians as the starting point for a project. The project below (the 3rd in a series of 3) using Terry Tao’s public lecture at the Museum of Math as a starting point to talk about math with kids was really fun:

Using Terry Tao’s MoMath lecture to talk math with kids: Part 3

You can also search on the blog for similar project based on Jacob Lurie’s Breakthrough Prize talk, a Bryna Kra talk at the Museum of Math, several different Evelyn Lamb blog posts, and many ideas from Fawn Nguyen and Patrick Honner.

2: Reacting to Dan Anderson’s “My Favorite” post:

Probably no blog post I read all year made me as happy as this post from Dan Anderson. He asked his kids to give short talks about the kind of math that they’d like to learn more about, and they came up with an amazing list of topics. If there’s one post I would recommend to anyone interested in math education it would be this post by Dan. Wish I could have seen the presentations:

A list Ed Frenkel will love – Dan Anderson’s “My Favorite” project

3: When my evening was similar to Fawn Nguyen’s evening.

Just after finishing up the Kip Thorne project – like 5 minutes after finishing it! – I saw a couple of posts on Twitter from Fawn Nguyen. It turned out that we’d be working on really similar projects with Zometool sets that evening. I’ve used so many of Fawn’s blog posts with my kids and it was fun to think that we were working on the same ideas that night without even knowing it! Also, the stuff you can do with kids using Zometool sets is amazing 🙂

When my evening was similar to FAwn Nguyen’s Evening

4: Adding in binary with Duplo Blocks

A couple of years ago when my older son was learning about binary we built a “binary adding machine” out of Duplo blocks. It was a super fun project and I got to return to it this year with my younger son. It is such a fun way to learn about place value and build number sense:

Adding in binary with duplo blocks

5: Jordan Ellenberg’s “Algebraic Intimidation”

I enjoyed Ellenberg’s “How not to be Wrong” and the section where he discusses the standard proof that 0.999… = 1 caught my eye as a potential fun project to work through with the boys. I especially liked Ellenberg’s concept of “algebraic intimidation” that he talks through in this section of the book, and decided to revisit the old Numberphile video showing that 1 + 2 + 3 + . . . = 1/12 with the boys as an example of “algebraic intimidation.” This project was particularly fun because my younger so is so bothered (as in actually physically bothered) by that result. He knows in his heart that it can’t be true, but the algebra all seems right – hence the intimidation part of Ellenberg’s idea.

Jordan Ellenberg’s Algebraic Intimidation

Looking forward to more fun projects in 2015!!

A cool geometry problem shared by Tina Cardone

Saw this neat geometry problem shared by Tina Cardone on Twitter earlier today:

One little spoiler in the first video below just in case you want to think through the problem first.

There are a couple of algebraic solutions to the problem that aren’t too difficult. The result is so cool and so simple, though, that I thought there must be a nice geometric solution, too. It took me a while to see it, but eventually I stumbled on the right idea. It made for a fun Sunday afternoon 3D printing project.

Here’s the first part:

 

and the second part. Spoiler warning again, though the warning this time is that I don’t give the full solution. Showing why the last two shapes have the same area is the fun part of seeing the geometric solution to this problem 🙂

 

A “Mowing the Lawn” problem with Legos

Today for our Family Math project we looked at a problem about rates. Half motivated because I was looking for a fairly light project and half motivated by a rate problem that my older son struggled with last week. See here for the more complicated problem from an old AMC 10:

The 2002 AMC 10A Problem 12

The specific problem we looked at today was a “mowing the lawn” problem and we used some of our lego pieces as props:

UniKitty

The first thing I did was introduce the situation. We have three people who mow a lawn. One person mows at a rate of 4 units per minute, a second person mows at a rate of 6 units per minute, and the third person mows at a rate of 8 units per minute. If the lawn is 288 units, how long does it take each person to mow the lawn?

One reason for introducing the problem this way is to get a little bit of arithmetic practice. The second reason was to show that rate problems aren’t that hard when you know the rates. This second point comes into play later in the project.

 

The next part of the project was to see how long it would take to mow the lawn if all three people were working together. Since the kids knew the rates for each individual, it wasn’t too difficult for them to add those rates together and calculate the required time.

After we finished that calculation, I changed the question a little bit:

Person A takes 72 minutes to mow the lawn, person B takes 48 minutes, and person C takes 36 minutes. How long will it take the three of them to mow the lawn if they all work together?

This version of the question caused a bit of difficulty.

 

Since we were stuck on the second problem from the last video, I decided to start a new movie to try to get a new approach. We talked through why this problem was similar to the first problem to see if the similarity gave us any clues as to how to solve this problem. The kids recognized that in the first version of the problem we knew how much work each person did in a minute. Figuring out how to translate the new problem into the old framework was the key idea:

 

One last challenge problem to end the project – if Unikitty and Buzz Lightyear work together, will they be able to mow the lawn faster than Sensei Wu working alone? I asked this question because I was curious how they would approach it – the way from the start, or the way from the last video, which is more complicated. They chose the way from the second video first and then noticed the easier approach after they solved the problem:

 

So, a fun project showing two different approaches to the same problem. One approach leads to a relatively straightforward solution, the other is a little more difficult, but knowing that there is an easier approach helps. Understanding how to transform difficult problems into slightly easier problems is an important step in learning about problem solving. It is fun for me to watch the kids learn some of these problem solving techniques.

Some fun conditional probability problems for kids

Lat night I was flipping through Art of Problem Solving’s Intermediate Counting and Probability book and found a couple of ideas that looked like fun problems to talk through with kids. Today we talked about two of them for our family math project.

The first problem is about rolling dice. You throw two six-sided dice and all you know is that neither one shows a 6. What is the probability that the sum is equal to 8? It was interesting to hear their approach to this problem since we haven’t talked about probability in a couple of months:

 

The next problem was about flipping coins. You play a game that involves taking turns flipping two (fair) coins at the same time. The game ends on a turn where there is at least one head showing. When the game ends, what is the probability that the other coin was also a head?

As with the last problem, it was interesting to hear the ideas that the kids had for this one. My older son guessed that the answer would be 1/2 and my younger son guessed 1/4. I think that it was a little more difficult for them to understand the right answer on this one (compared to the first problem), but we did eventually get there.

 

So a fun morning talking about some introductory probability problems. I like both of these problems because they are easy to understand even though they both have slightly surprising answers. Hearing the expected solutions and then getting to the actual solutions made for a great morning math talk with the boys.

Adding up perfect squares with snap cubes

We did a really fun project based on this tweet the day before Christmas.

After posting the project (whose link is embedded in the tweets below), James Key pointed out two videos that he thought my kids would like, and he was right!

and

 

This morning my older son and I had a great time working through the ideas in the two videos. Here’s how it went:

(1) First we reviewed the three pyramids that inspired the first project and discussed a little bit about how the project related to adding up squares:

 

(2) Next we moved on to a slightly bigger example: a geometric interpretation of the sum of the first 5 squares. We show how you the shape we get by combining these three large pyramids is starting to look more like a cube:

 

(3) The final piece of this project was moving on to the algebra. We come up with a counting formula for the number of snap cubes in our three pyramids and compare it with the formula for the sum of squares that I mentioned in our previous project (well, technically 3x that formula). We don’t work through the algebra – that’s an exercise for the reader! – but we do check that the two formulas give the same answer for the first 5 squares.

 

So, a nice follow up project, and also a fun little project for the morning after Christmas. Thanks to James Key for pointing out the additional videos!

A neat geometry project inspired by a James Tanton / James Key tweet

So much fun with this project – some great things to notice about geometric shapes, multiple manipulatives to use to help see the geometry, and then an ending that makes a cool connection with algebra. All from . . . this tweet I saw via James Tanton earlier in the week:

Looked like a really fun project and today we sunk our teeth into it.

The first thing was having the kids build the models out from our Zometool set. I was at work when they built the models, and I was happy that they were able to construct the models from the pictures on Twitter. It was interesting to hear what they thought about the shapes:

 

I wrapped up the first video too soon, unfortunately, because I wanted to ask them some questions that lead into making the shape on Mathematica:

 

Now we went over to the computer to take a look at the code we used to make the shape. I’ve not talked about this topic with either kid, so this part is more just showing them that you can make this shape rather than teaching them how to do it:

 

Making the shape on Mathematica allows us to export a version of it to print on our 3D printer. While one of the pyramids was printing I asked each kid to tell me what they noticed about the shape. First my older son:

 

and then my younger son. I was really excited to hear him say that he noticed that it looked like the shape was being built up out of squares!

 

Next we went back to the living room to show how the 3D printed shapes fit together. You get a slightly different perspective with the 3D printed shapes compared to the Zometool shapes, and this 2nd perspective was nice. I asked the kids about the volume of the pyramids expecting to hear them say that the volume was 1/3 of the cube. However, they caught on to the idea that the shapes were built up out of squares and that the volume of the pyramids was somehow related to the squares. Fun! Before diving into that, though, we did talk about why each pyramid had a volume equal to 1/3 of the cube.

 

We finished by talking about how you could sum up squares and get 1/3 of a cube. Mostly because I couldn’t resist, but it was a fun way to end this project, too. My younger son got a little mixed up with the algebra, but we got back on the right track after a minute or two. It is amazing to see how the sum of squares formula produces an answer that is 1/3 of a cube. Super fun way to end this project and a neat way to connect a little algebra with the geometry we just saw.

 

Learning Math isn’t always a straight line part 2

This morning I picked what I thought was a medium level difficulty problem for a movie with my older son. The problem was based on the first challenge problem in the back of the chapter on triangles in our Introduction to Geometry book. Here’s the problem:

“The angle bisector from one vertex of a triangle passes through the center of the circumscribed circle. Prove that the triangle is isosceles.”

By no means a simple proof, but I thought it would lead to a good conversation. Instead he got stuck and I didn’t do a great job helping. The entire 13 minute process is, I hope, a good problem solving lesson, though. Learning math isn’t always a straight line.

Part 1: Introduction to the problem (about 4 min)

The three things that happen in this video are:

(1) We talk through the problem,
(2) We talk a little bit about properties of perpendicular bisectors, and
(3) We eventually arrive at a reasonable picture for the problem

So, a pretty good start to the problem:

 

Part 2: The beginning of the proof (about 4 min)

There’s a tiny bit of overlap with the last video. I cut these videos by time rather than by content.

The four things that happen in this video are:

(1) He realizes that the perpendiculars from the circumcenter to the sides are equal,
(2) He has a good guess for which two sides are likely to be equal,
(3) He gets a little confused about the picture. In the picture that we’ve drawn, one of the perpendicular bisectors looks similar to the angle bisector and that causes some confusion, and
(4) He identifies two triangles that are congruent

 

So, it turns out that the last movie ends in a spot that is just one step away from the final proof. That’s good. But that last step turns out to be a little bit elusive. Part of the difficulty is that in getting to this point my son has lost some of the information that is important from the beginning of the problem.

That happens all the time in problem solving, and is why I think this exercise is a good example. I was reminded of a similar thing happening to Tim Gowers when he live blogged a solution to an International Mathematics Olympiad problem. I wrote a bit about Gowers’s live blogging here:

Problem Solving and Tim Gowers’s live blogging an IMO problem

For the point I’m making here, the important line from Gowers’s blog is this self-deprecating one: “You idiot Gowers, read the question: the a_n have to be positive integers.”

When all problem solving in math is presented as an easy straight line process, you forget that everyone from kids to Fields medal winners sometimes lose the thread of a problem they are working on.

Part 3: (about 5 min)

Here is where I struggle. We are so close to the solution which made me reluctant to point him in the right direction. Unfortunately we were also up against a time deadline since our neighborhood dog walk happens at 7:00 am and going on that walk is important to my kids. May seem silly, but the time pressure was real to my older son.

In this video:

(1) I try to help him see that the perpendiculars also bisect the sides, but that’s the point that he’s forgotten, unfortunately

(2) I have him go back and read the problem again,

(3) Once he realizes that the circumcenter is the intersection of the perpendicular bisectors, he finds the thread of the problem again and we finish the proof.

 

Part 4: (about 5 min)

After he returned from the dog walk I thought it would help him get a better understanding of the proof if we went through it again. This was just to help him learn, but I thought it would also be good to film it to show what a totally misleading presentation of what learning math looks like. Everything goes perfectly smoothly in this one!

 

A great morning talking math with the boys

Some days working with the boys in the morning goes really well and I’m left super charged up for the rest of the day. Love days like this.

Today I started a new chapter in Art of Problem Solving’s
Introduction to Number theory book with my younger son. This chapter is about modular arithmetic, which is a topic that we’ve touched on a little here and there but never in any formal way.

The book introduces modular arithmetic in a pretty natural way – with clock arithmetic and also by looking at rows of integers and pointing out which numbers are in the same columns. The only real speed bump in this introductory section came when we looked at negative numbers. It wasn’t immediately obvious to my son that, say, -1 was the same as 7 modulo 8, but the analogies to clocks helped get us over this hump. I really enjoy watching kids digest new math topics.

While I was working through this chapter with my younger son, my older son was working on some challenge problems from an old AMC10. Problem #14 from the 2000 AMC10 gave him some trouble:

2000 AMC10 Problem #14

Here’s the problem:

“Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the last score Mrs. Walter entered?”

So, by funny coincidence, my older son had a modular arithmetic problem, too. In talking through this problem with him we noted that:

(1) There’s nothing to learn from the first number,
(2) The second number has to have the same parity as the first number, and
(3) Then things get hard 🙂

I showed him that looking at the five numbers modulo three helped with the third step. The numbers become 2, 1, 2, 1, and 1. There’s only one way to have the numbers add up to be divisible by 3 – you have to take the three numbers that are congruent to 1 mod 3. The interesting thing is that you don’t know the order, but you do know the first three numbers.

The same approach works for the 4th number since 76, 82, and 91 add up to be 1 mod 4, we need a number that is 3 mod 4 to be the 4th number. The two choices are 80 and 71, so 71 is the only choice. That means the last number must be 80.

So, wonderful start to the day talking introductory modular arithmetic with my younger son, and then some slightly more advanced ideas in the same topic with my older son.

With that problem behind us we moved on to our Introduction to Geometry book. We are finishing up the review section in the chapter about triangles and stumbled on an absolutely wonderful problem:


Problem 7.46
: Medians AX and BY of triangle ABC are perpendicular at point O. AX = 12 and BC = 10.

(a) Find AO and BY.
(b) Find the length of median CZ.

Part (a) led to a good discussion about properties of medians, but did not require a deep dive into the math. Several of the other review problems have been about medians, so basic properties of medians are pretty fresh in his mind.

Part (b), though . . . great question! His approach was really interesting to me, but it was pretty difficult algebraically. He understands that the three medians chop up the original triangle into 6 smaller triangles with equal area. He used this fact, the fact that he knew two legs of one of the triangles that had CX as a third leg, and Heron’s formula to try to solve for CX.

This is obviously some messy algebra, though I think it is important to be able to learn to work through algebra like this. We went through it slowly and did eventually arrive at the answer.

An interesting geometric solution involves not looking at CX, but at XZ and the side AB of the original triangle. In this approach you can use the idea that the median to the hypotenuse of a right triangle has length equal to half the length of the hypotenuse to solve the problem. This fact is a slightly advanced geometry fact, but one that was fun to talk through.

The combination of these two solutions made for a great 30 minutes of talking through this problem.

This whole morning left me really happy and serves as another fun example of why I love teaching my kids math.