Today was a pretty blah day until my last period class. My first three classes all had assessments so I had no fun conversations and I watched work pile up. As I came in to my last class of the day – my Geometry class – one of my Geometry teammates was waiting in my room to share that his students had been making some great strides in GeoGebra. He told me that a number of his students were really beginning to dig into what GeoGebra could do for them, especially now that we are talking about transformations. I used Geogebra extensively when writing my text and I borrowed from resources around the web for activities. One of them was an activity called A-Maze-Ing Vectors which had been created by the amazing Jennifer Silverman (@jensilvermath) and we used that activity the past two years. My teammate who had been waiting to share his good news had asked me this past summer about modifying this activity. We had had trouble completing the activity in one day and it did not take up enough for two solid days. He also had an idea about combining vector transformations on objects more complex than points. He created a pretty wonderful adaptation of the activity (you can find it here) and my students worked through it yesterday. I opened class today by projecting the last page on my AppleTV where we had to navigate a triangle through a maze and I invited a student to come up and draw on the TV (with a dry erase marker, don’t worry!) and I cannot tell you how great the conversation was in class. I sat down – a commitment of mine based on my #TalkLessAM session at TMC16 – and just watched the fireworks unfold. Kids were challenging each other, going up to the TV to draw their ideas, debating distances, talking about slope, worrying about vertices colliding with walls and discussing the option of rotating the triangle as it moved. I was SO thrilled with the engagement and the level of conversation. I credit this to a number of factors. The original activity was terrific and my colleague’s rewriting of it is creative and concise. Kids like drawing on a TV – it feels naughty or something. I sat down and got out of the way. Kids had worked this through the day before in their table groups and were invested in both supporting their teammates and making sure that their memory and their perspective was clearly heard. They were supportive of each other and slightly defensive if someone else had a different approach. After a pretty uneventful day at the end of the week it would have been easy to just limp tot he end of the day, but these kids brought each other to the finish line for the week sprinting. I am optimistic that we can pick up with a similar level of energy on Monday.

# Tag: Geometry

## The Language That We Use

I recently engaged in a spirited discussion prompted by Patrick Honner (@mrhonner) on twitter and on his blog. The original post that started this whole discussion can be found here and it is well worth your time. Engaging comments there an on the twitters and a friendly suggestion by Patrick himself has me writing here, thinking out loud. To set the stage for this post, an image from Patrick’s post is important.

A quick glance at this certainly suggests that these are congruent figures until you look more carefully at how the question is worded. This is a pretty classic example of the kind of question that makes students think that test writers are gaming the system to catch them in a mistake. We are looking at two figures that are equivalent to each other. A rigid transformation maps one onto the other. However, that mapping does not map them in the order suggested. A classic mistake that I lost points for as a student and one that, sadly, I admit that I have probably deducted points for when grading. The debate on the blog and on twitter raised some really challenging questions about our goals with this type of specificity. Yes, mathematics is a precise language and precision is a powerful habit to try to help develop. However, I keep thinking about my fun Geometry class from last year. When we were discussing how to determine whether a triangle with given side lengths was acute, right, or obtuse we worked out a strategy where we assumed that the Pythagorean Theorem would hold and we decided what the consequence was when it did not. This led to my students saying things like this; “If the hypotenuse is bigger than we thought it would be, then the triangle is obtuse.” Now, I know that the largest side of an obtuse triangle is not called the hypotenuse. When pressed on the issue I suspect that *almost* all of my students knew this as well. Optimistically, I want to say that they **know** this as well, but is is early August… My concern here is that I was letting them down by letting them be a bit lazy with their language. What I did at the time was to gently remind them that hypotenuse was not the best word to use there but I understood what they meant when they said it. Should I have made a bigger deal about this at the time? Was I being understanding and flexible? Was I being undisciplined and imprecise? I suspect that there is a decent amount of both of these in my actions and I have to admit that I did not think too deeply about it at the time. In the wake of the conversation that Patrick moderated, I am thinking deeply about it. It is also early August (again, I note this) and it is the time of year that my brain reflexively starts dwelling on teaching again. I am also thinking about a distinction that I got dinged for as a student but this time it is one that I do not ding my students for. I remember losing points in proofs if I jumped from saying that if two segments were each the same length then they are congruent. This is, obviously, true but I was expected to take a pit stop by making two statements along the way instead of jumping straight to congruence. I know that equivalence of measure and congruence of segments (or the same argument with angles) are slightly different meanings. A nice explanation is here at the Math Forum. But I feel pretty strongly that my 9th and 10th grade Geometry students are not tuned in to the subtle differences and I think I am prepared to defend my point of view that they do not need to be. I want my students to be able to think out loud and I DO want them to be careful and precise in their use of language but I do not want them to think that this is some sort of ‘gotcha’ game where I am looking for mistakes and looking for reasons to penalize them.

I am thankful to Patrick for getting this conversation started and for gently nudging me to try and work out my thoughts more thoroughly on this issue. I am interested in hearing from other teachers – particularly Geometry teachers – on how they try to navigate these conversations. How precise should our high school students, especially freshmen and sophomores, be when discussing these issues?

As always, feel free to jump in on the comments section or reach out to me through twitter where I am @mrdardy

## How to Succeed

Feedback from my students at the end of the year touched, in part, on the idea that many of my students take some time to adjust to my expectations in our course. Years ago, I wrote a document called *How to Succeed in Calculus*. This was adapted from a document I found online by a teacher I never met named Dave Slomer. I have modified that document for my Geometry class and I want to share my first draft here. I shared it with my Geometry team and we have a nice conversation started about how to introduce and integrate this document. The first reaction from one of my colleagues is that the document might be a tad too long and students might easily put it aside. I agree that it is a bit wordy but I also feel that there is not much that I want to cut out. I would love any constructive feedback either here or through my Twitter account over @mrdardy

Here is my first draft –

How to succeed in Geometry

KEEP UP WITH THE ASSIGNMENTS.

Over the years, I have found that the best indicator of a student’s success is whether they keep up with their assignments. Students who keep up will likely do well – students who don’t likely won’t. We will be together for a good amount of time this year and we will routinely refer back to ideas and skills that we have discussed together. If you do not keep up with your assignments then it will become increasingly difficult for you to master new skills.

REMEMBER THAT THE GOAL OF AN ASSIGNMENT IS TO UNDERSTAND THE MATERIAL – NOT JUST GET THE PROBLEMS DONE.

You understand the material best when you can do the problems – and get them right – BY YOURSELF. There is absolutely nothing wrong with asking questions or seeking help from me, from other teachers, or from your fellow students. Everyone will need help sooner or later in this course. However, you must have the integrity to realize that the goal of the assignment is NOT just to get the assigned problems done. When we write our problem sets we are aiming to make sure that there is sufficient practice for all of our students. However, there will be times when you will need more practice than this, and you must have the courage and integrity to realize it. When you ask for extra practice, we can provide you with assignments that will help you to master new skills.

TREAT ASSIGNMENTS AS “PRACTICE TESTS”.

If you take your homework problem sets seriously, if you spend time thinking and working through the problems we present to you, you will feel more prepared for tests and quizzes than if you do not. Hard work spent on daily practice pays off on test days. Athletes who take practice seriously are better prepared for game days. Musicians and actors who take rehearsals seriously are better prepared for performances. Students who take daily practice seriously are better prepared for assessments. We know this to be true.

USE EXTRA PAPER

Your problem sets have narrow spaces available. Do not try to squeeze all of your work in these spaces. It is unlikely that you will be able to read your own work when you look back at your work and it is very unlikely that I’ll be able to clearly see your work and understand your reasoning. Do your work on notebook or blank paper and give yourself space to draw and to think.

NEVER ERASE.

If you hit a “dead end” and want to start over, cross out the work you don’t want with a big “X” – do NOT erase it. It might turn out later to be correct. Also, if you come to me for help, the first thing that I will say is “Let me see what you have done so far.” If you tell me that you erased it, it will be much harder for me to help you. Erasing can be a big time-waster on tests (where time is very valuable).

READ THE BOOK.

This is important in every class, but in this class the text serves as a valuable supplement to what happens in class. Often your homework will be to read the book in addition to any of the problem sets that we have written. Read the book carefully with a pencil and paper nearby. Pay particular attention to the illustrations and examples. Study the examples carefully. All of you have access to a PDF of the text and some of you will also have opted to have a physical copy of the text as well. Use your physical copy, if you have one, for margin notes. Use your PDF regularly to follow hyperlinks to explanations and activities that have been built in to your text. These are valuable resources and we expect that you will attend to them when you are asked to read.

LEARN THE VOCABULARY AND SYMBOLS.

It is vitally important that we can communicate in the language of mathematics. As you read or participate in class, pay particular attention to the meaning of each new term and symbol. This is a course that is heavy on vocabulary, you need to spend time and energy on this aspect of your study of Geometry.

REVIEW REGULARLY.

Luckily for you, tests are cumulative, and we will review in class; therefore review is somewhat automatic. Don’t hesitate to go back to review or seek help on algebra skills or on earlier ideas from this course that you may not have mastered as well as you wanted to.

TAKE GOOD NOTES DURING EACH CLASS.

Good notes are essential for success in any technical field. They are essential for review – not only for tests, but also for the problems you will work that evening. It is far too tempting to sit and listen and watch during class. You may feel comfortable at times following our conversations this way. However, when you sit down at night to do your homework, you will be without a valuable resource and you may not remember well what the conversation was hours ago when we were together in class. Every study of learning that has ever been done suggests that the act of writing something down helps in strengthening our memory. It is my expectation that each of you will come prepared each day to take notes on our class conversations.

CLASS TIME IS VALUABLE.

You need to use the time at the beginning of class to get ready for geometry. Get out your books, assignments, notebooks, pencils, etc. I will usually have a question on the board or the TV monitor when you arrive in class. Get to work on that and get your mind in its math mode. Socializing may be more pleasant than math, but the goal is to make math more pleasant, and socializing often gets in the way. At the end of the discussion period, begin (or continue) the current assignment right away – what better time to get help if you get stuck? We only spend valuable class time on important topics, so take good notes constantly during class.

ORGANIZE.

Your success depends on your ability to recall (or find, relearn, and then remember) concepts and techniques that were introduced earlier. If your notes and assignments are scattered about, folded inside the covers of your book, papering the bottom of your locker or the floor of your bedroom, you’re sunk.

BECOME AS SELF-SUFFICIENT AS POSSIBLE.

There are many students, and just one teacher, and time is too valuable for you to just wait – stuck in neutral – for help. Look in your text and your notes for sample problems that might shed some light on your difficulty. Learn tenacity – don’t just “fold” at the first sign of difficulty. Is there another way to approach the problem? You can do it.

SEEK HELP AGGRESSIVELY.

Everyone, no matter how smart or proficient in math, will get stuck sometime this year. Perhaps there is a new concept or technique that just won’t fit into place in your brain. Tenacity and self-sufficiency are great attributes, but sometimes there is going to be a quiz on this stuff tomorrow. Sometimes there just isn’t time to be tenacious. Attend conference bells, ask questions in class, just be sure to get the help you need to succeed.

COMMUNICATE.

If you have a worry, complaint, suggestion, or concern of any kind let me know. I can’t fix it if I don’t know about it. Remember that just because a problem – or a solution – seems obvious to you, it may not be obvious to everyone. Speak up.

NOW, A PERSONAL NOTE…

There are some things I do in class that you may find unorthodox. If we understand each other early in the year, we’ll avoid a lot of stress later in the year. There are mathematical facts that I expect you to know and I will remind you that you should know them. There are times when you will ask a question and I may reply with a question. Or, I may redirect the question to someone else in class. This is not done to avoid answering a question, it is done to encourage a thoughtful discussion and to help you to develop important problem solving skills. I believe strongly that we understand ideas more deeply if we can explain our own thoughts to others. For this reason, we sit in groups facing each other, rather than having everyone face me or face the board. I expect that you will explain ideas to each other and that you will ask each other questions. Questions in this class will ALWAYS be answered; you may just have to be patient before the answer arrives.

A quote by Galileo Galilei

“Philosophy [nature] is written in that great book which ever is before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”

## Problems / Exercises

I wrote about this earlier today and I want to spend a few minutes trying to organize my thoughts.

A conversation on twitter today with David Wees (@davidwees) reminded me of a conversation with a former colleague. It also reminded me of a class I took in my master’s program. The course had the vague title Problem Solving and my professor (who was my advisor) had a long background in studying problem-solving. I remember (not clearly enough) that we had a working definition for what qualified as a problem. The definition revolved around the idea that there are certain questions we encounter in math where we immediately know what we are supposed to do – what formula to use, what definition or theorem to call upon – while there are other questions where we do not immediately know what we need to do. The first group we classified as exercises while the second group were called problems. It is not necessarily that problems are harder. I have certainly dealt with many challenging math questions where I knew exactly what I needed to do, it was just really hard to do it. I have a real fondness for problems in mathematics and I have developed the habit of writing homework assignments for my classes that should probably be called problem sets. For years, I was writing these for Honors Calculus, AP Calculus AB, AP Calculus BC, Honors Precalculus, and AP Stats. A few summers ago I wrote a Geometry text for our school and I wrote all the HW assignments as well. These students are fundamentally different in many ways from the students I was working with in those other classes. I do not necessarily mean that they are inherently less talented or anything like that. What I do believe is that they are younger, less experienced, and less patient in their problem solving. So, they are more likely to simply shrug off a problem and figure that we’ll talk about it the next day. Over the course of the year most of them have become more patient and they are aware that we will discuss these questions and that they will not be graded on their HW. After my twitter exchange this afternoon, I am (once again) rethinking this strategy and I am nowhere near a conclusion. I did share the tweets with my class after we struggled through the question I wrote about earlier today. I asked them to honestly share their opinion about whether it is a valuable exercise to struggle with questions like this one. A few were upbeat and said that they liked thinking about these questions and that it is helpful to try challenging problems. One student said something really striking. She said that it is really frustrating to work through these problems alone and that she wishes she could get the opinions/insights of others when she is struggling with these questions. This is certainly in line with what David suggested in our exchange and with what my former colleague (who I wrote about earlier today) mentioned as well. I have some thinking to do here. I do believe that it is powerful for students to wrestle with challenging questions. I do believe that by not grading HW I am helping to create a safer environment to struggle. I know that a number of students work together on assignments either here in our dorms or libraries at night or during study halls during the day. I also believe that conversations in class are richer when they have some ideas already thought out to toss around. However, I also recognize that this is frustrating for some students and may simply push them further away. I recognize that if I am going to say that I value collaboration that I need to commit to making the time for that when we are together. I also recognize that what works to motivate seniors in college level math classes might not work as well with 9th and 10th grade students in a required math class.

Lots of thinking to do, luckily the summer will afford me some valuable time.

## Platonic Triangles

Too long ago I started a Geometry post by suggesting that I might have a two post day in me. Needless to say, it did not unfold that way and some combination of malaise, exhaustion, and the irresistible momentum of the end of the year has kept me away from this place of peace and comfort for some time now.

I want to share something from our Geometry class this year that was largely motivated by the work of Sam Shah (@samjshah) and his colleague Brendan Kinnell (@bmk2k)

At TMC Sam and Brendan shared boatloads of ideas and docs that they had created for their Geometry class and I am still in the process of digesting them. One that jumped out to me immediately was a document that they called The Platonic Book of Triangles that they were kind enough to share and to allow me to share in this space. Sam wrote about their process here and here.

What I did this year was try to de-emphasize naming the trig functions and just concentrate on the inherent similarities tying together right triangles as a lead in to discussing the inherent similarities relating all regular polygons and circles. Part out of a whole has become a mantra in my class these days. So, what I did was I went to a local copy shop and had them print out a class set of bound copies of the above referenced book of triangles. My students are referring to it as the magic book of numbers. We reference it regularly to set up proportions to solve right triangles. I had the book laid out so that each page had complementary angles on either side. So the students recognized – with a little prompting – that the side lengths on the triangle with the 38 degree angle marked matched up with the side lengths on the triangle with the 52 degree angle marked. I have been SO happy with how they have taken to this reference. In a way it reminds me of the trig tables I used to look up in the back of my book but this has a couple of major advantages. First, it is far more visual and helps the students orient themselves. Second, it does not rely on memorization of a mnemonic about the definitions of the cosine, the sine, or the tangent of an acute angle in a right triangle. I have been careful when I do use those terms to say as clearly as I can that * for now* they do not want to talk about these functions for anything other than acute angles in a right triangle. There is a whole world of trig excitement waiting for them after their experience in our Geometry class is a dusty memory.

From this conversation about solving triangles and using this to lead into explorations of regular polygons I wanted to make sure to introduce the idea of radian measures to my young charges. I came up with what seemed like a clever idea. It was a chilly, drippy day here in NE PA so I called up the weather bug applet on my laptop. However, what I did before class was I changed its unit of measure to celsius rather than fahrenheit. A student mentioned that it was unpleasant outside – with a little prompting – so I called up my weather bug and expressed surprise that it was only 13 degrees outside. Students *quickly* pointed out that this was simply a different way to measure the same thing, that there is a way to jump from one representation to the other. Aha, the hook was baited! I then launched into a pretty unexciting, standard representation tying together radians and degrees, relying on my mantra of part out of a whole over and over again. I am not fully convinced that they are buying in and there is evidence that many of my students seem to think that attaching pi to a degree measure is simply some sort of stunt. I am also seeing evidence that simplifying fractions, especially those where the numerator is already a fraction, is a serious challenge to too many of my students. However, what I am convinced of at this point is that a seed has been planted that has a better chance of blooming in precalculus than for those students who did not see the concept of a radian presented to them before. We have our unit test on Monday and I hope not to be disappointed.

## A Geometry Explanation Idea

Today will likely be a two post day after too long of a layoff from this space. It’s funny, I know that this writing relaxes me and lets my brain breathe in important ways. However, I am too prone to let busy-ness in my life prevent me from taking advantage of that.

In Geometry we are examining relationships between angles and arcs. You know the drill, right? Central angles = intercepted arc measure. Inscribed angles are equal to half of the measure of their intercepted arc. Interior angles (other than those at the center) are equal to half of the sum of their intercepted arcs. Exterior angles equal half of the difference of their intercepted arcs. As I am explaining this yesterday in class I am emphasizing this half relationship over and over but I just kind of gloss over the fact that the central angle does not fit this pattern. I have gone through this explanation in the past but I do not think I was as explicit about the developing pattern with the one half scale factor in the past. So this morning I was thinking about how I can best help my students focus on this detail. I was motivated, in part, by a lovely blog post by Michael Pershan (@mpershan) which, in part, is about managing the processing load of our students. I was struck when reading by the fact that I was asking my students to juggle seemingly different rules yesterday. So I had this idea and it probably is not revolutionary in any way. But I think I want to simply say that central angles are interior angles. They happen to be interior angles that form two congruent arcs. This way I can ask my students to think about the one half scale factor for every angle/arc question that they think about.

Does anyone else out there do this? Do you agree with this idea? Do you think that there is a downside here that I am not seeing yet? I’d love to hear some thoughts in the comments or you can find me over on twitter @mrdardy

## Taxicab Geometry – A Brief Exploration

We are officially on spring break here at my school and we end the term with a week of what are called test priority days. The idea from the school’s end is that we want to protect students from having says with three (or more) major assessments as the winter term comes to a close. With a two – week break most teachers try to put a little bow on their material before taking off so as not to simply start off again on March 14 repeating a week’s worth of material. However, this leads to some awkward scheduling. My last test priority day was Monday and I met classes on Tuesday, Wednesday, and Thursday. I sent out a call for ideas on twitter (like you do, right?) and I received a handful of great suggestions. From a conversation with Henri Picciotto (@hpicciotto) and Becca Phillips (@RPhillipsMath) I decided to spend a few days with Geometry AND with Discrete Math on an intro to taxicab geometry. Henri shared a great link to one of his pages (I encourage you to download that file from Henri – the relevant ones for this discussion are labs 8.4 – 9.1) and I modified some of those ideas and created two handouts of my own (here is #1 and here is #2)

I want to take a moment here to reflect on how our two and a half days with this unit went. We worked Tuesday and Wednesday in each class and wrapped up our conversations before tackling the cool problem I wrote about here to finish our time together on Thursday. First, I want to comment on my documents and how I intend to tweak them before using them again. Then I will comment on the class action these days.

Handout #1 – First change I would make is that point B would be the point (5,4) instead of (4,5). I do not know if any student caught this, but when I imposed the map of Gainesville, FL on the situation I described, Anne is not at the point (4,5). This tweak would solely be for my comfort. I do know that students in all three periods had trouble deciding whether street location should be an x coordinate or a y coordinate. It should have been an easy decision, I think. I like the introduction of the Manhattan map as a way to discuss what a city block might mean, but I did too much talking this first day. I need to introduce the idea then get out of the way and let the students ask these questions. I also should change some of the coordinates I suggested. My students *really* wanted these points on the section of the grid I provided. I should probably adjust for that. Finally, I have to admit that I am pleased with the questions I asked here. I think that there is a pretty nice balance of practice, of comparing taxicab and Cartesian distances, and of asking some nice guiding questions. By the end of the day Tuesday I felt that my students were in a pretty good place.

Handout #2 – I like that I start off with the same image and the same text to reframe the conversation. I think I will take away the text here that defines a circle and make sure that this definition arises from conversation – either whole class or in small groups. I love the sense of discovery that emerges as the students begin to realize what a taxicab circle will look like. I had GeoGebra fired up on the projector and started taking ordered pair suggestions so we saw the shape emerge together. I am happy again with my questions here even though I am unsure of whether there is actually a clear formula for the number of lattice points inside the border of a Cartesian circle. We did stumble upon a formula for the lattice points inside a taxicab circle and it was pretty darned exciting to see this unfold. Since this was the last night of class work I had very little evidence that any of my students had entertained this question on their own. We had a nice enough conversation about it in class, but it would have clearly been more energetic if there had been some reflection on their own by any of my scholars.

My Geometry students seemed more engaged and interested in how the ideas unfolded in this exploration. Perhaps this is due to its clearer relationship to our ‘normal’ material for the course. My Discrete kiddos were willing to have these discussions, but they were clearly less excited/annoyed/engaged/frustrated/surprised by the discovery of the fact that circles are now squares. I felt pretty committed to the idea that we should agree on whether we wanted to limit ourselves to only considering lattice points when deciding about the nature of the taxicab circle. I had been rooting for a loosening of the idea of points here so that we would have a continuous boundary in the taxicab world as we do in our Cartesian world. Since I had so clearly framed the conversation the day before in terms of city streets and avenues almost all of my students wanted to stay with that restriction and they voted clearly to restrict to lattice points. There have been a few other places where the Geometry students were asked to agree on definitions. We agreed that a trapezoid should have only one pair of parallel sides and we agreed that kites should not have four congruent sides, they should have two pairs of congruent sides that were **not** congruent to the other pair. There is a clear pattern of wanting to agree to more restrictive definitions here. I have discussed this with one of my Geometry teammates and he seems a bit bothered by my willingness to allow these restrictive definitions. I understand his point about definitions later on in math, but I feel pretty committed to letting the students come to these agreements together at this level. I hope I am not undermining their future as mathematicians here. I like the placement of this material in a short, unconnected time span on our calendar. We could have this conversation at a number of times in the year and I want to keep this in my back pocket to uncover when time allows/demands a unit such as this one. I think that the fact that the students knew that this would not be part of an immediate assessment allowed them to relax a bit and just play with some of these ideas. I also think that this fed into the near complete lack of work done on finishing the questions I presented after class discussion time. I think I am willing to accept this limitation as long as the benefit of relaxation comes along with it.

I want to thank Henri and Becca for helping push me into this and I want to thank my teammate Mary who was willing to dive in and try this unit as well. My other two teammates tried some different ideas and I want to pick their brains to see how life went in their classes for these three odd days. I also want to say that I am fairly happy to have a bit of a break now and I hope to return to school on March 14 with at least a couple of weeks planned out carefully for both the Geometry and Discrete Math classes.

## More Similarity Adventures

Yesterday we had a two hour delay and I was looking around for an idea to engage my Geometry students at the end of the day. As I have been writing about for awhile now, we are engaged in conversations about similarity. We had some Kuta skills practice, we had some problem sets I wrote for the students for HW practice and today I wanted to have a little activity where I could introduce a question and get out of the way and listen to them debate/discuss/discover some important ideas. I looked at my own Virtual Filing Cabinet and rediscovered a great question posed by Nat Banting (@NatBanting) over on his blog called Musing Mathematically. The question I pulled was from a post last year looking at coffee cups. That particular post can be found here. Below are the two key photos that prompted the conversation yesterday.

I posed the following questions on my class handout :

- Show that these cups are not similar.
- If the small cup of coffee costs $0.99, how much would you expect the large cup of coffee to cost?
- Since they are not similar, change the height of each cup – maintaining the diameter of the top – so that the cups are similar to the small size.
- Now, instead change the height of each cup – maintaining the diameter of the top – so that the cups are all similar to the extra large size.

Before presenting these questions/challenges I prefaced the conversation by talking about the habit of upsetting, like at a movie theater concession stand, and pointed out that larger sizes are (almost) always the better value but usually not necessary. Another important note is that I allowed them to ‘cheat’ a bit by presuming that the coffee cups are cylindrical. We have not officially touched on much in the way of volume conversations so we needed to come to an agreement, which we did quickly, on what the volume of a cylinder ought to be.

I am so thrilled with how our conversation unfolded and with the ideas that popped up during our chat. The students were quick to notice that the large and extra large cups each have the same diameter, so similarity there is thrown out the window. A student quickly nominated the ratio of height to diameter as the scale factor that was important. They were shocked by the theoretical cost for the large cup of coffee. I suggested that we ignore the pi in the calculations of volume and at first they were happy with my reason why and then they balked at the idea of just throwing it out of the calculations. This seemed to be a nearly perfect length for an exploration on a silly thirty minute class day schedule. I only hope that they remember nearly as much of the conversation as I do.

## Eyes on the Prize

In Geometry today we were reviewing for tomorrow’s test. A student asked to go over a proof about kites that we did together last Friday. It got me thinking about which proofs are really essentially interesting in Geometry AND it got me thinking about some former colleagues in Florida. I worked with a history teacher who had an interesting habit. The day after any test he handed out the essay question that was going to be on the next test in a few weeks. He checked back in on the question during the unit and used this question as a guide to their discussion along the way. At the same school I worked with a photography teacher. At the beginning of any project assignment she would hang up the best photos from previous years and referred back to these as a guide for her students.

So, today it occurred to me that there might be ways for me to model for my students what the goal posts are as we move along through the course together. I think I wrote about this already, but one of the changes I have made this year is that I am handing out previous tests that I have written a few days before our test day this year. My students take this HW assignment more seriously than any other assignments. I wrote the Geometry book we use and I have written all of the problem sets we use. I hope that my students feel that this problem sets are meaningful and worth their time. However, I understand that there are calculations to be made about how to spend time and my students feel that available time is at a minimum. I think that I will not wait until the week of my next test, I think that tomorrow I am going to hand out last year’s test even though our next test does not occur for a few more weeks. I will check back in on this test every couple of days to give the students a sense that we are making progress. I also want to spend this summer compiling about twenty or so proofs that I think are particularly interesting. I will not include the proof itself, I will simply put together a set of diagrams and given information along with the conclusions to be drawn. I think that I want to hand this packet out at the beginning of the year next year and use this as a regular reference during the year.

I would love to hear any opinions about the benefits or drawbacks of these ideas.

## A Fantastic Day of Wrestling with Problems

A former colleague of mine, Lisa Winer (@Lisaqt314) tweeted a problem at me last night. Wednesday night I s my basketball night and then I curled up to watch some Netflix with my wife, so I did not see the problem until this morning. It has since been making the rounds a bit. It is called ‘The Hardest Easy Geometry Problem’ and you can find it at https://www.duckware.com/tech/worldshardesteasygeometryproblem.html

I started working on it on my side board today and it caught the attention of my BC students. One of them found a solution using trigonometry and I constructed the triangle in GeoGebra to confirm that he is correct. However, I still have not found a way to solve it geometrically. I reduced the problem to four equations with four unknowns but the matrix is singular and I could get no solution. I did, however, have fun playing with it and watching a number of my students dig in.

In Calculus BC today we talked ourselves into the area formula for regions bounded by polar curves and we had great conversations about it in both of my sections. In each class I had at least one student remember some area formulas for triangles that are rarely used and that help serve as the basis for the integral involved. I was pleased with each of those classes today.

In geometry we are working with quadrilaterals and a recent HW problem presented the students with a parallelogram and some algebraic expressions to deal with. Most of my students made an assumption regarding the intersection of diagonals for the parallelogram. They correctly assumed that they bisected each other. I was pleased that they made this assumption but I made sure that they felt comfortable with an argument supporting that fact and then a series of questions erupted that carried us through the end of the day. Do diagonals bisect each other for all quadrilaterals or just parallelograms. A couple of quick sketches at their desks implied that it was not always true. A quick visit to GeoGebra seemed to convince them. Then a student asked if a quadrilateral could have congruent diagonals if they do not bisect each other. A few more sketches and then the guesses started flying in. It did not take long to guess that an isosceles trapezoid would fit this bill. Again GeoGebra confirmed our guess. What next? How about the triangles for,Ed when the diagonals cross? Are they all congruent? Are they congruent pairs even? Quick feelings that the ‘side triangles’ are congruent but the top and bottom ones are not. Right again! But my favorite part came next. I did not plan on talking about area for a couple of days still, but the moment felt right. I asked if we could deduce an area formula for this trapezoid. Now, last year at this point I had a student suggest drawing one diagonal to find two triangles. Standard and clean. I also had a student suggest dropping two altitudes from the ends of the shorter base. Again, a nice standard solution. I had one student suggest rotating the trapezoid 180 degrees to create a parallelogram twice the size of the trapezoid. Not standard at all, but also kind of confusing for his classmates. This year, I had a student named James make a suggestion I had not head before. He asked me to draw segments from the end of the shorter (upper) base down to the midpoint of the lower base. This created three triangles all with the same height. I took a picture of the sketch we Made on the board. That is the photo on top of this post. I must say that I am completely delighted at this clean and clear way of looking at this area problem.

A pretty good day overall, I must say.