## Segment Addition Postulate: Hanging Pictures

September 14, 2018

I just did an activity with my Pre-AP Geometry classes that I am fairly proud of. It occurred to me this summer, when I was hanging a bunch of pictures, that I was actually doing a practical application of the Segment Addition Postulate.

Origin Story
It all started when I was hanging my Captain America picture display.

I decided I wanted the five pictures centered on the wall and centered vertically on each other. I thought the math involved in making that happen was interesting, so I came up with a slightly simpler version (DOC) (PDF) for my students.

The Setup
When I introduced the problem, I brought one of the pictures from home to show them what I was working with. I realized that my back blackboard was 12′ long, the same width as my wall, so I decided to make full-size “pictures” for them to arrange once they had their measurements.

What I Learned
I have two Pre-AP Geometry classes on alternate days. The first day, I introduced the problem and just told them to get started. This led to quite a bit of confusion, and I had to spend some time answering the same questions over and over. I thought it was fascinating that two approaches emerged — some groups were making their calculations using the information I gave them, which was pretty much the way I intended them to go. Another group, however, decided they were frustrated with the numeric approach and went directly to the full-size shapes to arrange first. They finally got everything arranged, but the whole “display” was too far to the left, so they had to shift everything over. Once they had, they went back and measured where everything was “supposed” to go. Their numbers weren’t exactly the same as mine, but I was pleased by how close they were.

On the second day (especially since my principal and evaluator were coming), I decided to give them a little more direction before turning them loose. We went through the process of finding the center of the display and how to line that up with the center of the room. I also explained how to align the pictures vertically with each other so that the tallest was 14″ from the ceiling. Overall, it went much more smoothly, and I had fewer students get frustrated and just quit.

Both my principal and evaluator complimented me on the lesson design, and on the “real-world” aspect of the problem. Admittedly, I’m not sure too many people would be as anal as I am about getting pictures aligned in this fashion, but I think the principles can be applied to a lot of practical situations.

September 23, 2017

One of my favorite sessions at TMC17 was the CO+DE=MATH session by Tamar McPherson (@teachme124) and Stephanie Reilly (@reilly1041). They introduced me to trinket.io, which I really like because you can write code on the left side of the screen and run it on the right side of the screen. I am going to be attempting to do a couple of classes with my Pre-AP Geometry classes this week on coding. We’re going to do some of the Hour of Code problems on code.org on the first day, and then I’m going to attempt to introduce them to Python using trinket on the second day. In the process, I decided I’d have some fun and try to write a program to simplify radicals. I wish I could embed it, but WordPress won’t do frames.

Enter a square root you want to simplify (ex. $\sqrt{75}$), and it will give you the simplified radical (ex. $5\sqrt{3}$). Here it is: Simplifying Radicals

## First Day Plans (#SundayFunday) (#MTBoS)

August 12, 2017

I want to do all the things! Arrgh!

This year, especially after all of the inspiration at TMC, I’m having trouble narrowing down what I want to do on the first day of school for my two Pre-AP Geometry classes. The last few years, I have tried to get away from reading through the syllabus/procedures and have kids actually doing math. That has come back to bite me a few times because I think I didn’t spend enough time setting norms while we were doing the math. The 1-100 activity by Sara Van Der Werf has been really popular with the MTBoS this year, and I think this setting of norms combined with doing math has a great deal to do with it.

Here’s what I have to get done on the first day:

• I have a flipped classroom, which means that I really need to explain the flow of homework, classwork, and quizzes to my students. The biggest mistake students make in a flipped/mastery learning classroom is getting behind, and I really need to pound that into their heads!
• For the first time, we gave all of the Pre-AP Algebra I students a summer assignment. It wasn’t a huge amount (two pages of problems), but if I’m going to honor those who did the work and take this seriously, I need to do some sort of assessment on that first day. Our plan is to give them a quiz on the first day, and anyone who made less than 80% would then re-take the quiz (different version) on the second day.
• Gather student information. For the last few years, I have used a Google form to collect information such as preferred name, whether they have internet access, favorite subject, how confident are then math, etc. I always have some students who are shocked when I call them by their preferred name almost immediately, and I have to remind them they told me what they wanted to be called!

I’m figuring that will take about half of the class (about 35-40 minutes). With the remaining time, I can’t decide whether to do the 1-100 activity or have them read an excerpt from Have Spacesuit, Will Travel. I’m currently leaning toward doing the 1-100 activity, and then after my 8th graders leave (they have to leave about 20 minutes early every day to catch a bus back to the junior high), use the remaining time to have my 9th graders read the excerpt. Or we might just use the 20 minutes attempting to explain the wacky schedule we’re going to have this year — we’re going to have a 23-minute rotating advisory period every day.

For my three Astronomy classes, I like the plan I’ve used for the last couple of years:

• Let them sit where they like (these are mostly seniors, with a few juniors mixed in), although let them know I reserve the right to change them up if they can’t behave. I usually use a paper form to gather names to make my seating chart, but I’m thinking of integrating it into my Student Info form.
• Collect student information. I have a form similiar to my Geometry form that I collect similart types of data.
• Create a “Pocket Solar System”. I give each student a 3-foot long piece of adding machine paper. We label one end as “Sun” and the other end as “Pluto” (or “Kuiper Belt”), and I ask them to lay out the planets according to how they think they are spaced out. When they are finished, I have them flip the paper over, label the ends the same way, and I show them how the planets are really spaced out. We can then talk about how little mankind has really explored. To give them even more perspective, we will work through calculating how far away the nearest star is.
• I then go over the basic format of the class, how the grading is set up, and what my expectations are. This year, I think I’m going to try using Consensus Rounds to have them come up with respect agreements for Teacher-Student and Student-Student interactions.
• If I have time left, I’m contemplating using the 1-100 activity to get them thinking about group work because they will be starting their first group project the next week.

My final “class” is a class babysitting students as they attempt to make up Algebra I, Geometry, or Algebra II credits using ixl.com. Aside from getting them registered on the site and explaining how the process works, I’m not sure if I’ll do anything else with them.

## Summer of Conferences: TMC17

August 8, 2017

I didn’t quite mean for it to turn out this way, but apparently I only go to Twitter Math Camp (TMC) in odd-numbered years, so this was one of those years. The last two TMCs I attended (TMC13 in Philadelphia and TMC15 in Claremont, CA), I rode the train because I generally find it a more relaxing way of traveling. Unfortunately, the only way to go by train from Fort Worth to Atlanta was through Chicago (!), so I flew this time. Because I was attending the Desmos Pre-Conference, I arrived on Tuesday. One of my goals for this conference was to force myself to be socially outgoing because I know there are things I have missed out on in the past, so when Heather Kohn (@heather_kohn) offered to lead a pre-pre-conference tour of Atlanta, I signed up for a tour of the Georgia Aquarium. I wasn’t necessarily interested in seeing the Aquarium (although I’m very glad that I did — it was incredible!), but I wanted the opportunity to meet some people ahead of the conference and so have some built-in familiar faces for when the conference started.

The Desmos conference was very interesting. I ended up in a two-person session on Polygraphs with Chris Danielson, and we actually ended up playing the polygraph I created for my Astronomy class — which was great, because it made me realize that I needed to edit the pictures to make the names more readible. We ended up working on a possible polygraph idea for parallel lines and transversals, which I’m definitely going to work on this year.

The other session that I found interesting was First Steps with the Computation Layer. This has kind of been my summer for programming, and I thought it would be interesting to drill down into what makes Desmos tick. It was fun, in a frustrating kind of way, but I’m not sure the additional flexibility is really worth my time investment right now.

Again, trying to be more outgoing, I spent the evening playing games with a table of guys — thanks Chris (@Plspeak), Jonathan (@rawrdimus), Bill (@roughlynormal), and Josh for making me welcome!

Thursday, the actual TMC17 conference began. Here are my recaps and takeaways:

Morning Session: Playing with Exeter Math
I already wrote up how much of a blast I had in this session. Part of the reason I chose this session is because my summer conference load has been pretty heavy and I wanted some time just to play with math. The other reason I chose this was to give me an opportunity to experience math from a student’s perspective — expecially the frustrations that happen when you can’t quite get things to work the way you want. I was very glad I picked this session!

Session #1: “An Object to Think With”: The whole body as a tool for mathematical sense making by Malke Rosenfield and Max Ray-Riek
This didn’t really turn out to be what I thought it was. We ended up making 3-D structures out of rolled-up newspaper sticks and tape. What I found interesting was the way Malke and Max facilitate the session: they started us up very open-ended and added structure as we went along. It was also fascinating how different the three groups’ structures were (my group’s was very free-form). I’m not sure what takeaways I have for my classes, but it was a lot of fun.

Session #2: Expos: Student Presentations in Math Class by Matt Baker and Kat Glass
I really found this session to be interesting because they use student presentations to (a) review before tests and (b) practice giving presentations. I especially liked the rubric that they shared with us. I already do presentations in Astronomy, and I think I’m going to add giving students grades on how they work in their groups on the presentation. Part of me really wants to have my Geometry students do presentations for review and part of me is a little nervous about the time commitments by both the students and me.

Session #3: A Trig Exploration: Exact Values and the Golden Triangle by Rachel Kernodle, Jamie Collins, and Molly Tanner
This was a whole lot of fun! After refreshing our memories of 45-45-90 and 30-60-90 triangles (and the corresponding trig values), we were then given a 36-72-72 triangle and asked if there was any way we could use this to find more exact trig values. Very cool!

Session #4: CO + DE = MATH by Stephanie Reilly and Tamar McPherson
This might be one of the most useful sessions I attended at TMC. One of the reasons I’ve been getting back into programming is because I think there’s got to be some way to correlate the logical thinking from programming with the logical thinking in Geometry. This session may have given me a really good tool by introducing me to Trinket.io. Until my laptop’s battery died, I was able to take their templates and build a tool for finding the third side of a right triangle, as well as one for calculating the distance between two points. I’m definitely going to be playing with this as school gets started!

Session #5: An Hour of Codebreaking by Bob Lochel
This was a fun session to play with different types of code makers and code breakers. The simulation of the Enigma machine especially boggled my mind. I’m not sure how much (if any) I can use in Geometry, but it was a good session.

Session #6: Clothesline Math by Chris Shore
I was so happy they added this session on because this I wasn’t able to attend his earlier presentation. I’m not sure I can adequately convey how much my mind was blown by the simple arrangement of some paper markers on a piece of string. For the initial stages, it was interesting to discuss what was the absolute minimum amount of information needed to fully represent $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$ on a number line; and I also enjoyed the discussion generated by placing $-x$, $x+1$, and $x+2$ on a number line containing 0, 1, and $x$. It wasn’t until we were looking at solving an equation, $2x+9=3x-6$ that things exploded. On the number line, we placed $2x+9$ and $3x-6$ clothespinned together. Chris then had us place $2x$ and $3x$, and that’s when we realized that $2x$ had to be 9 units away from the equality on the left and $3x$ was 6 units away on the right. The gap between was 15, which was $x$. Wild! From a teacher standpoint, Chris also gave us some good words of advice, such as making sure the students not at the board have whiteboards or the equivalent, so they have something to do.

This post has gone on too long, but I don’t want to forget about my favorite My Favorites:
What Else Can Google Slides Do? by Jennifer Fairbanks
Dynamic Web Sketches by David Petro

Maybe next year, I’ll be able to break my pattern and go to TMC18, even though it’s an even-numbered year!

## Summer of Conferences: CAMT17

August 8, 2017

CAMT (the Conference for the Advancement of Mathematics Teaching) was being held in Fort Worth this year, and my principal agreed to pay the registration fee for any math teacher who wanted to go. Because of other scheduling conflicts, Brenda and I were the only ones who were able to go. I figured we would probably not be attending the same sessions, since she would be focused on PreCalc and I would be focused on Geometry, and that’s pretty much the way it turned out. What I did not expect to happen was to run into Tiffany, one of my good friends from my UTA Master’s program. We saw each other after the first sessions of the day, and pretty much hung out together for the rest of the conference. I also ran into Anita, a teacher who used to teach Pre-AP Geometry at my school, as well as two or three other teachers who had been there. It was somewhat surreal.

This is my recap of what I learned at CAMT this year:

Even though this session was listed for 6-8, I decided to attend because I have followed Andrew for years on Twitter. It was definitely worth it. One of the things I will definitely use this year is having students guesstimate how a bottle’s circumference compares to its height as a reminder about circumference.

Session: Plick Me! Flipping Geometry by Sarah Ashley
While I have been saying I wanted to use Plickers for a couple of years now, I always seem to find some reason why it’s never the right time. What I appreciated about this session was that she actually had us using the Plickers and showed us how she set things up. She also provided some links to some useful sites, most of which either required iPhones or I already knew about such as Desmos or Geogebra, but she did mention a few I hadd not heard of such as Splashtop.

Session: The Importance of Getting It Wrong by Michele Adams
I’m not sure this session really answered their question of how to get students to persist, but it did have some interesting activities that reminded me why I shouldn’t forget about the Shell Centre for Mathematical Education resources. The activity that I especially want to remember when we hit volume was comparing bottles and graphs of their volume vs. height.

Session: Engage and Motivate All Students by Aaron Daffern
This was one of those sessions where you make a pre-judgement based on your initial impression of the presenter that turns out to be completely wrong. Aaron is an interesting guy. He’s currently a curriculum direcctor for a charter school which he had previously served as principal. One thing he said that really stuck with me was that as principal, he should be able to “walk into any classroom and teach any subject.” I know my current principal also has that philosophy, which (although it can sometimes come across as arrogant) is such a change from having principals stand up and publicly state that they aren’t that good at math (laugh, laugh). Aaron had us model a strategy that I’m probably going to use for my first day classes: consensus rounds. He initially had us come up with three to five characteristics of a good math lesson. At the beginning of the session he had given everyone slips of paper that had three different groupings on it. He then had us find our first group (which got rather chaotic in a room of ~100 people) and come to a consensus of three characteristics. He insisted that each member of the group needed to “own” the list. It was also important that we each write the list down because after two minutes, we regrouped and did it again. After two minutes, we regrouped one last time. By this point, most groups had talked through different ideas, but the main consensus of the whole room for the main characteristic of a good math lesson was “student engagement”. Of course, how we achieve engagement is the tricky part.

Session: Revamping a Classic: Interactive Notebooks Gone Digital by Amanda Packard and Rita Gongora
Although this was billed as a commercial product demonstration, Tiffany and I decided to attend because we thought it might give us some good ideas, which it did. We saw some good uses for OneNote for Classroom, as well as using Google Slides to make a template that students would then use to fill in for their notes.

Session: Breaking Down Barriers – Geometry Redesigned by Julie Kidder and Caroline Robb

This was a very interesting session. These two teachers are fortunate enough to teach at a private school that let them completely redesign their Geometry curriculum. One of the most fascinating decisions they made was to eliminate Pre-AP (Honors) Geometry as a separate course and put all different skill levels in the same classes. How they then handled the Pre-AP part was to designate parts of their assignments as “Honors”, and students had to accumulate a certain number of points to have their course credited as honors. Students were given a choice of ways to accumulate those points, which also included things such as competing in UIL events. While I certainly can’t implement that sort of thing in my classes, I really did like their lesson design.

## Playing With Exeter Math 2

July 29, 2017

I have been at TMC17 in Atlanta since Tuesday, and my morning session has been “Playing With Exeter Math”. In my group, we have been working on the Math 2 problems, which although I have worked through these before, I have enjoyed thinking about them more indepth.

I have been sitting next to James Cleveland, and he has a knack for asking interesting questions, which really make me think. For example, on problem 33, we have to show that a triangle is isosceles by setting the two distance formulas equal to each other. James realized that once you square both sides, you have two circle equations, and connected that to how we construct perpendicular bisectors by drawing two circles and marking their intersections. It became even more clear when I graphed the equality on Desmos, and Desmos produced it as a straight vertical line. Cool!

We also looked at problem 21, which asked us to divide the shape into two congruent shapes. We conjectured that any shape with a 180° rotational symmetry would have an infinite number of ways that it could be divided. In order to satisfy the conjecture to my satisfaction, I set it up in Sketchpad:

Yesterday, we looked at problem 45, in which the corner of a paper is folded down to the bottom of the paper. The original problem asks us to find the area of the triangle created. It then asks us to find the value of x that would maximize the area of the triangle. Working it out algebraically, we found that for a side length of 10, x would be $\frac{10}{3}$. This led James to conjecture that it would always be $\frac{2}{3}$ of the length of the side. After a lot of hairpulling, I got the following set up in Desmos that shows this to my satisfaction (I used a slider for $t$, and the equations were $y=\frac{1}{2}\left(t-x\right)\left(\sqrt{2tx-t^2}\right)$ and $x=\frac{2}{3}t$ along with a parabola to fit the maxima as $t$ varied.):

Today, which was our last day, I had thought to finish up problem 45 (I added the vertical line at $x=\frac{2}{3}t$.) Once again, James was working on a problem, but this time, he got almost the entire group intrigued by it. Problem 133 sounds simple: Dissect a 1-by-3 rectangle into three pieces that can be reassembled into a square.

We quickly realized that since the area of the rectangle was 3, that the length of each side of the square would need to be $\sqrt{3}$. We immediately went to 30-60-90 right triangles, but we couldn’t figure out a way to make the cuts. Finally, about 20 minutes before we were supposed to leave, Annie Perkins figured it out using Sketchpad. This is my re-interpretation of her solution:

I’ll probably write up a full reflection on my experience at TMC, but I wanted to get my morning session experiences written down before I forgot about them.

## Volume and Surface Area

May 12, 2017

I’m trying to get back in the mode of sharing what I’m doing in my class. The worksheets that I use for volume and surface area are probably some of my favorites because of the way that I scaffolded them. I first saw this technique of using shapes to code what information goes where when I was paired up with a CTE (career and technology) teacher several years ago, and I adapted it for my sections on solid geometry.