Going through the NY Regents Algebra exam questions in the NYT

Saw this article about the NY Regents Algebra exam in the NYT today:

NYT Article on the NY Regents Algebra Exam

The article made me sad on a lot of levels, but it did include 5 sample problem and I thought it would be fun to go through those problems with my older son.

Here are the problems and my son working through them:

Problem 1 – a problem about lines and slopes:

Problem 2 – a problem about algebraic inequalities:

Problem 3 – a problem about systems of equations and arithmetic (this problem makes me cringe, but my son has a clever idea about how to simplify the calculations):

Problem 4 – a problem about the graph (and the roots) of a polynomial equation:

Problem 5 – A problem about arithmetic and geometric series

So, I thought all but the 3rd one were good questions. I honestly have no idea why the exam writers felt it was important to write that question using the amounts of dollars and cents that they did. It seems to me that this choice made the problem needlessly complicated.

An Abstract Algebra question from John Golden

Saw this question from John Golden on Twitter a few minutes ago:

My immediate idea was two prior Zometool projects that we’ve done that touch on rotation groups, but they require a Zometool set.

My two next thoughts were a bit more technical – Galois Theory and Elliptic curves. On reflection, though, I feel like both are pretty tough tasks for one class.

So my next idea related to three things I’ve seen on Twitter recently.

(1) Start by watching the first 10 minutes or so of this wonderful public lecture by Jacob Lurie from last year’s Breakthrough Prize:

In the first part of the talk he discusses rings and touches on Emmy Noether’s work on the subject in the early 1900s.

Here’s how I used this video with my kids last week (we did not explicitly dive into abstract algebra, but we did talk about clock arithmetic):

Using Jacob Lurie’s Breakthrough Prize Lecture to inspire kids

(2) Next check out this video linked by Steven Strogatz last week:

In this video you learn about a few incredible ideas related to abstract algebra. For example, when you adjoin i to the integers, you get new primes, but you still have unique factorization. However, when you adjoin the square root of 5, you lose unique factorization. These ideas are just one step removed from what Lurie touched on in his lecture.

Oh, and the punchline of this video about the square root of 163 is pretty amazing!

(sorry not TeX-ing this, I’m writing in a hurry)

So, even just stopping with the ideas in this video you’ve got some neat facts that are pretty accessible (and cool!).

(3) Finally, if you have time, take a look at this “new to me” proof that e is irrational that Dave Radcliffe tweeted about last week:


Essentially this proof looks at numbers of the form A + B*e where A and B are integers. This set of numbers isn’t a ring, but it is at least another example of expanding a number system. For a one day lecture it seems close enough to what’s going on in part (2) above to keep the class flowing. Plus, it is sort of fun to see this proof that e is irrational.

It is also easy to skip of the first two parts take longer than expected.

Anyway, that’s my “pondering this Twitter question for 20 minutes” idea.

Summing up Squares

Earlier in the week my son’s math team had a practice problem with this sum:

2 + 5 + 10 + 17 + . . . .

The question was how many terms would it take in order to be above 650?

I though that this problem would make a good starting point for reviewing the formula for the sum of squares:

1^2 + 2^2 + 3^2 + \ldots + n^2 = (n)(n + 1)(2n + 1) / 6

This project didn’t go as well as I’d hoped, but maybe there’s a way to come back to it again later to reinforce some of the ideas. The difficulty the kids had – arranging 6 stacks of 1^2 + 2^2 + 3^2 cubes into a box – comes in the 3rd and 4th videos below.

We started by talking about sums of integers. Of course the infinite sum came up at the end!

With the sums of integers out of the way, we introduced sums of squares. After we wrote down the formula, we started exploring that formula with snap cubes:

Now we studied the slightly more difficult geometric cases of 1^2 + 2^2 and 1^2 + 2^2 + 3^3. The second case gave the kids a lot of trouble.

In the last video we weren’t able to construct the box relating to the sum of the first 3 squares. We kept trying here. I didn’t expect this piece to give them as much trouble as it did, so, unfortunately, I didn’t have any good ideas to help them see what to do.

One thing that was interesting to me is that the pattern with the colors seemed to hinder their progress as much as help them:

As prep for this project yeterday, I 3d printed 6 of the block sets for 1^2 + 2^2 + 3^2. This was a fun 3d printing project all by itself, and came in handy in this project to see if they were able to build the shape without the colors.

Although, as I mentioned above, the color patters seemed to give them as many difficulties as help, my older son thought it was actually harder to build the shape without the colors:

To wrap up the project we took a look at the three pyramids that fold together to make a cube. This idea was a fun project all by itself last year:

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

We also looked to see if the sum of squares was approximated well by n^3 / 3 for a few values of n.

So, although this one didn’t go exactly as planned, it was still pretty fun. After we finished I had they try to build the box made out of 6 1^2 + 2^2 + 3^3 + 4^2 pyramids, and they built that shape pretty quickly.

I like the connection with the sum and the 3 dimensional box and am excited to come up with ways to revisit that connection. I’m also trying to think through ways to show a connection with the sum of cubes and a 4D box – I bet that will be fun!

Dave Radcliffe’s Geometry problem

Saw a neat tweet from Dave Radciffe earlier in the week:

I was reminded of this problem again this morning when I asked my older son what he wanted to talk about for our Family Math project today. The topic he chose was similarity and congruence. I was a little surprised since it has been a while since we’ve talked about any geometry, but Dave’s problem fit the bill.

Although this problem is a little to difficult for my younger son to understand, I thought it would be fun to talk through it with both kids. He did struggle with a few of the ideas, but I think that overall it was an interesting problem for him. For my older son, this was a nice geometry review.

We started by talking through the problem and finding one easy idea:

Next we returned to the more general case to explore the ideas of similarity and congruence that help solve this problem:

For the next part of the project we tried to apply the ideas from the last part of the talk to our problem. This piece was difficult for my younger son, but we went slowly. It was nice to see him begin to understand some of the algebraic expressions.

Finally, we wrapped up by finishing the calculations and found that the area the shaded region in the problem was 4 \pi no matter where the segment of height 4 is placed!

So, a fun project and a excellent bit of luck that Dave had posted this problem earlier in the week. I still don’t know what made my son think of today’s topic, but it was a good geometry review for him and a nice algebra / geometry introduction for my younger son.

Our year in Math

I guess this is more of a November 2014 to November 2015 post, but I had some down time today and was thinking about all of the math-related things that I seen as well as some of the ideas that made for fun projects with the kids throughout the last year. Here’s what came to mind.

(1) Starting on a sad note – the death of Virginia Lee Pratt

Virginia Lee Pratt built a math powerhouse

Ms. Pratt built an incredible math program at my high school. Although she retired before I got there, her influence lived on at the school through my teacher, Mr. Waterman, who took over from her as the head of Central High’s math department.

We had Mr. Waterman and Ms. Pratt over for dinner when we lived in Omaha back in 2000. I remember asking her what it was like to be Saul Kripke’s math teacher –

“He rocked,” she said in a seemingly quite hip response for an 80 year old, “oh, yes, I remember he would just sit at his desk and rock back and forth as he was thinking about problems.”

I keep a little note that she wrote to me back in 1987 when I won a book prize:


VLP letter

She touched so many lives. RIP Ms. Pratt.

(2) The growing love that I have for using 3D Printing and Zometool to help my kids see both 2d and 3d geometry.

I hope that these technologies are in all schools soon, but I fear it’ll be a long time because of the costs. The ability to hold objects in your hand opens up geometry to kids in a way that I couldn’t possibly have imagined ahead of time.

One of the most amazing 3D printing projects I saw this year came, of course, from Laura Taalman:

I used Taalman’s pentagons to talk a little math with the boys here:

Using Laura Taalman’s 3D printed pentagons to talk math with kids

Another fun project was trying to use some of Dan Anderson’s calculus prints with kids:

All of our 3D Printing projects are here:

A link to the 3d printing projects on our blog

As for the Zometool work, I’m continually amazed at how versatile and how accessible this building set is:


My favorite project wasn’t even a real project. One of the kids on our street was over and built some amazing shapes just playing with our set:

A fun Zometool story

Another fun project this year was building the 5 tetrahedrons that live inside of a dodecahedron:

Five Tetrahedrons in a Dodecahedron

Finally, one project that used both 3D printing and the Zometool set was our study of the Gosper curve – this might have been my favorite project that we did this year:

A fun fractal project – exploring the Gosper curve

All of our Zometool projects are here:

Our Zometool projects

(3) Numberphile

Numberphile produced another set of incredible math videos this year. The work they do promoting math is absolutely amazing.

Their video on the fold and cut theorem totally totally captured the imagination of my kids as well as some other kids on the block:


Our one cut project

One super fun thing about this project was that I was able to share it with Martin Demaine, who helped prove the one cut theorem!

Another fun project inspired by a Numberphile video was this one on the Pythagoream Theorem

Using Numberphile’s Blob Pythagorean Theorem video in a lesson

One mathematician that’s done a lot of work with Numberphile is James Grime. He invented some non-transitive dice which we bought and used for a fun project:

Non-Transitive Grime Dice

Finally, Numberphile’s interview with Ken Ribet is something that I think every budding mathematician should see:

Two pieces from this year I wish I would have seen in graduate school

(4)Some great math books for adults that came out this year

The 2014 edition of The Best Writing in Mathematics was fantastic and introduced me to John Conway’s neat twist on the Collatz Conjecture:

The Collatz Conjecture and John Conway’s Amusical Variation

Speaking of Conway, Siobhan Roberts’ new biography – Genius At Play + a Jim Propp blog post about the surreal numbers led to two incredibly fun projects:


Walking down the path to the surreal numbers with kids

Walking down the path to the surreal numbers part 2

Although probably not going to inspire too many projects with kids, the new Princeton Companion to Applied Mathematics has been extremely enjoyable. The short articles have been especially nice for times when I drop the kids at music lessons or other times like that when I’m just waiting around.

Finally, Eugenia Cheng’s How to Bake Pi, CΓ©dric Villani’s Birth of a Theorem, and Frank Farris’s Creating Symmetry were wonderful “popular” math books from this year.

(5) Math books more for (and to use with) kids

Count Like an Egyptian is a book we ran across through an Evelyn Lamb blog post. It led to three really fantastic projects with the boys. The link below is to the 3rd project and it has links to the other two:

Count Like an Egyptian Part 3

Really Big Numbers by Ricahrd Even Schwartz is an incredible book for kids. It has ideas ranging from simple to extremely advanced and kids at any level of math will find something interesting. Our project touched on 3 ideas from the book:

A few projects for kids from Richard Evan Schwartz’s Really Big Numbers

Finally, the loop-de-loop idea from Anna Weltman’s This is not a Math Book have totally captured my younger son!


Here’s the 2nd of our two projects using Weltman’s idea:

Anna Weltman’s Loop-de-Loops part 2

(6) Math writing and blogs:

Evelyn Lamb’s blog Roots of Unity is absolutely amazing. Lamb’s blog (as well as her tweets) have inspired tons of projects:

Projects inspired by Evelyn Lamb

Erica Klarreich’s math articles on Quanta Magazine’s website are must-reads. As are Natalie Wolchover’s physics articles

Another amazing source of math writing I found this year was Richard Green’s Google+ account

Two of his posts inspired fantastic projects with the kids:

Using a Richard Green Google+ post to talk about Geometry with my son

Another great piece of math to share with kids from Richard Green

Finally, one of the most amazing blog posts of the year came from Lior Pachter

Unsolved Problems with the Common Core

I had a great time thinking about the ideas he presented:

Reacting to a Wonderful blog post from Lior Pachter

(7) Textbooks and other educational ideas

For sure the most amazing new textbook I ran across this year (thanks to Kate Nowak) was Michael Serra’s Patty Paper Geometry

Patty Paper Book

I need to use this book more, but we’ve already done a couple of really great projects from it:

Our Patty Paper geometry projects

Although I’d used it before with my older son, this year I gained a much greater appreciation for Art of Problem Solving’s Introduction to Number Theory going through it with my younger son.

We’ve done about 50 Number Theory projects so far. Both kids love number theory and Art of Problem Solving’s book shows how fun the subject is. It is a great way for kids to learn some math that is probably a little bit outside of the normal school curriculum while also getting tons of practice with arithmetic and opportunities to build number sense.

Here are our number theory projects

(8) Lectures / academic stuff

Moving to Boston has given me the opportunity to attend interesting academic lectures when I have time. Three of these opportunities led to two fun projects with the kids, and one interesting night at Harvard.

Back in September I attended the first lecture Harvard’s gender inclusivity in mathematics series. The speakers were Cathy O’Neil from mathbabe.org, and Moon Duchin from Tufts. I follow O’Neil on mathbabe.org and also on Slate’s Money podcast, but I’d was not previously familiar with Duchin. They gave a great talk, and I’d say that Duchin has the most “take complete command of a room 10 seconds after walking in” as anyone I’ve ever seen. My notes for that talk hare here:

Harvard’s Geneder Inclusivity in Mathematics Talk

Another neat talk I attended was Larry Guth’s lecture about the “No Rectangles” problem. I’d not heard of this problem before, but Guth’s lecture was fantastic and really interested me. The neat thing about this problem is that even though the general solution is complicated (and well beyond my understanding) a few of the basic cases are accessible to kids. I played around with the problem with the boys the following weekend:

Larry Guth’s “No Rectangles” Problem

Finally, I had the lucky opportunity to meet Peter Fisher, who is the head of the MIT physics department. He gave me two “dark matter” boxes which each contain one dark matter particle. Well, supposedly, it looked like they were both empty to me! These boxes led to a really fun conversation with the boys:

Talking a little math and physics with MIT’s Dark Matter Boxes

(9) The rise of data science

I actually don’t know what to make of this field just yet. I’m sure there are going to be people who make amazing contributions, but there’s also so much publicity around data science now that I feel like there’s much more form over substance.

But, there is some really neat stuff out there if you look in the right places. One of the more incredible articles – math or otherwise – I saw this year was this post about Taxi cab fares from IQuantNY:

How Software in Half of NYC Cabs Generates $5.2 Million a Year in Extra Tips

If you want to know what great data science work looks like – study that post.

(10) Twitter

I’m extremely happy that there is a big math community on Twitter. It has allowed me to follow, and learn from, people work in k-12 math education, college professors, and people who use math in their work. It really is an amazing collection of people. There are so many incredible ideas floating around!

I have tons of projects inspired by tweets – inspired by these people, for instance:

Fawn Nguyen

Patrick Honner

Alexander Bogomolny

Dan Anderson

Kate Nowak

Steven Strogatz

Tina Cardone

Tracy Johnston Zager

James Tanton

and I could keep going and going and going.

One neat thing that happened this year was that the NCTM conference was in Boston and I got to meet many of these people in person, which was super cool – and even the president of the Mathematical Association of America, Francis Su, came by to hang out:

My surprise afternoon with the MTBoS and MAA President Francis Su

Finally, there are always neat philosophical questions posed by math folks on Twitter – a series of blog posts from John Golden’s class essentially talking about what they thought math was inspired me to write some thoughts back to them:

To John Golden’s class

So, a fun year all around in math for us. Can’t wait to see what 2016 brings πŸ™‚

The Birthday “Paradox”

I asked the kids what they’d like to do for a math project today and my younger said that he’d like to do something birthday related. Well, that’s easy πŸ™‚

In the first part of the project I introduced them to the famous Birthday Paradox and asked them what they thought the answer was. The question is easy to state:

How many people do you need to have in a room so that you have a 50% chance of 2 people sharing the same birthday.

Their initial guess for the number of people was 180 (younger son) and 240 (older son):

After the short introduction we went on to talk about how to calculate the probability. My younger son had the nice idea to start from some simple cases of 1, 2, and 3 people in the room. Eventually we got to the idea of looking at the problem via complimentary counting.

Now that we had some experience with the complimentary counting approach, we did one more example – this time with 4 people. The kids noticed that the probabilities are increasing pretty quickly.

Finally we went to my work computer to look at the probabilities for various cases using Excel. The kids were really surprised to see how few people were required to solve the problem:

So, a fun project because the result is so non-intuitive. Also a neat project to use with kids because there are lots of basic math ideas we use to solve the problem – percents, decimals, and counting techniques to name a few. Nice little birthday morning!

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!

What learning math can look like: A challenging problem about runners on a track

This problem from the 2012 AMC 10a gave me son a lot of trouble yesterday:

Problem #16 from the 2012 AMC 10A

You can see from the beginning that he’s really confused about how to even approach it:

So, after the introduction to the problem, we move to the whiteboard to begin solving the problem. His first idea is to try to look at the least common multiple of the speeds – this is a tricky and difficult approach, unfortunately.

Next he tries to find the time it takes for each runner to run a lap. This, again, is a difficult approach, but at least this approach uses the idea of speed.

Finally I ask him to think about what it means for two of the runners to meet.

Next we looked at how long it would take for two of the runners to meet. We find that these two particular runners meet after 2,500 seconds.

He’s still a little confused about how to describe the situation in the problem. For example, he initially thinks that one of the runners will have completed 5 laps in 2,500 seconds. But, it does feel as though a few of the ideas are beginning to sink in.

Now we come to the final step in the problem. We know that two of the runners meet for the first time after 2,500 second. What about the 3rd runner – when does that runner meet the other two?

For the last part of the problem we returned to the original problem to see if we could have learned anything at all from the choices. We decided to check if the three runners were together after 1,000 seconds:

So, this is a really challenging problem. Even in the last video – which was after talking about the problem in the morning and then again for 20 minutes at night – he’s still a little confused.

It was interesting to me that his initial thought on how to approach the problem revolved around least common multiples. The change that running around a track – and thus having differences that were multiples of 500m – made a huge difference in terms of level of difficulty of the problem. Feels like we are going to have to review a few more problems similar to this one to help the ideas from this problem sink in a little more.

Looking for math-y games for kids?

Here are three that we’ve really enjoyed:

(1) Equilibrio

This game has been especially popular with the kids on our street and is visiting its 3rd house since we got it!

A review of Equilibrio by Fox Mind Games

Game Pic

(2) Prime Climb

I learned about this game from a Dan Anderson tweet:

This is a neat game that has lots of fun strategies relating to simple arithmetic.

A review of Primb Climb by Math For Love

(3) Terzetto

A fun game that helps kids play around with patterns and rotations.Β Β  This game is my younger son’s favorite game πŸ™‚


A Review of Terzetto by Gamewright


Dividing Fractions

Ran across an basic fraction division problem on an old AMC 8 today.

Problem #12 from the 1989 AMC 8 (AJHME).

The problem is easy to state:

Evaluate \frac{ 1 - 1/3 }{1 - 1/2}

When my son dove in to the problem he immediately remembered the “flip the denominator” rule. That rule makes pretty much takes care of solving the problem. BUT, after he got the answer I asked him if he knew why the rule was true. He said that he didn’t know.

So, we spent the last couple of minutes in the video talking about why it was true:

When I first introduced fraction division to him, we looked at flipping the denominator and also a second approach involving snap cubes. I guess the “flipping the denominator” rule is a lot easier to remember, but I still remember the fun we had making this video πŸ™‚