Author: samjshah

I am a high school math teacher in Brooklyn, New York. I enjoy getting students excited about math by being math’s loudest and most passionate cheerleader.

9th Graders Final Exam Prep / 11th Graders and College Recommendations

This is a two part post, but it’s going to be short. The first part is about final exams for freshman, and how to help them. The second part is about teaching students how to properly approach teachers for college recommendations.

First Final Exams in High School

I’m teaching freshpeople (9th graders) for the first time. And I’ve come to learn how important structure is for them. I’ve realized how useful it is to make topic lists for them (next year, I’m going to ween them off of them and show them how to create their own!). I’ve learned how important it is to be explicit with them about everything. And I’ve learned that many don’t quite know how to study.

In exactly a month, my kids are going to have their geometry final. So I whipped up a guide to explain how they might go about facing this daunting task. It’s not perfect. I hate the fact that it is so long and text heavy. But I want to get it out to my kids soon — so editing will have to wait for next year.

The truth is I don’t know if any of them are going to use it. But I’m going to at least provide them with some ideas — and maybe one or two things will resonate with them. Here it is below (and in .docx form). If you have any additional advice you give to your young ones that would go well in this, please throw them in the comments. Although I might not be able to add them for my kids this year, I can revise it for next year.

11th Graders Asking for College Recommendations

I am teaching a lot of juniors this year, which means I will be asked to write a lot of college recommendations. I never learned how to formally ask for a recommendation until I was in college — but when I was taught by a professor (who was helping guide me in the grad school application process) it was enlightening. I crafted a cover letter, got my best work together, and set up a time to meet with my professors who I was asking for aletting of recommendation from. At that meeting, I outlined why their classes were important to me, what I took away from them, and things I was proud of — and why I would really appreciate if they would be willing to take the time to do this huge thing for me. In other words, I was “pitching” this. It was thought-out, respectful, and professional.

When I first started teaching, kids would ask me for recommendations as a “by the way” in the hallway, or in a short one line email. I don’t allow for that anymore. I make sure they sit down with me and we talk through it. I ask them to fill out an extensive set of questions which often helps me frame the kids in my recommendation (if I don’t yet have a framing device in mind), and lets me learn about kids in a different way.

This year I sent an email out to my juniors, being as explicit as possible. It isn’t to make their lives harder. It is to teach them skills that are usually never explicitly taught. And all of this helps me craft a better recommendation.

Hi all,

I know it’s about the time that y’all are going to be thinking about soliciting college recommendations. If you are thinking of asking me to craft one, you should read this email. If you are certain you are not, you don’t need to read past this!

I know early in the third quarter I talked briefly about this in class, but I figured you should have it in writing too. First off, you should talk with your college counselor before approaching teachers about recommendations. They will be able to help you figure out if you’re asking the right people, who can write about the right qualities, for the colleges you are considering.

If you are going to approach me about being a recommender, there are some things you need to know. I am not a teacher who is grade-focused. I’m a teacher who values reflection, growth, hard-work, and demonstrated passion. If you’re a student who struggled but has shown a transformation in how you see and appreciate mathematics, or in your approach to effectively learning mathematics, or in how you communicate mathematics, or in your ability to work effectively and kindly in a group, or something else—all that is important to me. On the other hand, if you have done well on assessments, that is all well-and-good… but it is important that you are more than that… it is important to me that you have shown a passion to go above and beyond (inside and outside of the classroom and curriculum), or an enthusiasm for the material, or a willingness/eagerness to help others. In other words, it is important that you have thought about yourself, and can talk to me about how you are more than just grades.

That all being said, just a few reminders of what I said in class about recommendations:

· I do not write recommendations in the fall, so if you’re going to ask for one, you must ask me this year. Fall is a very busy time and is too far away; I like to have students fresh in my mind when I write. You also cannot approach me after our last day of classes (May 22).
· I never learned how to properly ask for recommendations until I was in college. So I want to help you learn that skill. (I’ve had to ask for recommendations in high school, college, grad school, and as a teacher.) If you’re going to ask me, send me an email to set up a meeting to talk formally about it. You need to plan this meeting, because you’re going to be in charge of leading it. Think about what you’re going to ask and how you’re going to pitch it.
· I said in class that you should start keeping a list in the back of your notebook of specific moments that you’re particularly proud of (large and small!), and things that you’ve done that might set you apart or make you unique or interesting! You should be sure to bring that to our meeting. If you have specific things you’ve done throughout the year that you are proud of (large or small!), you should bring those too.

As you might suspect, I write recommendations with great integrity—meaning I am honest and specific in what I write.

In the past I’ve been asked for a lot of recommendations from juniors. This year I may have to put a cap on how many I’m writing for, unfortunately, as each recommendation takes a number of hours from start to finish. After we meet, if I agree to write for you, you will be asked to fill out an extensive reflective questionnaire. I recognize that I ask a lot of students who request a recommendation, but I also know how important these recommendations are – and to do justice in the recommendation, these are important to me.

Mr. Shah

Do Kids Really Understand Trigonometry once Sine/Cosine/Tangent are Introduced?

This year I’ve been doing a lot of work with my geometry kids to get them to build up a deeply conceptual understanding of trigonometry. Right now we’re still in the part of the unit where the the terms sine/cosine/tangent haven’t been introduced, and kids are building up their understanding by thinking of ratios in specific triangles. But soon we are going to introduce the terms, and I’m afraid they are going to go to their calculator and use it blindly, and forget precisely what sine, cosine, and tangent really mean.

For my kids, at this level, I want each term to be a ratio generates a class of similar triangles — which all look the same, but have different sizes. And I want kids to conjure that up, when they think of \sin(40^o)=0.6428. But I fear that 0.6428 will stop losing meaning as a ratio of sides… that 0.6428 won’t mean anything geometric or visual to them. Why? Because the words “sine” “cosine” and “tangent” start acting as masks, and kids start thinking procedurally when using them in geometry.

So here’s the setup for what we’re going to do.

Kids are going to be placed in pairs. They are going to be given the following scorecard:


They will also be given the following sheet, with a clever title (the Platonic part refers to something we’ve talked about before… don’t worry ’bout it) (.docx form). This sheet has a bunch of right triangles, with 10, 20, 30, … , 80 degree angles.


Then with their first partner, on the front board, I project:


The kids will have 3 minutes to discuss how they’re going to figure out which two triangles/angles best “fit” these trig equations. (I’m hoping they are going to say, eventually, something like “well the hypotenuse should be about twice the length of the opposite leg, so that looks a lot like triangle C in our placemat” for the first equation.)

They write down their answers. If they finish early, I have additional review questions from the beginning of the year that will be worth some number of points — to work on individually.

When time is up, they move to a new chair (in a particular way) so that everyone has a new partner. I throw some other equations up. And have them discuss and respond. Then they move again, and have new equations up.

I’ve scaffolded the equations I’m putting up in a particular way — so I’m hoping they lead to some good discussions. And I’m hoping as soon as a few people catch onto the whole “let’s compare side lengths” approach, the switching will allow for more discussion — so soon everyone will have caught on.

At the end of the game, we’ll have some discussion, and through those discussions we’ll reveal the answers. And of course, the student with the most correct answers will win some sort of fabulous prize.

The questions I’m going to ask are here:

The discussion questions are here:


I’m super excited to try this out on my kids next week sometime.

Stuffing Sacks

Matt Enlow (math teacher in MA) posted a fascinating problem online today, one he thinks of when storing all those plastic bags from the grocery store. You shove them so they all lie in a single bag, and throw that bag under the sink. Here’s the question: how many different ways can you store these bags?

For 1 bag, there is only 1 way.
For 2 bags, there is still only 1 way.
For 3 bags, there are 2 ways.

Here is a picture for clarification:

Can you figure out how many ways for 6 bags? 13 bags?

You are now officially nerdsniped.

A number of people had trouble calculating 4 bags correctly, so I’ll post the number of ways 4 bags could be stored after the jump at the bottom, so you can at least see if you’re starting off correctly…

Additional Information: Matt and I figured the solution to this problem together on twitter. It was an interesting thing. We didn’t really “collaborate,” but we both refined some of our initial data (for 5 bags, he undercounted, and I overcounted). It seemed we were both thinking of similar things — one idea in particular which I’m not going to mention, which was the key for our solution. What blew my mind was that at the exact time Matt was tweeting me his approach that he thought led to the solution, I looked at my paper and I had the exact same thing (written down in a slightly different way). I sent him a picture of my paper and he sent me a picture of his paper, and I literally laughed out loud. We both calculated how many arrangements for 6 bags, and got the same answer. Huzzah! I will say I am fairly confident in our solution, based on some additional internet research I did after.

Obviously I’m being purposefully vague so I don’t give anything away. But have fun being nerdsniped!

Update late in the evening: It might just be Matt and my solution is wrong. In fact, I’m now more and more convinced it is. Our method works for 1, 2, 3, 4, 5, and 6 bags, but may break down at 7. It’s like this problem — deceptive! I’m fairly convinced our solution is not right, based on more things I’ve seen on the internet. But it is kinda exciting and depressing at the same time. Is there an error? Can we fix the error, if there is? WHAT WILL HAPPEN?!

The number of ways 4 bags can be stored is… (after the jump)


A Semi-Circle Conjecture

At the very start of the school year in geometry, we started by having students make observations and write down conjectures based on their observations. We had a very fruitful paper folding activity, which students — through perseverance and a lot of conversation with each other — eventually were able to explain.

However we also gave out the following:

And students made the conjecture that you will always get a right angle, no matter where you put the point. But when they tried explaining it with what they knew (remember this was on the first or second day of class), they quickly found out they had some trouble. So we had to leave our conjecture as just that… a conjecture.

However I realized that by now, students can deductively prove that conjecture in two different ways: algebraically and geometrically.


My kids have proved* that if you have two lines with opposite reciprocal slopes, the lines must be perpendicular (conjecture, proof assignment).
My kids have derived the equation for a circle from first principles.
My kids have proved the theorem that the inscribed angle in a circle has half the measure of the central angle (if both subtend the same arc) [see Problem #10]

Two Proofs of the Conjecture

Now to be completely honest, this isn’t exactly how I’d normally go about this. If I had my way, I’d give kids a giant whiteboard and tell ’em to prove the conjecture we made at the start of the year. The two problems with this are: (1) I doubt my kids would go to the algebraic proof (they avoid algebraic proofs!), and part of what I really want my kids to see is that we can get at this proof in multiple ways, and (2) I only have about 20-25 minutes to spare. We have so much we need to do!

With that in mind, I crafted the worksheet above. It’s going to be done in three parts.

Warm Up on Day 1: Students will spend 5 minutes refreshing their memory of the equation of a circle and how to derive it (page 1).

Warm Up on Day 2: Students will work in their groups for 8-10 minutes doing the geometric proof (page 2).

Warm Up on Day 3: Students will spend 5-8 minutes working on the algebraic proof (page 3). Once they get the slopes, we together will go through the algebra of showing the slopes as opposite reciprocals of each other as a class. It will be very guided instruction.

Possible follow-up assignment: Could we generalize the algebraic proof to a circle centered at the origin with any radius? What about radius 3? What about radius R? Work out the algebra confirming the our proof still holds.

Special Note:

Once we prove the Pythagorean theorem (right now we’re letting kids use it because they’ve learned it before… but we wanted to hold off on proving it) and the converse, we can use the converse to have a third proof that we have a right angle. We can show (algebraically) that the square of one side length (the diameter of the semi-circle) has the same value as the sum of the squares of the other two sides lengths of the triangle. Thus, we must have a right angle opposite the diameter!

I’m sure there are a zillion other ways to prove it. I’m just excited to have my kids see that something that was so simply observed but was impossible to explain at the start of the year can yield its mysteries based on what they know now.

The two semi-circle conjecture documents in .docx form: 2014-09-15 A Conjecture about Semicircles 2015-03-30 A Conjecture about Semicircles, Part II

*Well, okay, maybe not proved, since they worked it out for only one specific case… But this was at the start of the year, and their argument was generalizable.

Add yourself to the MTBoS Directory!

Jed Butler (@mathbutler, blog), in the past week or two, has worked to create a beautiful directory for math teachers who use twitter and who blog. We have had a few spreadsheets out there trying to do the same thing, but they tend to get outdated and lost. This directory is the real deal.

The point of this post is to get you to add yourself to the directory. If you’re already convinced, do it now. If not, read on to why you ought to…


It not only is beautiful, simple, and sleek, but it has the following features which blew me away:

(1) For each person, it creates a little index-card-like profile, which not only has our twitter picture on it but also has links with our interests. I confuse people easily (and really, why are 30% of math teachers named Chris?), and having a little picture icon, and all of their information easy for me to look at is going to be so so so helpful.


(2) It has a map which each person in the directory can easily add themselves to, and this map is searchable. I can, for example, zoom into NYC to see who the NYC educators are… or type my friend’s name into the search bar to remind myself which part of the country (world!) they are in.


(3) The directory itself is crazy searchable. Say you wanted to find teachers who have been teaching since 2000 who are in the Northeast US who teach Geometry and are interested in Groupwork. Done.


(4) If you want to quickly update your information, you can… no muss no fuss it is super easy!

Which is all to say: take 5 minutes and add yourself to the directory.

My Introduction to Trigonometry Unit for Geometry

I’ve been mulling over how to introduce trigonometry to my geometry students. I think I’ve finally figured out a way that is going to be conceptually deep, and will have kids see the need for the ratios.

I don’t know if all of what I’m about to throw down here will make sense upon first glance or by skimming. I have a feeling that the flow of the unit, and where each key moment of understanding lies, all comes from actually working through the problems.

But yeah, here’s the general flow of things:page

Kids see that all right triangles in the world can be categorized into certain similarity classes… like a right triangle with a 32 degree angle are similar to any other right triangle with a 32 degree angle. So we can exploit that by having a book which provides us with all right triangles with various angle measures and side lengths. (A page from this book is copied on the right.) Using similarity and this book of triangles, we can answer two key questions. (1) Given an angle and a side length of a right triangle, we can find all the other side lengths. (2) Given two side lengths of a right triangle, we can find an angle.

By answering these questions (especially the second question), kids start to see how important ratios of sides are. So we convert our book of right triangles into a table of ratios of sides of right triangles. Students then solve the same problems they previously solved with the book of triangles, but using this table of values.

Finally, students are given names for these ratios — sine, cosine, and tangent. And they learn that their calculator has these table of ratios built into it. And so they can use their calculator to quickly look up what they need in the table, without having the table in front of them. Huzzah! And again, students solve the same problems they previously solved with the book of triangles and the table of values, but with their calculators.

Hopefully throughout the entire process, they are understanding the geometric understanding to trigonometry.

(My documents in .docx form are here: 2015-04-xx Similar Right Triangles 1 … 2015-04-xx Similar Right Triangles 2 … 2015-04-xx Similar Right Triangles 2.5 Do Now … 2015-04-xx Similar Right Triangles 3 … 2015-04-xx Similar Right Triangles 4)

It’s a long post, so there’s much more below the jump…


Time Travel

This quarter, I’m letting kids — totally optionally — do a more in depth Explore Math. This time it isn’t getting a “taste” of a bunch of little things, but rather it’s explore one thing in detail. Anything math related that kids are interested.

Today and yesterday, I had three different meetings with a few different kids who wanted to discuss options. These conversations revolved around:

  • Park Effects on baseball batters (sports statistics)
  • Understanding why a particular algorithm creates the math art pictures it does
  • The Goldbach Conjecture and the Collatz Conjecture
  • Time dilation (and time travel)
  • How restaurants do their finances and stay in business

Super fun conversations, with kids who just want to learn stuff that they’re fascinated by. For example, the kid who wanted to talk about the Goldbach conjecture said that he wanted to work on proving the Goldbach conjecture (“I will not give up!” he wrote) — and the reason he wanted to do this is because he always had trouble with prime numbers and understanding them. Melting! MELTING!

Previous posts about Explore Math:

The site that launched Explore Math (mini explorations) last quarter for my kids: