Click on the image to go to the journal and see the cool math and science things kids at my school are working on!

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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.*

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.

Always,

Mr. Shah

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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 . 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:

Fin.

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

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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)

4. I promise you this is right, so if you only have 3, you’re missing one!

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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.

**Background:**

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.

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**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**.

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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:

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…

**Similar Right Triangles#1: The Forward Question **

I am going to begin by having students start seeing every right triangle they see as similar (or, to be technical, close to similar since these only have integer angle measures) to one of the right triangles in this book, titled the Platonic Right Triangles book. (If you want to know where Plato comes in, check out the packet…)

That’s how we will start. So they are going to use the Platonic Right Triangles book along with what they have learned about similarity in order to find missing side lengths of triangles. At the end of this, students are going to measure the height of the organ in the chapel in our school, using what they have learned.

The big idea of this first packet is getting kids to see that all right triangles are similar to this “Platonic” set of right triangles, and given a right triangle with a particular angle, they can use that fact to find missing sides of triangles with very little effort, by setting up a proportion.

**Similar Right Triangles#2: The Backwards Question **

This is the backwards question: Given a triangle with some side lengths, can they find the missing angle? At this point, all they have is their Platonic Right Triangles book. Mainly, the first few pages of this packet has kids try to figure out how they can use the book to find out the missing angle. They’re going to struggle at first, and have various approaches to it, some of which will be more efficient than others. But I see the first four pages of this packet, their struggle, and most importantly *the group and class conversations we have based on their approaches,* to be the key thing here.

Hopefully, at this point, we’ll converge upon the idea that the ratio of sides is a super awesome way to find the missing angle. If they are given the side opposite the angle and the hypotenuse, life is easy. If they are given the side adjacent the angle and the hypotenuse, life is easy. (This is because the Platonic Right Triangles book gives all the triangles with hypotenuse of 1.) But if they are given the side opposite the angle and the side adjacent to the angle, life is hard. And that’s okay. But it would be so much easier if for each of the triangles in our Platonic Right Triangles book, we had a list of all the ratios of the opposite side to the adjacent side.

The important of these ratios gets highlighted with some simple work with this simple Geogebra applet.

From here, we collapse our Platonic Right Triangles book into a simple Table of Right Triangle Ratios. An eighty-nine page book that can be represented in a two page table.

Students hopefully will finish this worksheet seeing the usefulness of ratios in a right triangles, and being able to use a table of values to help them find missing sides and angles.

**Similar Right Triangles #2.5: Why Three Ratios?**

This is a classroom opener. It came about when my co-teacher recognized that we never actually need all three ratios… the only reason we have all three ratios is for *convenience*. And so I whipped this up to get kids to recognize that fact.

**Similar Right Triangles #3: Naming the Three Ratios**

In this packet, students finally get names for the three ratios we’ve been using (sine, cosine, tangent), and hopefully the questions asked will require them to understand they are basically doing what they’ve been doing — but with fancy names. Specifically, I want students to leave understanding that *all a calculator is doing when calculating the sine/cosine/tangent of an angle is that it is going to a super extensive internal table of values and finding the appropriate ratio of sides for that right triangle*. And that *all a calculator is doing when calculating the inverse sine/cosine/tangent of a ratio of sides is that it is going to a super extensive internal table of values and finding the appropriate angle for that right triangle*.

Then there are some challenging conceptual questions that students will hopefully be able to answer, which require them to understand the geometric understanding of the trigonometric functions.

**Similar Right Triangles #4: Special Right Triangles**

I don’t think that special right triangles are all that important, but I know that I ought to expose my kids to them. So I made this…

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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:

http://samjshah.com/2014/02/12/explore-mathematics/

http://samjshah.com/2014/04/11/explore-math-reprise/

http://samjshah.com/2014/04/25/explore-mathematics-part-ii/

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

http://explore-math.weebly.com/

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*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.

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.

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**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):

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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):

“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:

Fin.

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Did you watch it? No? Seriously, watch it!

Okay, good. Ever since I saw this video, I wanted to try it in my class. I wanted to have my kids guess the rule. I wanted to be the one saying “yes” and “no” to their three numbers. And today, finally, I did.

In geometry, we’re starting a short excursion into proof and reasoning. Yes, we’ve done a few proofs. And my kids are learning to justify their ideas. But we’re about to embark on a few days where they think about proof, the importance of proofs, assumptions, and other such things. And in our next class, we’re going to start talking about induction and deduction. So today, this was a perfect warm up.

I gave each group a whiteboard. They threw up three numbers. I said yes or no. After they got 3 yes/nos, they were allowed to guess the rule. Then they did it again. Some groups were looking to get the rule in the least amount of guesses. Others were guessing willy nilly. It definitely took longer than I thought. No group got it, but they were making interesting choices with their three numbers in order to figure out the rule.

After about 7-8 minutes of this, I stopped them. I started playing the Veritasium video. It was awesome. Why? Because the random beach people that Veritasium interviewed gave almost identical answers as my kids gave! They saw that the way they were approaching the problem was the same. They heard a few more sets of three numbers that worked/didn’t work, and then I paused the video. Why? Because I heard kids whispering and murmuring that they think they had it. I gave those kids an opportunity to share what they thought the rule was (I did not confirm or deny their guesses). And then I finished the video.

I loved doing this because the kids were totally engaged. And when we start talking about induction and deduction, counterexamples, and keeping an open mind when problem solving, we can use this exercise as an activity we can refer back to.

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I told the groups to mix themselves up — for a change of pace. And then I handed this out.

I told each group that they had *one* opportunity to call me over so I could give them A Big Hint. Then I let them go, giving them giant whiteboards to work on if they wanted.

In my two classes, I had: one group solve the problem explicitly (they had a formula that worked) and one group come up with a recursive solution that really impressed me! The other eight groups were at varying stages of understanding. Most all the groups were gung ho about working, and most all the groups started discovering all these patterns.

Only one of the ten groups asked for The Big Hint, which means my kids have perseverance! I did give varying degrees of mini-hints to kids as I saw them progress, to nudge them this way or that way.

At the end of the second class, a few kids said how they are now braindead, because they did so much thinking. They were exhausted. As a teacher, I call that winning!

I’m still at a bit of a loss as to what I am going to do tomorrow. Since kids hadn’t really finished, I thought I would have them work a bit more. I’m not good at debriefing. Also, many kids have different observations, and I don’t have time to really dwell on this. This wasn’t supposed to even take the whole day!

My plan is to give all my groups collectively A Big Hint, give each group 12 minutes to figure out how to find the *n*th pentagonal number, and then after those 12 minutes are up, I will give them this:

We will go over the triangular and square numbers together as a class. And then WHAM, I leave them with the pentagonal and p*-gon* figures to figure out on their own.

Wish me luck that it goes well tomorrow. I can see it crashing and burning, or being a good wrapup.

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Imagine you’re a geometry teacher, and you want students to **discover** a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to **understand (deeply)** and **articulate why** this method gives you *the* center of rotation.

Although not perfect, I have whipped some stuff up in the past hour that I hope will get to the heart of this question. Of course in my head, I have class discussions, and we gently get at this. These sheets alone don’t get us there. But if you’re interested, feel free check ’em out.

(Also, since I just whipped them up, there might be some things that need fixing/tweaking…)

There are five of them (all combined).

- Rotations of two points [after this, through discussion, get student to make the connection between
*the perpendicular bisector as all points equidistant from the endpoints of a segment*and*the radii of a circle*] - Rotation of a line segment [after this, through discussion, get students to recognize that they are really considering
*two*perpendicular bisectors… we are looking at one perpendicular bisector to find all possible points which will rotate one end of a line segment to the new point*and*a second perpendicular bisector to find all possible points which will rotate the other end of a line segment to the new point… for both endpoints to simultaneously rotate to their new location, we have to look for the intersection of the perpendicular bisectors!] - Rotation of three points [after this, make it clear that this is
*not*the same as saying that you can trace a triangle on patty paper in two different places and find a center of rotation that will bring the first triangle to the second triangle… in fact, maybe I should have this as an exercise…] - Center of rotation practice
- Rotations of a complex figure

(.docx)

PS. If you’re talking about multiple “center of rotation”s, do you say “centers of rotation” or “center of rotations” or “centers of rotations”? It makes me think of culs de sac, which indeed is the plural of cul de sac. Thank you *Gilmore Girls*.

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Setup: We’ve been talking about *perpendicular bisectors *in various contexts in geometry. But they were just making observations and working on some simple proofs.

Last night in Geometry, students were tasked with the following:

It turns out that #3 is impossible, and #4 is possible with some guess and check. This sets up the background to have kids see something neat.

Then they are asked:

And now they see that for a triangle, the perpendicular bisectors of the sides all meet at a point. And that *is* rare and weird. They then were asked to look at the point that the perpendicular bisectors meet at and the vertices of the triangles and make a conjecture.

Only one student “saw” it. It was fascinating for me that it was so hard for everyone else to see it! Others had conjectures that might have been true for right triangles or isosceles triangles or equilateral triangles… but not that were universally true.

For the rest of the class, to get them there, I did the following:

This was a huge setup for my “one good thing.” There were gasps, and one student said, and I quite, “MIND BLOWN.”

This weekend they are going to try to figure out what the what is up?!

PS. Yes, I am fairly certain that the setup of having students see the rarity of perpendicular bisectors meeting at a point, as well as having them look and fail to see something inside a set of seemingly random points was crucial for the big reveal. In fact, the fact that they *didn’t *discover it on it’s own was so powerful when they ended up seeing it.

**UPDATE:** The file I used is here.

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