Cutting a double Möbius strip

Saw this sequence of tweets last week and thought that the paper cutting examples from the video in video linked by Patrick Honner would make for a great activity to go through with kids. I even recruited a few kids besides my own to try out the exercise.

Here are the four shapes that we cut in half:


The main purpose of the exercise was to hear kids talking about the shapes, and especially what they thought the shapes would turn into when cut. So, not as much discussion as usual, just ideas.

First up – my kids going through the four exercises:





Next up – a 2nd grade son of one of our friends:





I goofed up the construction of the double Möbius strip (you have to twist the strips the opposite way to make this work), so the 4th video is really our second time through, though 1st time with the correct shape.

Finally – a 4th grade daughter of one of our friends:





I loved hearing what all the kids had to say, and all of them seemed to really love these activities. Would love to go through this exercise with a larger group of kids – it would be amazing to hear all of their ideas about what shapes they thought they would see, and well as their descriptions of the shapes after we cut them. Lots of fun and lots of great opportunities to talk and share math ideas with kids here!

Two interesting problems from the Julia Robinson math festival

A comment on yesterday’s blog pointed me to some fun problems from the Julia Robinson Math Festival:

Julia Robinson math festival problems

It was exciting to see these problems and I really enjoyed played around with them a little bit yesterday afternoon. With nothing specific planned for our Family Math today I thought picking two of them to work through with the boys would make for a fun project this morning. Since my younger son is just starting to learn some basic geometry, I picked two of the problems that didn’t require any advanced geometry, but I’ll definitely keep the complete list in mind when I’m looking for other geometry projects in the future.

The first problem we looked at was problem #5 in the link above:

A circle of radius 15” intersects another circle, radius 20”, at right angles (see below). What is the difference of the areas of the non-overlapping portions.

On thing that made this problem attractive is that had some similarities with the problem we looked at yesterday. The boys found a nice solution and also avoided the trap set by the problem writers!

The second problem I picked from the above link was #8 – a neat Pythagorean identity that I’d not seen before.

This problem gave them a bit more difficulty, though the did remember a few things from Numberphile’s Blob Pythagorean Theorem video, so that was nice to hear. After struggling to figure out how to get started, they eventually decided to check out what would happen in a 3-4-5 triangle – a great way to get started!

The interesting thing about this video, though, is just listening to the kids trying to get their arms around the problem. Similar to many previous examples, the path to the solution isn’t a straight line.

We paused the last video because we’d gone over 5 minutes, but we just turned the camera on and off. The next video just picks up on the calculation we were doing for the 6-8-10 triangle. At the end of the last video there was a bit of confusion between the radius and the diameter of the circles. With that confusion out of the way we get to the solution of the problem without too much trouble:

So, thanks to Ben for pointing out this great list of problems from the Julia Robinson Math Festival. Can’t wait to use a few more of these problems in future math projects!

Talking through David Wees’s square problem

Yesterday David Wees posted this problem on twitter:

Since my younger son was on a little trip today I didn’t mind using this slightly more advanced geometry problem for our Family Math project. I hoped that the solution would be accessible to my older son, and we did indeed end up having a really great conversation about the problem.

We started by looking at the original question and seeing what he noticed. His first thought was to look at two simple cases:

After studying two of the easier cases we moved on to the general case. He struggled quite a bit to see what to do in this next step. This struggle was great and exactly what I was hoping for when I chose to go through this problem. Although we don’t make a lot of progress in the 4 minutes of this video, this part shows what learning and thinking about math often looks like – no straight line to the solution, and a lot of ideas that are good, but don’t quite work:

After about 4 minutes of struggle in the last video we turned the camera on and off after he asked “what else can we do?” The next couple of ideas after that question don’t lead anywhere, but then there’s an “aha” moment around 1:30 – “well . . . I see something now.” That “something” turns out to be a really nice congruent triangle argument.

So, a great problem from David Wees led to a really nice struggle + solution for my son. Hopefully a nice example of what a kid learning and thinking math can look like.

Kirstin Gray’s Shadowcon Talk

A conversation I was having with Michael Pershan on Twitter today included this tweet:

I’d watched the talk when it was originally published but took Michael’s lead went back and watched it a few more times yesterday. It definitely has me thinking.

It is certainly clear right from the beginning of the talk that we have many thoughts in common:

“I am genuinely curious about the content I teach and how I teach it.”

“I am genuinely curious about my students’ math conversations.”

“I am genuinely curious about the math work my students do each day.”

I also share her thirst for conversations about math. Similar to what she’s describing around 3:10, I’m constantly looking for new ideas to share with the boys on twitter, from blogs, and just about anywhere I can find something. It amazes me how many great math ideas and projects are out there if you just look!

Her discussion about fractions and twitter reminded me of an experience I had when I was talking about dividing fractions with my younger son. In our video that day I introduced the topic to him by defining division as multiplication by the reciprocal. That was one of the few videos I’ve put out that generated some negative reactions (it is MathProblems81 on youtube if you are interested), so I searched for a different explanation:

So, maybe not exactly the active search for advice that she is describing, but my thinking was certainly influenced by the advice on twitter in this case.

Around 5:30 in the talk she’s discussing wanting to know what her students think about topics using “talking points,” “always, sometimes, never” and by just asking them what they know. The specific part about decimals reminded me of how I began the decimal and fractions unit with my younger son last fall. Her comments from her students also reminded me of some of the ideas my son had (“I don’t think fractions are rational” for example):

Also, we just happened to do a little “always, sometimes, never” game with quadrilaterals earlier this week! Maybe watching her talk the first time around planted that seed:

Finally, the comment from her student about fractions in fractions reminded me of one of our old continued fractions projects. I love continued fractions – they are a great way to sneak in a little extra fraction practice while showing a really interesting math topic (and they also remind me of my high school math teacher who taught them to me):

Continued fractions and the square root of 2

This part shows how fun the topic can be (starting at 6:30 just in case it doesn’t embed properly): “ohhhh . . . that’s cool”:

Finally, her call to action is to start a math journal. My journal consists of the videos that we put out (almost) every day and this blog. I love using the videos to back and seeing how each kid’s thinking has changed over the years (and how my thinking and teaching has evolved, too). Sometimes things like Kirstin Gray’s talk remind me of old projects and I can review how we approached similar ideas. So, I definitely agree that starting a math journal is a great way to help you reflect about your own teaching and learning.

So, thanks to Michael Pershan for the suggestion to look more carefully at Kirstin Gray’s talk, and thanks, of course, to her for giving this amazing talk. It definitely made me think.

Some fun geometry this morning

For the last couple of weeks I’ve been teaching geometry to both kids. Sort of a coincidence since Art of Problem Solving’s Prealgebra does a little intro geometry at the end, but it is definitely a fun coincidence.

I’m enjoying watching my younger son think through some introductory ideas and enjoying watching my older son start to think about some ideas in 3-dimensional geometry. Today they each made great progress working through two challenging problems.

For my younger son the problem was a fairly complicated area problem. His approach to geometry has not involved a lot of calculations before, so I was surprised when he started calculating right from the beginning here:

Although I was impressed with his ability to work through the calculation from start to finish, I also wanted to show a more geometric approach. Fortunately we had some Magna Tiles laying around that were the exactly the right shape for this problem. These tiles allow you to see the solution without doing nearly as much calculating:

Hopefully this extra exercise with the Magna Tiles helps him build on his geometric intuition. I certainly leaned heavily on calculating rather than intuition as a kid, so I feel that I need to be extra diligent about showing ideas that help both kids build intuition.

The problem that my older son was working on this morning involved finding the distance between the centers of two faces of a tetrahedron. This problem is pretty challenging and requires you to visualize some difficulty 3D geometry. Unluckily (or maybe luckily, I’m not sure) we didn’t have our Zometool set laying around to help with the visualization – it all had to be in his head or on the board.

We probably talked about this problem for 30 minutes leading up to the video. One great moment was when he realized that the angles between the faces of the tetrahedron might not be 60 degrees even though all of the faces are equilateral triangles. Thinking about how to determine that angle led him to the solution.

After we finished our conversation I asked him to do a little recap of the problem and then threw in one extra question at the end just to mix it up a little:

So, a great morning. It is nice to watch my younger son make progress thinking through some of the ideas from elementary geometry, but also a little surprising to see him approach the problem by calculating. It is also interesting to watch my older son think through some of the ideas from 3 dimensional geometry. It isn’t easy to see all of the angles in 3 dimensional geometry, but hopefully today was a productive struggle for him.

What I learned from Grant Wiggins

I’m not even remotely knowledgeable about education theory, but I enjoyed reading Grant Wiggins’s writing. He had a wonderful ability to translate from abstract to concrete when talking about ideas in education and thanks to that ability I always had something to take away from his pieces.

One example in particular made a lasting impression on me. In his exchange with Patrick Honner roughly two years ago, he used the problem below as an example of a difficult problem:


The difficulty of the problem surprised me – only about 5% of US 12th graders answered it correctly. His writing forced me think about what made the problem so difficult. Part of that thinking was working through the problem with my kids:

Wiggins actually left a fun comment on the video which was a nice surprise for me.

Just a few weeks ago I was talking with my older son about cylinders and returned to look at the problem much more carefully:

A deep dive into a problem from Grant Wiggins / Patrick Honner

What I learned from this experience was that my own judgment of the difficulty of a problem isn’t relevant to anything. When my kids are struggling with a problem or concept, I give it more time and try my best to understand their difficulty. I’m now also much more suspicious when I see comments like “this isn’t a hard concept” floating around on line.

I don’t know enough about educational theory to know what theoretical framework his this cylinder problem fits into, but his use of this concrete example led to a pretty important step forward for me in thinking about how to talk about math with kids. I’m lucky to have seen it.

Rest in peace Grant Wiggins.

A lucky follow up to yesterday’s Mathcounts project

Yesterday we did a fun project checking out the outrageous speed of some of the Mathcounts participants:

The Insane Speed of the Mathcounts Final

One of the questions we studied in yesterday’s project involved finding the area of a rhombus. By lucky coincidence today my younger son and I started a section in our Prealgebra book about the area of quadrilaterals. I thought it would be fun to revisit the Mathcounts problem now that we have actually studied how to find the area of a rhombus.

First we just recapped how you find the area of a rhombus.

Next up, we took another look at the problem (and the insane speed) from yesterday:

Finally, we took a second crack at solving the problem now that we’ve learned a bit more about rhombuses. Really happy with the way my son talked through the problem here.

So, a pretty lucky coincidence starting a section related to one of the problems from yesterday’s project. It was really fun to see him work through the problem, and, of course, nice to have another gasp at the speed involved in the Mathcounts final.

3d Printing and Platonic solids

[note: written in 5 min running out the door, sorry . . . .]

We have a couple of old fold up platonic solids that we printed from Laura Taalman’s Makerhome blog. For example:

Laura Taalman Platonic Solid example

These models proved to be extremely helpful when we were talking through the section in our geometry book on regular polyhedra today. They even help you see the dual properties of cubes / octahedrons and icosahedrons / dodecahedrons.

Happy to have these props – another great example of how helpful Taalman’s blog has been for us!

A simple review of Element by Dragonbox

I was traveling at the end of last week and looking for something for my younger son to work on while I was away. I happened to see this (re)tweet from Keith Devlin:

and decided to check out Dragonbox. Turns out that they have a geometry game called Element that seemed to be exactly what I was looking for. Until tonight I’ve not seen much of the game, but here are some thoughts from my younger son (3rd grade) and then some quicker thoughts from my older son (5th grade) who played the game for the first time tonight.

I can tell you for sure that the game is engaging!

Game Pic

First up is my son’s description of the game and some of the things that he’s learned:

Here’s a 5 minute stretch of him playing the game. I don’t understand the game well enough to help him or even say what’s going on here, but there’s certainly some interesting problem solving.

Finally, here’s a quick recap from him of some of the things he’s learned and some of the things he’s hoping to learn:

Finally, here are some quick comments from my older son after he played the game for about 30 min.

So, despite not having a full understanding of the game, I’m happy with the experience that both kids have had with this game so far. There is certainly no doubt about the enthusiasm they have for it – even trying to put the game away to go to bed wasn’t easy!

I’m also glad to hear both of them talk about shapes with such enthusiasm, too. Hopefully I can figure out a good way to incorporate resources like this into more of the traditional work we are doing in geometry.

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.