Semicircle Puzzle

Matt Enlow posted an interesting geometry puzzle on twitter (tweet here), and I think the thing that got me intrigued was his initial challenge: “I can’t tell how hard this problem I just made up is.” Not knowing if there is an elegant/easy/obvious solution or not got me hooked.

I’m going to try to outline my approach/solution, because I sometimes like deconstructing my thinking to see how I actually think/learn… so from this point on… SPOILERS.

Some things that stood out to me… First, it looked like there was initially a single circle in a square, and the circle got cut in half and then it started sliding. So I initially drew the full circle in the square (before sliding), I drew the diagram shown, and then I drew the two semicircles in a rectangle after they fully “slid”… I saw the cut circle “in motion” — but after a short while I didn’t see how that would help me.

Then I drew the image and solved the problem and felt proud about it. But then I realized I drew the picture wrong. I circled the wrong part in my diagram, so you can see. I had the “slice” hit the corner of the rectangles, and then I was able to use similar triangles to come up with a solution.

I was proud but for some reason, probably because Matt’s initial tweet suggested to me that it would be harder than this, something was nagging me about it. So I went back and quickly saw my error. But I have always found that taking a wrong approach can help eliminate pathways to a solution, but might also help me see possible tools to use in a solution. And in fact, this idea of using that “cut line” and similar triangles was important in my pathway to the end.

So when I went back to the drawing board, I wanted to really see how this diagram worked… Some things were fixed (the 12 by 19 rectangle, the fact that the semicircles sort of “slid,” and importantly, the fact that the semi-circles were tangent to the rectangle at two places). So I decided to build this diagram in geogebra (with only one of the semi-circles), and as I built it, I saw that everything hinged on the movable point “G.”

I made the line where the semi-circles touched movable, based on the location of point G. Play around with moving point G here on this web-based geogebra page, and try to get it so the semi-circle on the bottom is tangent to the right and bottom side of the rectangle!

So to me, everything hinged on location of point G, or in other words, the distance from A to G (which is the same as the distance from H to C). We are looking for the location of point G which makes the semi-circle perfectly tangent to right and bottom sides of the rectangle. So to me, those appeared to me as “keys” to the problem. [1]

Sooooo I drew my diagram, and importantly labeled the distance from point A to point G with a variable, a. And then I labeled lots of things in my diagram in terms of that variable and the radius of the semi-circle, r.

I had two variables, so I needed two independent equations. And here is something nice… because I initially went down a wrong path earlier with my mis-drawing, I had already gotten similar triangles in my head! So I got one equation from that.

I hunted and hunted, and found another equation I could get… using the Pythagorean Theorem!

So now I had two equations and two variables.

… and since I knew this was going to be a beast to solve, I just used Desmos, and got that the solution is a=1.5 and r=7.5.

I did a little of the algebraic gymnastics to try to work this out by hand, but it was pretty uninteresting to me and I was pretty convinced that if I really wanted to, I could. To me, getting the equations was the interesting part, and the rest felt like pencil-pushing. So I stopped there. It was nice that the geogebra applet I created seemed to confirm my answer for me:

So that was my process to solving this mathematical puzzle. Who knows – I could also be totally wrong! I’m left thinking of the following:

(1) Is there a more elegant way to come up with the answer? Because the answer is so nice (a diameter of 15?!?!) but it comes out of such an ugly set of equations, I bet there is a nicer way. In other words, is there a better “conceptual” approach that gives a stronger insight into the geometric nature of the setup?
(2) How did Matt come up with this puzzle? How did he come up with the 12 and 19, so that the answer worked out so neatly to a diameter of 15 (radius of 7.5)? Based on my playing around with this puzzle, I wouldn’t have expected a nice answer — so that shocked me. I would have anticipated nice side lengths and an ugly diameter, or ugly side lengths and a nice diameter.

Finally: If you like puzzles like this, you might want to google “Sangaku” and look at the twitter feed of Catriona Agg.

[1] At this point, I had a small detour where I briefly tried to work this problem on a coordinate plane, where I was finding the intersection of the two lines to find the location of the center of the circle, point I, based on the coordinates of G… but when I realized that once I had the intersection point, I’d still have to find find the right coordinates for G to make the circle tangent to the edges, I realized that would be annoying. So I abandoned the coordinate plane work, though I could always return to it if I needed.


Two Problems that Got Me To Think

Here are two problems that have gotten me to think a lot.

The first one came from my Precalculus co-teacher James. We had been finishing up our unit on combinatorics and also creating new groups, and he devised a great question. So here’s the two-part problem I posed to my kids:

First Problem: We have a class of 14 students, with two groups of 3 and two groups of 4. If I were to have a computer program randomly create new groups: (a) what is the total number of different configurations/outcomes we could have? (b) what is the probability that your entire group was the exact same if you were in a 4-person group? 

I thought I solved it successfully and was feeling really confident. Then James told me I was wrong. Then I tried but didn’t understand his logic. So I made a simpler case, and then I thought I understood it. My brain hurt so much. I kept switching back and forth between a couple different answers. It was marvelous! Finally, I felt like I understood things and felt confident. I shared it with my class, and lo and behold, a couple students got what I got, and a couple students didn’t. But the students who didn’t convinced me with their logic. And then I shared their thinking with James, who didn’t have the same answer, and he too was convinced. And I thoroughly enjoyed being wrong and telling the kids that this problem messed with my head, and they helped me see the light!

The second problem came from a student who emailed me about wanting to become a better problem solver. And they shared this old entrance exam for this summer camp they were thinking of possibly applying for, and wanted some guidance. The problem that I got nerdsniped by and ended up spending hours working on over Thanksgiving break was as follows:

Second Problem:

This is from the 2019 entrance questions for a summer program. I think I was able to successfully solve (a) and (b). And then I think I solve (c) for n=3 and n=4 (and got an answer for n=5, but haven’t proved it is optimal). And I have no way to even start thinking about (d). But what I thought was lovely is how many different places my brain when went trying to think through this problem. And the neat geometric structure that arises out of the setup. (Even though I wasn’t able to fully exploit this structure in my thinking.)

I hope you enjoy thinking about these!

An excerpt from an essay

I received an email from a former student (R.L.) who I taught a few years ago. She’s a senior now in college and is taking an education class. She wrote a paper that she wanted to share with me, because half of it centered on her time in our Advanced Precalculus class during her junior year. I know I’m a warm-demander teacher (or at least that’s what I strive for). I try to make my classes a little bit harder than kids think they can do (but exactly at the level I know they can do). Reading her essay made me feel a lot of things, but I love how in so many ways she captured some of the things I strive for and do in class. The fact that she noticed them and remembered them years later means a lot.

She said I could share that part of her essay here on my blog when I asked. I like to archive good things in teaching, and this is something I’d like to archive. So here it is.


I’ve been thinking about ways that coaching, questioning, and telling played out in my education at Packer. Packer was the ideal setting for these methods of learning, as we had small, seminar-style classes and teachers with the capacity to work with students one-on-one and develop individual relationships. Upon reflection, my eleventh grade math class, Advanced Pre-Calculus with Mr. Shah, exemplified coaching, student-telling, and questioning, and also included exhibition-style projects.

This was the hardest class I took in high school. The content was very challenging, and Mr. Shah’s approach to math drove me crazy. Mr. Shah sees math as a creative field, one that demands critical thinking and a deep conceptual understanding of topics that many see as surface-level and robotic. Mr. Shah’s packets used broad questions as the benchmarks of understanding, pushing students to explain concepts using their own words. His problems had a playful tone, and we spent most of our class time working through material in small groups while he played music in the background and buzzed around answering questions and challenging students to think more deeply. Why are conics important? What is the meaning behind this geometric sequence? Why did you choose to solve this combinatorics problem this way? At the time, these questions made the class a nightmare for me – the content was already challenging enough (it was in this class that I failed the only test I’ve ever failed), and the way he forced us to think about it made it even harder. But, after reading Sizer and thinking about coaching, telling, and questioning, I see what Mr. Shah was doing.

In addition to trying to make math more fun and meaningful, he was pushing us to develop mathematical skills that built upon each other. Sizer writes that the “subject matter chosen should lead somewhere, in the eyes and mind of the student” (Sizer, 111). The curriculum progressed when we made “mathematical discoveries,” and those discoveries led us to mastery of complicated skills and a deeper understanding of concepts. Mr. Shah never talked at us. Discoveries came through telling, but, it was table-mate to table-mate telling. As the teacher, Mr. Shah’s role was to encourage our discoveries and offer support as we worked through our own questions and explained concepts to one another.

Through his approach, we were also practicing broader skills like critical thinking, creativity, perseverance, and thoughtful reflection. He centered the class around group work and we spent a significant amount of time reflecting on our individual contributions to the group and the strengths and weaknesses of our team. I honed these skills in my other classes at Packer, and I am sure they are part of my academic success at Tufts. Mr. Shah’s class was enormously challenging for me, but in writing this paper I have found an appreciation for his approach, and I know I owe him a thank-you.

Lastly, Mr. Shah also incorporated exhibition-style projects into our curriculum. He called them “Math Explorations,” and we had to do four of them throughout the year. They were not as big as these example exhibitions and they did not center around a presentation [like someone mentioned earlier in the paper], but they provided an opportunity for individual exploration in a subject area of our choosing. Being the English-lover I am, for one of my Math Explorations I wrote a series of math poems – a sonnet, some haikus, an ode, and a free-form poem. I was proud of these poems. It was exciting to take ownership in a class in which I often felt overwhelmed, and pursue something that made the content relevant to my interests. These projects were empowering, and they helped me feel connected to the material and confident in the class. If this is the power of exhibition-style learning, then I’m in full support, because the takeaways made a difference in my learning. If only more schools had teachers like Mr. Shah and the resources and the capacity to make classes like his more widely available.


Hispanic Heritage Month – Math

I am going to type a super quick post, because I need a break from thinking about school and classes, and I decided that this will be it. I realize that some of you might be celebrating Hispanic Heritage Month in your schools, and wonder how you can bring that into your math classrooms. I’m sure there are lots of ways. But I wanted to share — or remind some of you, if you forgot — of this amazing website called

It has profiles and posters of Latinx and Hispanic Mathematicians! And so many interview podcasts, where kids can hear what mathematicians do, but also hear stories that either might resonate with them because they’ve experienced similar things as the interviewee, or inform them of something they might not have experienced or even knew was an experience.

At our school, another teacher and I created a “Math Space” outside of our math office. (Read more about that here.) Since we’re now in COVID-times, we don’t have the table and stools there for students. But we still have the bulletin board. So I printed out two of the posters from the Lathisms site, and created my own poster of a passage from Evelyn Lamb about why we should care about these stories.

It’s a little barren, but this is what it looks like:

I haven’t decided what I’m going to do in my classes yet… But I am thinking of having them leave class 7 minutes early to come to the hallway, and have different students read different paragraphs aloud for the poster I made and also for one of the mathematicians. (And of course share the Lathisms site with them, and encourage them to listen to a podcast or read more.)

Wait, maybe the reading aloud in the hallway thing isn’t a good idea because with the masks it’s just hard to hear things… I might need to re-think! But now, I have to go back to planning for classes and looking at nightly work. Adios!

Mathematical Habits of Mind

I haven’t been blogging for a long time. As you can imagine, the pandemic took a toll on teachers, and at least for me and my teacher friends, we were working insane amounts of time, and it was so hard. Emotionally, physically, intellectually. At the time, I just didn’t have it in me to blog about the experience.

But now we’re about to start a new school year. And I’m vaccinated. And my students are vaccinated. And we’re wearing masks. And my classes are going to be with all my kids together in a single room [1], which is such an awesome thing compared to last year.

One of the classes I’m teaching this year is Advanced Precalculus. Another teacher, my friend James, is also teaching the same course. And he’s new to my school this year, and so when talking about the course, he shared with me how he formally incorporated Mathematical Habits of Mind in his teaching in previous years. And interestingly, last year, I toyed with the idea of formally getting kids to be metacognitive about problem solving strategies — but decided to focus on something else instead. So when James shared this idea with me, I got excited.

Right now I have an inchoate idea of how this is going to unfold. Hopefully I’ll blog about it! But for now, I wanted to share with you posters I made using James’ Mathematical Habits of Mind. Most importantly, here is a link to James’ original blogpost with his habits of mind and rubric.

Photo of the posters hung up in one of my rooms:

I know, I know, the lighting is terrible. The key words are:

Experimenter, Guesser, Conjecturer, Visualizer, Describer, Pattern Hunter, Tinkerer, Inventor

If you want these posters, the PDF file is here.

And here are all of them shared as a single sheet, and not as a poster.

Of course, if you’re a math teacher, you know there are a lot of lists of mathematical habits of mind. We agreed to use the ones James had already been using. But there are many alternative or additional things we could have included.

At the very least, I know that as we get kids to think about what strategies they’re using to solve problems, we’ll also see where there are lacuna in our curricula in terms of using those strategies. Or maybe we’ll discover it doesn’t have as much problem solving as I imagined in it. All entirely possible, since we — the kids and James and I — will all be looking through what we’re doing through our metacognitive Mathematical Habits of Mind lens.

[1] The reason I note this is because at the end of last year, I was teaching students live simultaneously in three places: they were in two different classrooms and there were a few at home on zoom. Yes, seriously. When I mention that to teachers and non-teachers alike, they asked how that was even possible. It was… a lot.

Concerns about MoMATH

I was so excited with the Museum of Mathematics opened in NYC. I attended a zillion lectures they had with math teacher friends before they had the space ready and a zillion lectures after they had their space ready. I even convinced some of my students to attend with me and another teacher, and we got to bond over math.

I encouraged some of my students to enter the contest for the Strogatz Prize for Math Communication at the end of last year. Even recently, I went to a wonderful virtual event they held called Bending the Arcwhich featured a panel of Black mathematicians and scientists and allowed participants to talk with them more intimately in smaller groups in breakout rooms. The name for the session came from Martin Luther King Jr.’s quotation which I sincerely hope is true: “The arc of the moral universe is long, but it bends toward justice.”

(The reason I mention that is because previously the museum had planned an event in honor of Martin Luther King Jr. day which somehow connected his “Letter from a Birminham Jail” to a session devoted to the Prisoner’s Dilemma problem. For so many reasons, this was a poor decision that I was surprised no one caught when setting it up.)

Also, over the years, I had heard grumblings about the museum from people who worked there or volunteered there as interns. Recently someone shared with me a letter that was sent to the Museum of Math’s Board of Directors that was concerning.

Letter to MoMath Board

My understanding was it was written a while ago, but only recently shared with the Board. I’ve been told that it is now officially okay to make this letter public. It’s short, but packs a punch. Here is the top-line conclusion, written in a signed letter by two former “Chiefs of Mathematics” at the Museum along with others who work/ed there. 

“With respect to our educational mission, race and class discrimination are embedded in the Museum’s practices.”

To me, knowing that there are multiple people — including people who were high up in the organization — who felt the need to write an open letter to describe some of their concerns speaks volumes to me. They didn’t have to. It is easier not to. It puts them at some risk, publicly speaking out. 

To me, one of the most problematic charges in the letter is that students from Title 1 schools who visit MoMATH often get lessons that end up being 20-25 minutes instead of the normal 45 minute sessions. The letter states “We cannot remain silent while the Museum chooses to offer sub par services forthe least fortunate students who are vastly more likely to be people of color.” I hope that with this letter, those at the Museum take a close look at their practices to ensure there is equity for all the students visiting the Museum. To me, more than anything else, this is of paramount importance before the museum opens its doors again. 

Also damning is this paragraph:

Unfortunately, the Museum actively discourages any form of negative feedback, and the staff has virtually no autonomy. This repressive culture has been described on social media and in the many letters you have received from former employees about mismanagement, abuse of hours, and general lack of respect for staff. In fact, the Museum’s formal Employee Policies Document warns staff members against contacting the Board. Staff members who have advocated for improvements in the Museum’s operations have seen obstacles set up in their paths and have been pressured out or fired. The rapid turnover of MoMath staff, which has an average tenure of less than a year and a half, is evidence of this. Joe Quinn, former Chief of Mathematics for MoMath, was fired shortly after expressing his opposition to the discriminatory provision of education services to Title 1 schools. We believe this to be unlawful retaliation against him

It is important for an organization which promotes diversity and inclusion to make sure that concerns can be heard safely, that feedback can be given. To these letter writers, it sounds like that hasn’t been the case and there is a culture at the museum which sounds, frankly, oppressive.

I went to GlassDoor, to see reviews of the museum from people who work there.


Most museums I looked at had star ratings in the 3+. (I recognize that those who leave reviews on GlassDoor are likely to be those who have a lot to say in either direction.) It was disheartening to read through the reviews

I only hope that this letter prompts some sort of investigation into the working conditions at the museum. How long do people work there? Why do they leave? Are employees being taken advantage of in terms of their hours worked? At the very least, this letter suggests that the Board can and should look into this. 


How to have kids set up an individual zoom meeting online for office hours using google calendar

This is going to be a really quick blogpost. On twitter, I was talking with someone about how to set up individual check in meetings with kids. I shared briefly how I had kids sign up to see me during office hours, and I thought I’d quickly write a blogpost about it to make it clearer. A lot of high school teachers at my school did this, and found it helpful. (I learned it from them.)

First, you need to install the Zoom Scheduler extension for chrome.

Then, all you do to set up appointment slots for kids is to go to the day you want to on your google calendar and create a new event. Now to make appointment slots for kids to use, it’s just this simple!


That’s literally it. You’ve done it! Once you’re done, your google calendar will look like this:


So how do students sign up for an appointment? You just send them a link to your appointment calendar, and they click on the appointment slot they want. To be clear, your appointment calendar is different from your regular calendar. The appointment calendar only lists the things that you have set up appointment slots for, not your entire calendar. (So you don’t have to worry that kids will see the happy hour that you included on your google calendar!)

To get the link to your appointment calendar to share with kids, just click the appointments you just created on your google calendar and you’ll see this:



Clicking “go to appointment page for this calendar” will lead you to a page with just appointments. The URL on that page is the link that you give to your kiddos. So they use that one link for the whole year! That’s it!!!

But if you’re like me, you want a bit more info… so…

This is what the appointment page looks like for your kiddos:


They click on the time slot they want, and it will reserve it in their name. This pops up when you click on it [note: it shows my name and not a student name because I signed up for an appointment with me! It will really show your student’s name.]


Now on the appointment page, that slot has been taken away so no other student can claim it:


When that happens, I as a teacher get an email that alerts me to the fact that a student has signed up. That email gives me the zoom link to the meeting — but I can get that zoom link when I go to my regular google calendar and click on the time slot that was taken. This is what my calendar looks like after someone signs up for a meeting:


Also, the student who signed up gets an email in their inbox with the information for the meeting (including the zoom link)… and the appointment automatically shows up on their own personal google calendar. AWESOMENESS!

Okay, that’s all!

In case this is helpful! It worked for a lot of us at my school.

PS. A couple pieces of advice…

I did this during my required “office hours” which were usually 3-4pm. I created 15 minute windows for them to meet with me. But I told kids they had to sign up before 2pm on the day of. That way I wasn’t just waiting around for them in case they signed up at 3:44pm.

Even though these were individual meetings, I figured sometimes kids might want to come in a pair or trio (if they were working on something together and got stuck). I told them one of them had to sign up but they could share the zoom link with others, so we could all meet at once! No one actually did that in the spring, but I suspect that that’s because I only mentioned it once or twice… I might make more of an effort to have kids to do that in the fall.

UPDATE: Recurring Office Hour Meetings

If you want to create Office Hours that kids can sign up for every Monday, you don’t have to do this every week. You can do this when setting up your office hours…

After entering the basic information, click “MORE OPTIONS”


You’ll “Make it a zoom meeting,” and then you’ll click on the “DOES NOT REPEAT” option to bring up the options to create a repeating meeting. I always choose Custom — because even if you want office hours weekly every Monday, if you click “Weekly on Monday,” they’re going to appear on your calendar every Monday until the end of time. :)


So then you just decide how you want the office hours to recur! (Yup, you could even say Monday, Wednesday, and Friday if you want! Or every other week!)


This screenshot above shows office hours being created every Monday but will stop showing up on the calendar on December 14th (so, for example, at the end of a semester or some reasonable date).

But WAIT Sam! What if there is a particular Monday you can’t have office hours?

Just click on the office hours on your google calendar and press the delete button (trash can). It’ll ask you if you want to delete ALL of the office hours, or just this one day of office hours. Just pick “This event”: