# Exterior Angle Sum Theorem

I made this quick activity for students to discover the exterior angle sum. I just printed several quadrilaterals and pentagons, cut out the exterior angles, and put them mixed up in a Ziplock bag. I gave students 5 minutes to work with a team of 3 to somehow figure out what all the angles added to. I didn’t give them much direction beyond saying they needed to put the color edges together to discover it. Walking around the room, students kept calling me over to check their hypothesis as got really into it to be the first to figure out the theorem. It was a quick way to have students discover a simply topic rather than me just directly telling them…and it worked…two groups found it out in the 5 minutes!

# Lions, Tigers, Math, Biology, World Geography, English, and DIM, oh my!

Okay, I know that title is cheesy…

We took our freshmen students to the zoo last week for their first class field trip and I loved how it wasn’t just a field trip related to one content area, but in our planning we managed to relate it to every class. Here is the link to the assignment: Zoo Student Handout and the explanation is below…just in case you’re going to the zoo anytime soon with students and want to do a similar activity. 🙂

I’m looking forward to more opportunities that we can create interdisciplinary learning for students.

# Conditional Statements in Geometry and Computer Programming

While planning for the logic unit in geometry a few weeks ago, I wanted to increase the rigor and apply a more meaningful experience to the topic than I had in previous years. In the past, I have had kids do a project in which they found ads in magazines that used (or could be rewritten to use) conditional if-then statements. Then they rewrote the ads to include the converse, inverse, and contrapositive statements. I think this is a fun activity and it is a really great way to have students recognize if-then statements in the real world, however, I began to think that this experience might not be project worthy. I found that asking students to rewrite the statements did not have a lot of meaning to them and some of the mathematical logic got lost when they simply repeated a statement about shampoo or men’s deodorant.

So this year instead, I planned to use the ads as just a warm up by showing students an Allstate commercial and asking them to identify the if-then statements. Then, I started thinking about what else in our world uses conditional statements and how I could make a more meaningful and rigorous project. I realized computer science uses if statements for a program to do something if a condition is met. Several years ago I worked for a company where I used coding, but I knew I needed to brush up on my understanding before I could teach it to kids. In my search to make this project of relating conditional statements in geometry to computer programming, I stumbled upon two helpful resources: Pearson and Khan Academy.

In our Pearson textbook, there are enrichment activities and coincidentally, the textbook had a similar idea to mine for the logic unit. They gave students some code and asked them to identify the hypothesis and conclusion in it. However, they used GOTO which is a bit outdated and not used a lot anymore. So, I held on to their idea about dissecting code and rewriting the hypothesis and conclusion, but searched around for a more relevant platform.

In my mind I wanted students to actually write code, manipulate it, and see it work with this project. I soon found Khan Academy’s tutorials on if statements and as I was working through them, I found that it was the perfect match to what I wanted. The tutorials teach students how to edit code with different scenarios and then students do a similar challenge to try to master the code. In the first challenge that I had students do, they were shown a ball that drops off the screen and they had to use if statements to make it go back the other way, displaying a bouncy ball effect on the screen.

For the project, I created a worksheet for students to use while they watched the tutorials, practiced the code, and performed challenges to see that their code actually worked. It asks the student to rewrite the if statements as a hypothesis and conclusion (similar to Pearson), decide if there is a biconditional phrase, and then also use the inverse code to manipulate it again. (The worksheet is below and a Googledoc is linked here). Then after completing their code, I planned to have them reflect on their work by writing a blog post with two key aspects in mind: Connections: The student can recognize, explain and use connections among mathematical ideas and Problem Solving: The student can apply mathematical algorithms (series of steps), tools, and/or representations to accurately solve problems.

Finally, I showed my ideas to one of the deans at our school who is much more knowledgeable in computer programming than me to get some of his insights on the activity. He helped me create a short piece after students finish the challenges that helped them pre-write before their blog post. It also helped me create a rubric (below) to guide students as to how I would grade them including accuracy of conditional statements, problem solving throughout the activity, and the written expression of their blog post.

Overall this project went great…it has now become one of my new favorites!! Every student was engaged in the tutorials and the challenges; I saw them have a sense of ownership in their learning because they could self direct and work at their own pace. When students got the code, several of them got so excited and called me over showing off their successes. I also had a lot of really interesting conversations during and after the project…one student said they can now imagine how complex creating a video game would be with all the coding involved. Several students said they understood why their program needed both the if and the then components to make it work and really liked seeing their work actually do something. I am excited to read their blog posts soon (they’re due on Tuesday) and hear how they summarize the project and learning objectives!

# World Geometry

While planning for geometry last week I began thinking about how I could amp up midpoint and distance formulas. After learning the formulas, my students practiced with very abstract points. However, I knew I wanted them to see examples that had meaning. With the help of the English and World Geography teacher, I developed my students first interdisciplinary activity of the year which they named their World Geo-metry assignment.

In WorldEng (their combined World Geography and English class) my students had been learning about Burma. So, I gave them a map of the region with questions where they had to find coordinate points of key locations they had been studying such as Dhaka, Naypyidaw, Bangkok, and Yangon (I learned a lot just from the start of this!!) After finding the coordinates, I asked them to find the midpoint and distance between these locations and draw conclusions based on the map. To make it more complex and meaningful, I added some questions about the scale in miles for students to understand the actual distance it would be to travel from one place to another. The final question asked students to compare the size of the border of Burma and Thailand to other borders of Burma and justify why refugees might be immigrating along this region (something they had been discussing in WorldEng).

I really liked the discussion I heard between students as they worked through the activity. I decided to give each student a worksheet, but make each pair share a map and I think this helped students talk through the locations, make connections, and agree upon their answers. I also really liked how the answers for distance were not exact integer answers. They had to work with tricky numbers and understand if their answers really made sense. Finally, I knew this activity was successful after students commented on how they were combining three classes to do their calculations.

# First Week Highlights

I just finished the first week of school and looking back, it was one of my favorite starts to the year. Normally I’m not satisfied with my first day of school activities and either feel that they’re too cheesy or too boring. This year, however, I finally feel really happy with how the first day went because I had a high level of engagement from my students and the rest of the week followed in the same way. So, to recap the week, here are a few highlights, including many protocols for certain activities that can be used throughout the year and not just the first days of school!

Monday (first day of school part 2): After completing our quizzes, I explained our last activity: 31-derful. I found this activity from another favorite blogger: “Everybody is a Genius.” I displayed the same instructions she did and then let them go for it in groups. I loved seeing and hearing their thought processes with their groups. It gave me an insight into their problem solving and communication skills. Every class had 1-2 groups complete the puzzle and the other group were super close! Just like the activity above I knew this one was successful because on Tuesday (and Wednesday) several kids came into class asking if they could play the game again saying it was so fun!

Tuesday (part 1): I saved setting rules and going over the syllabus for the second day because I didn’t want to rush through either one and I knew the first day class times would be shortened. Normally, when going over my own rules and setting classroom norms I have done a chalk talk. I like chalk talks, but students don’t understand the value of silence during this activity, and it’s hard for me to facilitate without saying to stay quiet every 5 seconds when they are hyped up from the first days of school. So, I thought I’d save introducing chalk talks for later in the year…or maybe one of my fabulous colleagues will do one before me and be better at keeping them quiet :). Instead, I did a four corners activity to facilitate setting classroom norms. I loved how this went for several reasons: It got students up and moving, but in a structured way. Also, as we discussed agreements and disagreements, students were standing, which at first I thought might be a little chaotic, but in every class, they actually listened really well while standing…somehow it made them more self aware to who was talking and what they were saying. I also liked hearing students voice their opinions about how they learn best. I think students felt safe sharing how they felt because they often had someone else beside them that felt a similar way.

Tuesday (part 2): After we set norms and before we went over the syllabus, we jumped into a discussion of our summer assignment (a reflection about their own math understanding after reading the freshmen assigned book, Bamboo People). With the suggestion from a fabulous colleague, I used a Microlab protocol to facilitate discussion. This went well because it gave all students a chance to speak while keeping the conversation flowing in a productive manner.

Wednesday: We started the day with a WODB warm up that I’m going to do this every Wednesday…I love this activity! With it, students had a chance to communicate their thinking while producing some really interesting debates. In geometry, I used “shape 5” which gave students some new language and facts that they will be using later in geometry such as a dodecagon, polygon requirements, and composite figure. In algebra, I used “number 1” and a couple students gave an argument for something I didn’t even see…9 didn’t belong because all the others made 7 when you added together their digits. After the warm up, geometry played TGT to review algebraic concepts before moving on to geometry (I’ve posted about this game before). All students were engaged in this game because it was competitive, but safe. I think having students choose their comfort level with the material helped them feel at ease and confident in their competition teams. In Algebra we completed a KWL chart with a preview to their first quiz. I think this helped set the tone for why they need to know what they will be learning the next few weeks. Then, we reviewed patterns by doing this lesson. It was a great, low prep activity that helped students review patterns and formulate their own thinking without me directly telling them the sequence. The next couple days we did some book work from our Springboard textbook. I am really liking the reading required from the textbook, but I realized I need to work on my facilitation of teaching from a textbook (this is my first year directly using one). I’m not going to use it every single day, but definitely more than I ever have in the past because I think it is a really good resource for STAAR type of materials.

Friday: After taking some notes and doing practice on Thursday about points, lines, and planes, geometry played this sketch game. It was great to hear students communicate their learning again to each other. Many were saying the process was so hard, but kept at it and saw that the more specific they were, the more accurate their partner’s drawing would be. Algebra had their first “standards check” before moving on to non linear patterns. Geometry will have one Monday. I think the format of the SBG checks are going to be really good for myself and students. I especially love having students know exactly what their learning goal is and having them self assess their learning.

One last highlight: So far, students are doing really well with my grading breakdown of homework/classwork counting for 0%. I know it’s only been one week of school, but students seem less concerned about what counts for a grade and whenever I assign a task to complete, they all jump into it knowing it’s for them to practice their learning…hopefully the rest of the year follows the same way!!

I’m so thankful for all the great resources I’ve found through other blogs and am ready to take on the second week with a little finalizing of plans tomorrow…for now, time to relax! 🙂

# Chicago Trip and Math

My husband and I just got back from a great trip in Chicago to celebrate our first year anniversary. I loved every part of our trip from the delicious food, enjoying a no-hitter game at Wrigley Field, walking the city, seeing iconic sights, and so much more. Even though we were there for absolutely nothing to do with work, I couldn’t help but become inspired by the math of the city. I decided to jot down my notes here so I don’t forget!

1. Chicago Architectural Tour: This was one of my favorite things we did. It was a 90 minute guided boat tour that took us along the Chicago River as we learned about the history of the city through the architecture. I definitely want to show my geometry students pictures from this and hopefully convince them through photos that math is truly used in professions, appreciated in everyday life, and highly sought after for beauty and meaning in a city. It was incredible to stand and look up at the enormous buildings as I visualized what it would be like to build one. I also found the history behind each structure to be really interesting from a math mindset. One of the first things our tour guide reminded us about was the Egyptians were very influenced by geometry in their early architecture. As we went along the river, we saw the transformation from early styles such as Gothic, Renaissance, and Neoclassical, to modern day styles. Some key buildings are below. The first building is the John Hancock building which I found really interesting because we learned that the X’s were intentionally structured on the outside of the building as a way to provide the building stability, allowing it to have no poles throughout the inside, and thus giving it a completely open concept.The second was a very iconic apartment and multi-functioning building used in several movies (I still need to find out what movies…but I know there was one where a car crashes out of the building and falls into the river). The circular shape meant a lot to the architect, Goldberg, with it’s aerodynamic features, lack of any corner rooms (often thought of as reserved for high society), and enabled all rooms to be centrally located to the center. Next, was a triangular shaped building which was again, designed intentionally, to allow residents to have more lakefront views than a square or rectangular shaped building.Then, there was one that was built right over a train and so the builders were tasked with how to safely design such a building. They designed it narrow enough at the base such that the train could pass by it, but then it will become wider as it goes up with the use of triangular frames.

Finally, the last one was really aesthetically pleasing and it wasn’t until our tour guide explained that it was designed as a map of the river with the red feature symbolizing a “you are here” spot, that I really appreciated the creativity and brilliance behind the design.

2. The Ferris wheel at Navy Pier: While waiting in line for the Ferris wheel, I couldn’t help but notice the geometry behind the huge structure. I will definitely show these pictures to my students, and hopefully I can think of some cool project and/or investigation we can do with circles, arc length, area of a sector, etc. and Ferris wheels.

3. The Bean (Cloud Gate): This is one of the most well known areas of Chicago and I’d love to learn more about the shape and structure of it. When I show this to my students, I’m curious to hear the words they would use to describe its shape. I think there are also some interesting reflective properties my students and I could talk about. For example, when you walk in the middle of the structure you can see the same reflection 4 times (I tried to capture this in the picture below.) My husband and I had fun finding ourselves in the mirrors and then I suddenly realized we could be using words like translated and reflected…I think this would be a cool example to show students when we discuss these terms. I think proportions and similarity could also be referenced with this structure when you think about how your image changes depending on where you stand in relation to it.

Thanks, Chicago, for an awesome trip and lots of learning!!

# 3D Solids Lesson Using Jigsaw Cooperative Learning

Here is a quick lesson I put together using the Jigsaw Cooperative Learning technique for 3D solids. We learned the surface area and volume formulas for prisms in a direct instruction model and because we have such little time left in the year with testing and such, I decided Jigsaw-ing these concepts would be the best way. Overall, it went well…I did have to model how to have students teach each other and not simply copy, but after doing that, I was very pleased with how they communicated their understanding to each other.

I gave each expert group some cards of one of the solids I found here: http://math.about.com/od/formulas/ss/surfaceareavol_2.htm. With these cards, they had to talk through what the variables meant, what the lateral vs. total areas were, and how to use the volume formula. After each group finished completing their row of one assigned solid (writing the formulas, defining variables in their own words, solving a given example, and creating their own example), I mixed up the groups to have one solid represented in each new group. Then each expert taught the others how the formulas work and what they learned.

Here is front and back of the student worksheet I created, the rest can be found here: .

# Introduction to 3-D with Polyhedron Nets and Islamic Design

A couple weeks ago, I was planning for our last unit in geometry which is 3 dimensional solids, and as I was measuring nets and counting vertices, edges, and faces, I suddenly realized I was really bored. If I was bored, my students definitely would be! I knew I needed to amp up my curriculum to still teach nets, relationships between 2-D and 3-D, and constructions (TEKS G.6B and G.2A), but somehow make it more engaging.  What eventually fell out of my plans was a new connection between math and World History.

I started thinking that students could choose a 2-D net, decorate it, and then fold it into the 3-D polyhedron. However, this was not very exciting and required little critical thinking. Also, I knew I wanted students to decorate their models, but I didn’t want them to just draw flowers or smiley faces or simplistic designs with no reason behind it. So, I asked the World History teacher if there were any connections he thought I could make between our two classes. He told me they were about to start Islamic culture and history which was perfect because Islamic art incorporates a lot of geometric design. We talked about some questions to prompt some thinking for his class. By asking students to pre-think, my geometry students were able to be the experts the next day in World History.

Here’s the basic outline of the lesson plan.

1. Pass out the Islamic Art and Polyhedron student worksheet and talk with the students about the fact that we are going to make a connection between World History and geometry…yet again!! 🙂

2. Show this video (or any other video you find) and ask students to jot down anything they see that answers #1 (What patterns do you see in Islamic art?).

3. After the video, have students pair up and talk about what they saw. Then call on students to share out whole class. (Think, Pair, Share model)

4. Then, we went on to the questions #2-4 which talks about the history of the region and asks students to compare Islamic design to Chinese, Eurasian, and African Art, but you can add or take out any other questions that would be relevant to their World History class.

5. Show students the net templates they can choose from. Some chose simple nets like cubes, rectangular prisms, while others chose more complex such as octahedrons and stellated dodecahedrons. (There are many templates online…I decided to use this website. Warning: Some models are very tricky, but I think if students get to pick, they will have the buy in and motivation to complete it.)

6. After they have chosen their template, found the number of vertices, edges, and faces, we used the rest of the class to design their Islamic artwork. Remind students that they must use rules and compasses when drawing lines, circles, and arcs, because Islamic art focuses on very precise designs.

7. The next day, we came back and started class by having a chalk talk with this question: “What is the main focus of Islamic art…what does it include/not include?”

8. Then after a 3-5 minute chalk talk, I let them work on their design and fold their 3-D nets.

Overall, students enjoyed this hands on activity. One change I would make is to print out the larger nets rather than the single page nets, especially for the more complex types (anything larger than an octahedron) because folding and taping those got quite tricky! My plan is to hang these up with fishing wire between my room and the World History room as a visual connection between the two classes. Thanks, Mr. Sprott, for helping me dream up this mini-project!

# Thoughts That Keep Me Up at Night

I have been thinking a lot about grading, assessment, and the meaning behind these to both teachers and students. I have read Dan Meyer, Daniel Schneider, Educational Leadership, Matt Townsley, and Rick Wormeli while researching and talking with colleagues about mastery and standards based grading (SBG). I really like a lot of the ideas of SBG including more frequent and smaller assessments that allow one to know a student’s mastery on a standard. I also like the thought of a 1-4 scale and the language that is used to convey what each number means. One example I found that I really like is that the Solon School’s language in their rubric (1).  However, I am still struggling to wrap my head around SBG and mastery in the math classroom. I still have lingering questions that honestly keep me up at night. I want to do what is right for students and I want to push them to understand what they know and what they don’t. Even further, I want them to take charge of their learning and with my feedback, help them to know how to gain mastery on a concept.

Here is what I want to keep in my classroom regardless of grading…

Student communication and group work: I think when students talk out mathematical problems together, they cognitively grow a lot. A student’s ability to explain a topic further enriches their own understanding, and when they hear an explanation from another student, they relate to the language they’re using. So, regardless of how I grade and what I grade, I still want students to work together to solve problems.

Reasoning: I also want to be sure I am still allowing room for reasoning and processing skills beyond algebraic skills. I want to continue to provide opportunities for students to explain and justify their understanding of concepts through written and spoken dialogue. Whether this fits into a numerical grade or not, students still need to be pushed to think deeply and justify their reasoning.

Here are my questions that linger…

How do I create a balance between group practice and independent practice? How do I convince students that individual work time is just as, if not sometimes more, beneficial than group work? And finally, how do I convince students that individual assessments are meant to be informative not punitive, especially when we take points off for wrong answers rather than give points for correct attempts? Right now I give a lot of time for group practice. Again, I love the learning that happens when students talk through math. But I am realizing as I read more about SBG, I need to create more opportunities to show what they know individually. In that, I need to create time to give my feedback to them on an individual level beyond tests. I think by adding in more frequent assessments, this will do that and give students the opportunity to analyze what they know. These could just look like short quizzes done on note cards at the beginning of class. They could be graded on a 1-4 scale with more feedback than a regular assignment as it leads up to a cumulative summative assessment. The question then becomes, do I have the time myself to dedicate rich feedback more often to every individual student? How do I create that time, especially as a math teacher, when so many of my days are dedicated to teaching new material rather than refining knowledge students already know? Ultimately, can someone find me some more hours in the day? 🙂

Feel free to respond to any or all of my questions, share this with every educator you know, and continue the conversation of grading and assessment!

# MatHistory Part 2

This post has taken me a while to write with several revisions because I just haven’t known how to write it in a way that gives justice to everything I have loved about this assessment. Every time I sit down to write it, a new way to introduce the post races through my head. However, I think the best way to start out is simply thanking the AP World History teachers who had the vision and the enthusiasm for working with me to create this complex, unique, and highly successful interdisciplinary assessment. So, thank you, Mr. Freeman and Mr. Sprott! I hope this leads to many more mathistory and mathenglishistry (math-English-history-chemistry) ideas!

So, here’s how the project unfolded…

Last Tuesday, the World History teacher presented our students with the idea of a power scale timeline and explained how to create one. Each group of four students were given 16+ maps of various European empires that showed the time at which each empire owned land. After identifying the region, they transferred the area onto a larger scaled timeline. Then, when all the empires were on one map, students used their mathematical knowledge to identify key points and explain why they are important relevant to time and land area (we presented this part on the second day). At this point you may be thinking whhhat theee hecckkk is this lady talking about…don’t worry, our students were also a bit perplexed by the task at the beginning and in first period, we did have a brief time when we felt as if our students might revolt against us. However, after encouraging our students to just try it out, much to their surprise with a little patient problem solving (as referenced by Dan Meyer) they excelled at the task. After they got the hang of it, I asked one student to explain the process and here is his recording. I liked getting to be in the history classroom this day because I was a second person who could help facilitate and answer questions as students created their timelines. They also started to make the connection with me being in the room that this might have some math involved in it.

On the second day, as the students were finishing their timelines, we presented them with the math portion. As they identified key points historically, we heard them using mathematical language and vocabulary terms as they talked about undecagons, parallel slopes, parabolas, intersecting lines, exponential growth, etc. It was really cool to see students make connections and hypothesize using their knowledge from both classes. We encouraged students to work together because the questions we gave them ranged in topics from both Algebra II and geometry. Each group had students from both classes and therefore they were able to be experts in the subject they were taking. Walking around helping students work through the assignment, I overheard two students talking about interdisciplinary learning and caught the end of their conversation on tape. Hearing them voice their appreciation for interdisciplinary learning really made it all worth it!

Yesterday I finished grading the math portion of the timelines and was sitting with the English teacher during our monthly Saturday school when she asked me, “so, would you do it again?” My immediate answer was, “YES!!” I think that asking students to use mathematical evidence in their explanations of what was happening historically made them think deeper. I also think they were able to synthesize better by using logical mathematical thinking. Additionally the featured image at the top of this post is one that we were very impressed by. She spent time finding key images and icons from each empire and finished her timeline with detailed watercolors. It made us think of the possibilities…perhaps students could design a timeline based on different aspects of each empire (religion, architecture, art, etc.) I think there is so much more we can do and definitely lots to think about. Over the next few weeks students will continue to learn in their World History classes about each empire and add to their timelines in order to create a large final timeline in groups…so more to come!!

Here is the link to the math student materials: MatHistory