Euler Characteristic

Yesterday half of my Geometry students were out taking English 10 ECA's, and I thought it was the perfect opportunity to do some fun math things in class that I don't normally get to do. I chose to do a lesson on the Platonic solids from NCTM Illuminations, since we had just finished studying surface area and volume of solids. And I wanted to play the game Brussels Sprouts with them. I planned this out a few days ago, and the night before I looked over the rules of Brussels Sprouts and the math behind it. Lo and behold, Brussels Sprouts relies on the Euler characteristic that we were going to study when we explored the Platonic solids. I was blown away!

There is a special relationship between the number of faces, vertices, and edges of Platonic solids called the Euler characteristic: #f + #e - #v = 2. The NCTM applet lets students come up with that relationship on their own. That same relationship is what allows you to always win at Brussels Sprouts if you know the trick. Watch the video to find out how!

Game of Brussels Sprouts

Euler Characteristic

Stadel's Transversals Activity

In geometry, we're studying pairs of angles formed by two lines and a transversal. I used Andrew Stadel's activity for this lesson and set up 9 stations on my extra long whiteboards with painter's tape as the lines. I taped the problem sets next to the stations on the board and left out some dry erase markers for each group. We started with examples of each of the pairs of angles: alternate interior, alternate exterior, consecutive interior, and corresponding. Then I paired them up alphabetically (first time this year I didn't let them chose their own partner), and set them to work. They also had to use vertical angles and linear pairs, which they learned last week. The first problem is a good basic problem that shows if they understand each angle pair. The second problem is quite challenging. After four classes of geometry (about 30 groups of students), only one had solved the second problem. However, it doesn't matter to me that they didn't solve it because they were still working with the angle pairs and they were challenged by it. I loved seeing the kids communicating with each other. I had to prompt some to work together, but overall it was great seeing them up at the board solving a problem with their peers. We'll have to review the angle pairs over the next couple of days, but it was a great start to the section. Thanks Andrew!

Organizing

Day 1 of organizing school work went pretty well. I was surprised how organized I kept everything during the year because it wasn't too hard to go through. Today was organizing geometry papers by chapter, tomorrow I will start looking at the lessons and activities in each unit and how to improve them or what I could add to them. These are some pictures of my piles...

Box of geometry papers

piles of papers

organizing geometry papers

Posters

Talking Math With Your Kids

I don't have kids, but the title of this post comes from the title of Chris Danielson's blog by the same name.

I was at the laundromat tonight, and as I was pulling my clothes out of the washer, I noticed an elementary-aged girl working on math homework. She was telling the adult with her that she had to "count by fives from six to fifty-five." She was working on the list when she said with frustration, I hate math! The adult said you can't county by fives like that because it doesn't work out evenly. She ten started counting out loud: "five, ten, fifteen..." The woman then read the problem from the math book herself, and it turned out that it asked students to county by fives from six to fifty six. The girl started sharing her sequence of numbers: 6, 11, 17, 23, 29, ... Another woman with her said: How did you get 29 from that? The girl noticed that she was counting by six's instead of by fives. So she started over. She wrote down 5, 11, 16, 21, counting up from one number to the next. I wanted so much to ask her if she saw a pattern, but I waited patiently as she kept working. When she reached 26, she exclaimed, I got it! I asked her to share her list of numbers. She had erased them, so she started writing them again. The list came more quickly to her this time, and she pointed out that all of the numbers end with a 1 or 6. I wanted to ask her to explain the pattern more, but her parents were talking to me about other things.

Normally, I wouldn't poke my nose in other people's business. I would go about unloading my clothes from the washer, trying to listen unnoticed to the conversation, and leave when I was done. But because I have been reading Chris' work for a couple of months, I was intrigued by what I learn from this girl and her mathematical thinking. So I watched, and in the end got to see a girl go from frustrated to excited about her math homework, which made me very glad!

GeoGebra success!

At the beginning of the year, I had my Geometry students use the free software GeoGebra to model an orienteering situation. They really struggled with it for a number of reasons I won't get into now. We are currently working with circles in class, and I had gotten to a section of the book that includes four theorems about inscribed angles. I didn't like any of the types of problems provided by the textbook - I found them uninspiring and I felt that students could learn how to do the problems without understanding the theorems. So I created a short project in which students would create models of the theorems in GeoGebra. I hoped it would give them a sense of what the theorems say and how they work. I also hoped they would gain some familiarity with a new digital environment, which I think is important for students. They had to create the models so that even if I dragged some of the points around in the program, the theorem would still hold up. In addition, students had to write up the steps they took to create the model. I had many trepidations going into this project - Would the technology work? Would students be able to work in the environment of the software? Would it be too easy or too hard? Would it take more or less than the three days I was allotting for it? In the end, it worked out beautifully. It was the right amount of challenge for my students. They worked at their own pace, and few students had real difficulty figuring out how to use the tools in GeoGebra. I led them through the steps of the first theorem, and they created the other three on their own. I am including samples of their work, as well as the project description and grading rubrics.

For the students who got done early, I asked them to spend a little time on this awesome geometry construction website:  http://sciencevsmagic.net/geo/
 

Materials

Project Description and Rubrics
Steps for Theorem 10.6
Steps for Theorems 10.7, 10.8, 10.9

Logarithmic Math Date

Almost every day I write the date on the board using a mathematical expression. Today I was lazy and just wrote February 12, 2014. Sometimes I think my kids don't like the date being written in math language, so I just put the regular date. During my last class of the day, just as I was finished doing a proof at the board and we were about to play Mastermind, one of my students asked me to come back so she could ask me something. When I got there, she told me the date on the board was bothering her and and wondered if I could write the date as log(base 2)4096 = x. I said probably no one else would understand it, but ok. So I went up and wrote it on the board, then we went on to play Mastermind. At the end of the hour as students were leaving, another student came up to me and said that he thinks that x doesn't equal 12. I grabbed my calculator and showed him how to rewrite the equation so that 2^12 = 4096. The first student also argued that the equation was right. I couldn't believe I was having this conversation! These are two freshmen who are in both geometry and algebra 2 who are thinking about logarithms outside of their algebra 2 class. It was so cool! I also had to share it with the algebra 2 teacher because I thought it was so great.

Fractions

The internet was down at school so we couldn't do our normal Estimation 180 to start my 8th grade math classes. However, the state exam is coming up, so I thought I would refresh their memory about fractions. I put three problems on the board - one each of addition, multiplication, and division. Students got to work and very few had to ask me how to do the problems. As I walked around I was impressed with how many students were getting correct answers, and quickly too! I was expecting more than half the class to have forgotten what to do.

I credit it to the month we spent doing fractions every day at the beginning of class. I know this is an area where kids struggle, so I made sure we practiced a ton. Honestly, I didn't spend much time on the conceptual aspect of fraction operations; most of the work we did was drill work. Even after doing fractions almost every day for a month, there were still students who struggled. After that month, I let fractions go and focused on other things. I wondered how much all that practice would stick with them. I read somewhere on the MTBoS about a teacher's frustrating experience teaching a concept. He spent extra time on it, and even then felt that his students didn't get it. However, later in the year when it came up again, they all knew the concept. I hoped that this would happen with my students and fractions.

Today before we went over the problems together, I told my students how proud I was of them for remembering how to do fraction operations. One of the students said to me - That's because you're obsessed with fractions. I laughed and took the comment with happiness. I said - That's the point! I made you do it every day because I know how hard fractions are and I don't want you to leave here not knowing how to do them. So go ahead and think of me as being obsessed with fractions.

Mastermind

I read this post about using Mastermind to teach proofs and logical thinking. I've been meaning to introduce the game for a while in my high school geometry classes for a Mastermind Monday (along with Tangram Tuesday), but didn't get around to it until yesterday. As I introduced the game, my students paid close attention, which is not always the case when I am talking. I tried to use the language that Avery describes on his blog post, but I had a hard time with it. Instead, I asked students after their gave their guess, "What was your strategy?" or "What were you thinking when you made that guess?" If I heard students say something like, "There's not a 1 in this game" or some other conjecture, I would ask them to repeat it for everyone to hear. There were individual arguments over what to guess next more than whole class discussions. I need to work on making sure the class is all involved equally in the game, as well as work on having them make conjectures rather than just describe a strategy. Otherwise, the game was fantastic! The first time we played, every students was engaged. Their minds just latched on. In every one of my four geometry classes they asked to play it again today, and we did. This game is absolutely becoming part of my regular arsenal of activities.

Video for Class

I'm have to go to a training session at school tomorrow, which means I'll be missing one period of the day. Today's a snow day, so we're already missing a day this week. We have lots to get done in class and I don't want to waste tomorrow. So I had the genius idea (!) of making a video for them. But then I had to figure out how to produce it. I thought of using screen-cast-o-matic - that thing is amazing. It's like a screenshot, but it does videos. However, for my video I have to write math stuff and draw triangles, which isn't so easy with a trackpad or even a mouse. So no scree-cast-o-matic. Instead, here's what I came up with. I feel so low tech.