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:

Two Organizational Things I Do

I don’t know if I’ve blogged about these things before. These aren’t Two Classroom Ideas That Will Completely Change Your Teaching or anything. In fact, I’m willing to bet that many of you have tried or currently do something similar. But for me, these two things have made my life easier and my classroom run more smoothly. So in case this helps…

The First Idea

In Geometry, I want my kids to learn to use multiple tools, and find the tools that are the most useful to them at any given moment. One moment they might need patty paper to trace something. Another moment they might need eraseable (this is key!) colored pencils to emphasize different things. Another moment they might want to pull up Geogebra on their laptops. And another moment, they might need a ruler to draw a straight line. Who knows. So what I did at the start of the year was create geometry buckets, populated with the tools that each group might need at any given time.


I have a different bucket for each group. I color coded most of the items in the bucket (with the exception of the protractors, because I didn’t want to cover any of the angles!). I store the buckets in the room. At any point, kids are allowed to grab them. Sometimes they have to, because they are asked to measure an angle, or draw a circle. When I have them use the giant whiteboards, they have their dry erase markers and an eraser in the buckets. But most of the time, when a kid needs some patty paper, or a ruler to make a diagram, or colored pencils to organize their ideas or annotate a diagram, they’ll just grab the bucket and bring it to their groups. And at the end of each class period, the kids will just put them back.

I thought things would get lost or mixed up. But it’s been a semester, and I just went through the buckets and have found only a few colored pencils were in the wrong boxes, and only a single compass migrated from one box to another. I love these buckets of geometry tools!

The Second Idea

I do tons of groupwork in my classes. And I try to switch up groups often enough for some spice, but let them work together long enough so they can learn to work together (I try to do it two times a quarter). However, when kids are in groups, passing things out and collecting things can be annoyingly time consuming. And if my kids know one thing about me as a teacher: I don’t waste time, not a second.

So here they are: something I’ve been doing for the past few years. Folders. Specifically, each group gets one folder.

20150226_175913 (1)

On the front (not photographed), I have a label with the kids’s names on it. Inside are two pockets. The left hand pocket is for things I normally would hand out. (Mainly: the packets that I make for kids to work collaboratively through.) The right hand side has two purposes: (1) I have kids turn in nightly work sometimes, so they will put it in there, and (2) when I mark up the nightly work, I put it back in there and students collect it the next day.* There are also some “The Dog Ate My Homework” forms for when a kid doesn’t turn in their work. Instead of them calling me over and giving me a story explaining what happened, I just have them fill out that form saying why they didn’t have their work.

One huge benefit for having these folders is that it allows me to mix up where the groups sit each day.** When I walk into the classroom, kids aren’t sitting down usually. They are waiting for me. I throw down each folder on group of desks, and then kids sit at the group of desks with their folder on it. That way: kids are in different locations each day, mixing things up. The group in back won’t always be in back! Sometimes I give a kid the folders to put down, and sometimes the power, the sheer power of who sits where, goes to their head. (“Oh, you’re standing by this group of desks? Too bad, I’m putting your folder waaaay over at that far group of desks.”) Fun times.

Now you might say: each day you have to put in the packets you’re going to hand out the next day? Nope.Well, sometimes. But usually not. In classes I’ve taught before, where I have my ducks in a row, I do a massive photocopying of the papers for the entire unit. I lay them out, and fill up the folders. Then I’m pretty much set for a week or two (or more!). Below is a picture of me doing that today!


That is all. Go back to your regularly scheduled lives now.

UPDATE: I forgot to say: I color code the folders for each class. So red folders = my geometry class, blue folders = one of my precalc classes, green folders = the other one of my precalc classes. I also use a lot of file folders to organize things for me. And for those, I use the same color folders for each of my classes. So, for example, when I give a geometry test, I bring a red file folder to class. And then I keep the taken geometry tests in that file — and when I’m going home, I just throw that red file in my backpack so I can mark ’em up.

*The fact that there is one folder per group also has the added bonus that when one kid forgets to put their name on their nightly work, you know exactly whose it is, because it is in the folder for that specific group (and usually all the other kids put their name on it).

**Someone, somewhere, told me that there was some ed research that suggested that kids sitting in the same spot every day helped them learn better. I have my doubts about that.

Angle Bisectors of a Triangle and the Incenter (and lots and lots of Salt)

In Geometry, we’re about to embark on the whole triangle congruence bit (SSS, SAS, ASA, etc.).  Below is the plan we have outlined.

The TL;DR version: By learning about angle bisectors, we motivate the need for triangle congruence. We have students figure out when they have enough information to show two triangles are congruent. They use this newfound knowledge of triangle congruence to prove basic things they know are true about quadrilaterals, but have yet to prove that they are true. Finally, we return back to angle bisectors, and show that for any triangle, the three angle bisectors always meet at a point. As a cherry on the top of the cake, we do an activity involving salt to illustrate this point.


STEP 1: To motivate the need to figure out when we can say two triangles are congruent when we have limited information, we are showing a problem where triangle congruence is necessary to make an obvious conclusion. 

We’re going to see the need for triangle congruence to show that:

any point that is equidistant to an angle (more precisely: the two rays that form an angle) lies on the angle bisector of that angle.

This will be an obvious fact for students once they create a few examples. But when they try to prove it deductively, they’ll hit a snag. They’ll get to the figure on the left, below. But in order to show the congruent angles, they’ll really want to say “the two triangles are congruent.” But they don’t have any rationale to make that conclusion.

Hence: our investigation in what we need in order to conclude two triangles are congruent.

When kids do this, they will also be asked to draw a bunch of circles tangent to the angles. All these centers are on the angle bisector of the angle. This will come up again at the end of our unit.


Kids will be doing all of this introductory material on this packet (.docx)

STEP 2: Students discover what is necessary to state triangle congruence.

This is a pretty traditional introduction to triangle congruence. Students have to figure out if they can draw only one triangle with given information — or multiple triangles.

We’re going to pull this together as a class, and talk about why ASA and SAS and SSS must yield triangle congruence — and we’ll do this when we talk about how we construct these triangles. When you have ASA, SAS, SSS, you are forced to have only a single triangle.

I anticipate drawing the triangles in groups, and pulling all this information together as a class, is going to be conversation rich.

The pages we’re going to be using are below (.docx) [slight error: in #4, the triangle has lengths of 5 cms, 6 cms, and 7 cms]

STEP 3: Once we have triangle congruence, we’re going to use triangle congruence to prove all sorts of properties of quadrilaterals.

Specifically, they are going to draw in diagonals in various quadrilaterals, which will create lots of different congruent triangles. From this, they will be asked to determine:

(a) Can they say anything about the relationship between one diagonal and the other diagonal (e.g. the diagonals bisect each other; the diagonals always meet at right angles)

(b) Can they say anything about the relationship between the diagonals and the quadrilateral (e.g. one diagonal bisects the two angles)

(c) Can they conclude anything about the quadrilateral itself (e.g. the opposite sides are congruent; opposite angles are congruent)

We have a few ideas percolating about how to have students investigate and present their findings, but nothing ready to share yet. The best idea we have right now is to have students use color to illustrate their conclusions visually, like this:


STEP 4: This is a throwback to the very start of the unit. Students will prove that in any triangle, the angle bisectors will always meet at a single point.

Here are the guiding questions (.docx)

And we will finish this off by highlighting the circles we drew at the start of the unit. Notice we have a single circle that is tangent to all three sides of the triangle. The center of that circle? Where the angle bisectors meet. Why? You just proved it! That location is equidistant to all three sides of the triangle.


STEP 5: The reason we’re highlighting these circles is that we’re going to be cutting out various triangles (and other geometric shapes) out of cardboard, elevate them, and then pour salt on them. Ridges will form. These ridges will be angle bisectors. Why? Because each time you pour salt on something flat, it forms a cone. The top of the cone will the the center of the circle. We’re just superimposing a whole bunch of salt cones together to form the ridges.

The other geometry teacher and I both saw this salt activity at the Exeter conference years ago. Here are some images from a short paper from Troy Stein (who is awesome) on this:

The general idea for this activity is going to be: kids take a guess as to where the ridges are going to be, kids pour the salt and see where the ridges are.

As the figures get more complex, they should start thinking more deeply. For example, in the quadrilateral figure above, why do you get that long ridge in the middle?

My hope is that they start to visualize the figures that they are pouring salt on as filled with little cones, like this:


The material I’ve whipped up for this is here (.docx):

Polygonal Crystals

In Geometry, we’ve been trying to turn the course on its head. Recently, we’ve been working on reasoning and proof. One thing students weren’t able to prove was that the sum of the interior angles of a triangle always add up to 180 degrees. (That’s because we haven’t gone over anything involving parallel lines.)

They even blew up balloons and drew triangles with two right angles on them (using protractors and rulers).



[Note to self: students cannot tie balloons. Also, they will be found scattered around the student center later in the day if you let kids keep their balloons.]

But we said: assuming that you know that on a plane the sum of the interior angles of a triangle add up to 180 degrees, can you prove that quadrilaterals have a sum of 360 degrees for their interior angles.

And each group was able to latch onto the idea of dissection without me saying anything… breaking the quadrilateral into two triangles.

But then… then… they started to say something that scared me. They said “there are two triangles, and since each triangle is 180 degrees, the quadrilateral is 360 degrees.”

To you non-geometry teachers, this might not seem problematic. But I immediately thought: “oh gosh, these kids think of triangles and quadrilaterals and the like as having some inherent property that can be added to others. They aren’t saying the sum of the interior angles of the triangle is 180 degrees… they are saying the triangle has 180 degrees.

So I gave them a follow-up question (which I’m proud of):

triangle“The Blue Triangle is 180 degrees. The Pink Triangle has 180 degrees. So the Giant Triangle (the blue and pink triangles combined) must have 360 degrees. How is this possible? Did we just break math?”

One group had someone who figured it out right away, but the others took a good amount of time trying to figure out where this argument failed. I loved it because it really showed them a misconception they had.

It was the perfect question, because over the summer the other geometry teacher and I came up with the following (which we are in love with) involving triangle dissection:


Finally, to check if each group understood where these came from, we had them write a “triangle dissection expression” for the sum of the interior angles of this pentagon: