Tuesday, May 31, 2011

Grading: Road to SBG

From the excellent

So, my son and I (X=11) have been discussing starting sentences with "so."  It appears that it may be genetic.  In real life I'm quite awkward in conversation and transitions are especially difficult, and I think "so" is an all purpose connector.  Leads to something that fits or to a change in topic.

Of course, transitions being awkward is something that I think is true in general.  How to get students to transition from what they've known to something that may be completely new is the crux of Standards Based Grading, or more wholesome and holistic ways to evaluate as a whole.  Back When I Was a Worse Teacher (tm) I could sometimes be heard to joke (sortofjoke) that I wished I could offer my students a deal: take a C and leave the class, stay and take the class for a chance - no guarantee - at better.  I was known to be an easy grader though, and I wanted it to be the case that if students made a genuine effort that they would get at least a C.  I also use to jokingly propose a Survivor style grading system where we vote people out of class starting at the third week.  First out fail, but get the rest of the semester off.  Thought the tests could be like challenges where multiple people can earn immunity.

Strange Brew
my current favorite comic strip
does my favorite movie line to quote?
It was a joke!  The point of the joke was to get people to think of why they were in the class, and the grade might not be the point.

Working with Dave Coffey, who is our local assessment guru, I saw him use a portfolio to assess our Math for Middle School class, and we have long used a portfolio for our student teacher assistants.  In our introduction to mathematical reasoning class there's a summative assessment called the proof portfolio.  One time I was teaching it, at the end of the course one of the students asked: "I'm so much better than I was at the beginning of the semester - why does that writing count in my grade?"  Whoa. My only response was that THAT was a very good question.

I realized that more of my grading had less to do with where the student was at the end of the semester than I had wanted to think.  Mostly because of the reason I hate the most: that's the way it's always been done.  Furthermore, I realized I hadn't ever really thought about what I wanted grading to achieve, let alone whether it was accomplishing the job.  It's usually pretty quick for a room full of students to agree on some characteristics of good grading:
  • fair
  • measures real understanding
  • not fear or anxiety inducing
It usually takes some discussion to get to:
  • measures where the student is at the end of the course

My portfolios shifted to feedback only up until the end of the semester, and then the last time they are graded.  The grade is about half on completion (as a percentage) and half on exemplars.  The students choose the exemplars.  In early turn ins, the students ask for feedback on what they want to know, and I share what the grade would have been, and give feedback on their issues.  The would-have-been grade allows me to identify big issues that the students don't seem to recognize.  I collect at the 1/3 and 2/3 point, though some students want more frequently.  (So they can be more responsible.)  The final grading is relatively easy though visually daunting; how many boxes of grading?  The final exemplars need annotation about what makes them exemplars, and have been the best way for me to evaluate problem solving and communication.
What are we assessing?
brad_holt @ Flickr

So what about the content? That was hard for me.  How to evaluate content in a way that allowed for improvement right up to the end.  Getting involved in twitter around the same time I was reading more and more educational blogs exposed me to standards based grading.  Especially Sam Shah and Shawn Cornally.  They supported with good stories and honest difficulties, links to several educators trying or mastering the practice, and good resources, like the SBG wiki.  I definitely wanted my preservice teachers to be exposed to it, and felt like it was the missing piece of the puzzle.  Or at least better.  In the fall I tried running parallel systems - wow, was that a bad idea.  This past semester I went full out, and did better.  Learned a lot for the fall.

Next time: what I did, how it went, and how I'll adjust.

Sunday, May 29, 2011

Who Are the New Teachers? The Long Story

At our university, content educators are mostly in their respective content departments, which is why we have a dozen or so math educators in our math department.  In our secondary teacher prep, we have three courses that are our "Math Ed" courses: Math 229, which is HS content focused, Math 329 - which is MS content focused, and Ed 331 - which is our content seminar for teacher assisting, when the novice teachers are in schools for the mornings and teaching at least a unit.  We are in negotiations to see them during student teaching, which will be excellent.

This is another guest post from a student assistant: Brock Walsh.  He paused school for a bit, but then came back with a very clear motivation about wanting to be a teacher.  Dave Coffey already posted a bit from him, where he used the NCTM process standards as an outside resources.

In his teacher assistant portfolio, he reused a bit from his 229 class, and I thought it was a neat opportunity to follow a student from early on until later in their teacher education. As an add on, I also included his piece from this past semester on the Conditions of Learning, which Dave recently posted in the Learning Museum.

Equity - Insights from the Past

(The following is a paper that was written for MTH 229, in which I had looked into the principle of equity as I related it to my experience of a nine week observation.)

Articles: Excellence in the high school classroom is something that teachers strive for. Sometimes conducting a learning filled classroom can be easy, but other times a teacher might not fully see and take advantage of teachable moments for all students. Being aware of and preparing for these teachable opportunities for all students to learn at a higher cognitive level is vital and defined under the Equity Principle of the high school principles and standards.

The article “Focusing on students’ Mathematical Thinking” by M. Lynn Breyfogle and Beth A. Herbel-Eisenmann focuses on trying to understand the thought processes of a student’s reasoning instead of relying on a student’s answer. Reasoning occurs when a student has time to think and then explain their thoughts. The time that is given after a question and before an answer is known as “wait time” and within this time, a student’s cognitive thoughts will increase. In the article, the authors emphasize an important detail. They quote from their findings, “Although most teachers are aware of the importance of waiting after they have asked a question, the importance of waiting after a student responds has received less emphasis (Rowe 1986). This all relates to students maximizing their learning by having the time given to them so they can process ideas for themselves.

The article goes further to say that when a student has given a correct answer, we as teachers should question them as to how they arrived at that. Students will often learn the most from themselves or when another student explains their reasoning. Asking for justification is a great way to evaluate not only a student, but the class as a whole when they respond and get involved in the discussion. Putting both “wait time” and “justification” together strongly represents the idea of equity and its importance in class.

The article “Unveiling Student Understanding: The Role of Questioning in Instruction” by Azita Manouchehri and Douglas A. Lapp relates to the Equity Principle directly by emphasizing the point that we as teachers need to ask the right questions for optimizing a lesson. Our questions need to facilitate learning and with the right questions being asked we can pull out conceptual reasoning from the entire class.
Magoo0311 @ Flickr

Personal: The only class in high school that ever truly challenged my reasoning was AP Calculus. Not because it was a hard class, but because the teacher invested so much into our learning and asked questions that forced us to explain ourselves. The mathematics classes that I have taken in college act the same way. The professors ask questions that require my justification. Sometimes I don’t fully know how to justify my answer and that can be blamed on the fact that I never had to do it through grade school. The in-class illustrations from the articles represent teachers asking questions that facilitate class, but do not emphasize reasoning like the classes that I have taken in college.

One specific instance of equity that I can remember my AP Calculus teacher applying was related to group work. The class was split into groups of fours and had to present on asymptotic behavior. Each person in the group had to focus on one specific aspect to present to the class and the groups had to hold each other accountable for their work. I can remember that there was a ton of questioning that occurred which felt like a debate. Within that debate, a lot of reasoning was taking place and uncertainties were being explained! The class as a whole was involved, and that is something special when an entire class is participating in discussion. In general, any time a teacher is at the front of a classroom instructing or going through a worksheet, and maybe only asking questions that a few students answer is a case of poor equity and should be avoided.

Observation: I conducted my observation at a well funded school with nice facilities. I observed a teacher and her freshmen/sophomore Geometry class during sixth period on Tuesdays and Thursdays. The class was primarily of white ethnicity, but there was one black boy and girl, and a hispanic girl. There were 15 females and 12 males in the class. The three learning objectives that I observed were review on algebraic properties, theorems about angles, and the last day was devoted to preparing for an upcoming test.

The questions posed in class were probably 50/50 for being open or closed. I noticed that when a question was presented in open form, there would generally be justification with the response. I compared notes with Susie K. from class to find a comparison between in class questioning. She told me that from her observation at Jenison High School, students were asked open questions about half the time in an algebra class but in the geometry class there were generally more open-ended questions. This makes sense to me as it seems fit that the higher level class should be challenged by the questions they get asked. A teacher should expect that as a student’s cognitive level grows; then they should also be able to reason more in depth. The wait time in the class I observed was generally around 3-4 seconds. This is reasonably good, but like the article proposed, there was really no wait time after a student responded. A good way that the teacher made sure each student in the class would have time before a response was by saying, “Everyone think about the problem by yourselves and then compare with a neighbor.” This way, students have time to learn by themselves and from their peers.

Further identifying the questions asked in class, about 22 percent of them required justification. I consider this to be a relatively adequate amount for a Geometry class, but would be something I would like to see get higher in preparation for more advanced math classes. A good example of an open-ended question that required justification was, “What justification do we get for AB+BC=AC?” I realize this seems obvious but sometimes that is exactly what is needed. She also asked, “If both L1+L2=180 and L2+L3=180, then shouldn’t they equal each other? Explain how you know this.” This question set-up made the students think about the properties and theorems that apply to these statements. These types of questions force students to think about possibilities. When called to answer, then the student can explain their best reasoning for an answer. Justifying yourself will sometimes correlate directly with equity if an explanation is clear and insightful so that the whole class learns from the response to the question. All of this stems from the question though, if a good open-ended question was not asked to begin with, then the opportunity for and learning in general has been lost. A good open-ended question posed in class was, “When I say adjacent angles, can you picture that in your mind?” Another one was, “How do you prove something true? What does it take to accomplish this?”

Interview: I conducted my interview questions by simply asking a few questions after each class to get a general sense of what she expects form her students. About instruction and questions Cristina said, “When I generally ask questions, I expect the students to think before giving a response. I’ll ask for understanding from the entire class and if nobody has a question then we move on. I expect students to ask if they don’t know. For communicating, specifically in Geometry, I expect students to use the correct language and correct theorems/properties. It’s important for students to have this foundation.” For the workload she said, “Homework happens every night, and each student is expected to complete their work or at least give a good attempt towards answering the question. Of course, I want all of my students to do well. It is really up to the student to provide the effort, and I am here to help each individual student as much as possible.”

Outside Resource - Conditions of Learning

One Laptop Per Child @ Flickr
This is my opportunity to share my understanding for the “outside resources” portion of my portfolio. During the exit interview, I was asked to explain my reasoning for using the Process Standards from NCTM, and Cambourne’s Conditions of Learning. Somewhat confused by this inquiry, I responded that I included them because they are both a framework that I feel needs to be implemented in the classroom everyday. These are both a resource that I want to keep a focus on when I teach because when using them, I feel my learners will effectively learn more. I was told that these were not the usual types of resources that are used, but upon my explanation, John and Dave understood my intentions of having them included and commended me for seeing these outside resources as a means for having a framework that benefits me in the classroom. Including them in this portfolio is a way for me to have a constant reminder of them.

Monday, May 23, 2011

Spirograph 1

from Robert S Donovan @ Flickr
I have always loved Spirographs.  When I inherited my sister's I was in heaven - I think the exact box pictured here, supposedly from 1968.

Seeing this tumblr entry with an animated Spirograph picture when I had a few minutes sorta free, inspired me to finally put together a sketch.  So I thought I'd share it.  It's definitely good for experimenting to see if you can figure out what is going on.  Very fun to make.  The webpage lets you see the patterns, but doesn't give nearly the control that the GeoGebra file does.  It succeeded insofar as you can make some pretty pictures, and I didn't have to cheat by parameterizing the curves..

I'm interested in developing some lessons on ratios that could be visualized in interesting ways with the curves traced. If you have feedback on the sketch or on the lesson idea, I'd love to hear it.

The View menu should allow you to refresh views in the applet below, but it can get hung up.  I sometimes have to click pause several times before it pauses.  As I said, on this one I recommend the GeoGebra file, if you're going to do any serious playing around.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Wednesday, May 18, 2011

Algebra's Tiling

I don't think algebra tiles or blocks are a panacea for what ails student algebraists, but I do think they are powerful.  Even though as a mathematician I am pretty comfortable with symbolic reasoning, at heart I am a visual thinker.  Having a way to visualize algebra opens up many possibilities for learners, even when they had a good symbolic understanding beforehand.  The goal for me is not to replace symbolic manipulation, but support the concepts.

I designed the activity for students that may have not done a lot of investigation before.  So it starts with a lot of modeling, and then letting them try.  Students did an amazing job.

Even before doing the mental math (making the point that even number operations can be visualized, plus setting the context for the follow up activity), I asked the students to take a  look at the blocks to see what they noticed. Mr. Boeve had had the students play with the blocks the day before, which is an excellent idea.  They noticed corresponding dimensions, colors, different designs.  These were the tiles from Algebra Lab Gear, so there's 1/2 x blocks, 1/4 x blocks, 5 sticks and 25 sticks.

None of the students had seen the visual multiplication before, but they were willing to give it a try.  They made connections as to why the pieces represented what they did.  The verbal connection, x squared, was biggest, but then a few students recognized the x times x relation.  They picked up the symbol to picture representing quickly, and that gave an opportunity to talk about how there are many different ways to write things in math.  They had 2x+3+1x+2 and 2x +3 +x+2 and 3x+5.  We introduced what mathematicians call simplifying, which they connected to fractions.

I raised the problem of negatives - how could we show negatives, because algebra has a lot of those.  They thought we could have two color blocks, or use some kind of design.  How could we do it with the blocks we have?  Maybe we could separate the positives and negatives.  Nice thinking!

Sometimes I think of mini-lessons like these as equipping the students for problems.  The problem list offered practice, light extension and serious problems.  After 15 minutes to do their choice, which they loved having, we came back together and I asked if there were any they wanted to see me do?  They suggested problems, and if there was a student to explain them, they gave it a try.  One of the themes throughout was "give it a go."  I made sure to ask some students who had incomplete or incorrect thinking so we could talk about that, too.  Because the whole situation was different, it helped with making that safe.

In discussing the subtraction problems, they got to three separate ideas: taking away, zero pairs, and adding the opposite.  The students exploring these ideas were able to give reasons for why it made sense with the blocks.

We just worked with the document camera, but if you want virtual algebra tiles, there's the National Virtual Manipulative Library tiles, a two color version from the Michigan Virtual University, and NCTM's Illuminations Algebra Tiles.  None of them are ideal, but all are serviceable.  It's very possible to make homemade algebra tiles, and there was a good article about that in the Mathematics Teacher: "Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," Annette Ricks Leitze and Nancy A. Kitt, September 2000, Volume 93, Issue 6.

Thanks to Mr. Boeve and his classes for the nice opportunity!  Jill Beauchamp came along for experience with the algebra blocks, and she was a great support to the kids, so thanks to her, too.

Playing With Blocks

Photo credits: Eamonn @ Flickr

Tuesday, May 17, 2011

Who Are the New Teachers?

Guest post today from one of our student teachers from this past semester.  Sarah Cavazos will be student teaching in Fennville, MI this fall and is - to steal a phrase from Sir Ken Robinson - exceptional but not an exception.  She is bright, dedicated and passionate about teaching.  I find that many of our novice teachers have much in common with her.  This past semester I got to see her try a game of her own design in a classroom that had not done much of that, and adjust it on the fly to improve both gameplay and better address learning outcomes.  Then she gave a terrific Little Big at the end of the semester about teaching all students.

Triangle Rummy

Object of Game: Obtain a set containing the picture, the name and the definition of the same triangle.

Triangle Rummy is a game made for two to four players that contains 24 cards (7 pictures, 7 names, 7 definitions and 3 Free cards). There are 7 different triangles in the deck of cards, each demonstrated by a picture, a name and a definition. There are also three Free cards (Free Picture, Free Name and Free Definition).

To start the game, the dealer deals out three cards to each player and places the rest of the deck face down between the tables. The youngest play starts by taking the top card from the deck. In order to keep three cards in their deck, the player must discard one of the cards in their hand face up next to the facedown pile. The player to the left can chose to take the face up card in the discard pile or choose to take a chance and take the next card in the discard pile. In order to win, a player must obtain the same triangle demonstrated by three different cards.

If any player obtains a Free card, they can choose to use that card in place of a triangle name, picture or definition depending on the name of the card. For example, say a player had the picture of an isosceles right triangle and the name of the isosceles right triangle and then obtained the Free Definition card. In order to win the game, the player must say the definition of an isosceles right triangle. If the Free Picture card was played, the player must draw the correct picture as well as have the matching name and definition cards. If the Free Name card was drawn, the player must name the correct triangle and have the matching definition and picture cards.

The Unreachables

Photocredit: qthomasbower @ Flickr

Monday, May 16, 2011

Power Up

A math game for exponents.

A colleague gave me a game created by one of her students Steven Reynolds.  The idea was having the power as an exponent connected with the power of a superhero.  I liked the premise, as I'm a comic fan myself and have a son who's obsessed.  The timing of playing this with the 5th graders was perfect, the day before Free Comic Book Day, the closest thing the superhero community has to a holiday.

The fifth graders had had no experience with exponents before, so this was the introduction.  The game was good ground for getting them talking to each other about things like 3^4, how to calculate it, and noticing some of the properties.  The main confusion was typical: 2^3 = 2x3.  But they went from me reminding them to reminding each other to the vast majority being comfortable with the notation.

The most exciting thing about the game was how much people got into the context and then wanted to play.  Some students that had never gotten this much into a math game were engaged; not just in the creative side of it, but in playing the game, also.  The students asked if they could choose to work more on their origin stories for their writing time later and Mr. Schiller gave them that option.  Two students gave me permission to post their stories, which are after the revised game below. (As usual, good suggestions from the students.)

I launched the game by creating my character on the overhead, at the same time as they were creating theirs.  While people were still working on different steps, I had a few people share what they had so far.  My super hero was the Time Keeper, who got his powers by finding a watch that turned out to belong to Father Time.  He got super strength (base 4), super speed (base 3), and the magic watch (random).  Then I played a game vs Shadow the Hedgehog in front of the whole class in which he pounded me.  The game play is pretty quick, so kids were all over the room playing each other.  Overall, they rated this game a definite keeper.

Download from Google as pdf  I'm grateful to my son, budding comic artist at almost 11 yr.s, for the following illustration of the example from the game sheet. (Click for full size.)

Here's two of the origin stories.

Shadow the Hedgehog by Jim (created by science, doesn't know if he's a hero or villain)
At a space complex ark, he was looking out the window. "Shadow, you've been looking out that window all day," Maria said. Shadow replied, "I just want to know what it's like down there. I've seen pictures, but... I just want to know what it's like."

(3 years later) "Doc, why am I nervous?" asked Shadow curiously. "You will see," Doctor Robotnik said.

A blaring alarm sounded. Doc cried, "Intruders! Shadow, guard Maria with your life!"

Shadow and Maria were running away from the GVN soldiers when one yelled, "Stop or I'll shoot!" Shadow told Maria, "run! I'll try to stop them." One of the soldiers pulled out a pistol, fires, and Shadow yelled, "Maria!"  Shadow causes chaos control with Maria, and she says, "Shadow, before I die... Be friends with the people on earth." Shadow yelled, "NO!!"

The Doc says, "take out the gun soldiers and their leader and see how they feel when their loved one is lost." Shadow said, "yes, sir."

When the Doc was arrested, they did not know about Project Shadow.

The Changer, by Katrina
"Ah, the sweet smell of safari!" I said to the driver.  "Now don't go too far from the jeep, 'cause you'll get bit," the driver said.

I walked for hours.  I could not find the jeep, when all of a sudden a cheetah was face to face with me.  I ran and ran.  I could already see the blood coming out of me.

The cheetah bit me.  I screamed for my life.  A couple of minutes later, I was home in an instant.  I went to my lab, and mixed some potions together to get the cheetah out of me.  But instead, I was on fire.

I could feel the flames.  I felt like I was off the ground.  I looked ..."What the heck," I said, " What is this, a joke?" Something in my head told me it wasn't.  "Now I know what to do," I said with an evil laugh.  I lifted up my car with no trouble.  Then I knew that I was now a super hero with super-strength, super-speed and firepower.  I help fight crime.  I help police catch villains.

And my name is the CHANGER.

Photo credits: The Tick I have no rights to, but he is my favorite. Legal use, Fern R and Spencer77 @ Flickr.

Wednesday, May 4, 2011

Triangle Mosaic

Holy cow, have I been busy.  Sorry for the lack of new posts.  What makes it worse is that I have had several guest posts to get up that students were kind enough to send me weeks ago.  In addition to scads of things that I want to write up for myself!

The first is from a preservice secondary teacher named Jill Beauchamp.  She is active in coaching cheer, and in our local Dutch culture.  (And it's almost tulip time.)  I'm pretty sure she's a licensed wooden shoe dancer.

On an assignment that gave a choice of follow up options after playing with Pierre Van Hiele's mosaic puzzle in class (from “Begin with Play,” by Pierre van Hiele, Teaching Children Mathematics, Feb 1999), Jill chose to make a activity based on my triangle puzzle.  And she was willing to share it!  I like how she really captured Van Hiele's idea of beginning with play, and uses the puzzles to get at the triangle properties.  She makes the most of what I was designing the puzzle to do, have one triangle of each type.

The assignment:
Teaching Math – Mosaic Making

Choose one or more of the following to do for this:
  1. Analyze Van Hiele’s mosaic. What geometric properties of the pieces permit all the combinations we saw in class?
  2. Create your own mosaic puzzle and document your design process.
  3. Create a new lesson using PvH’s mosaic or my 7 triangle mosaic at http://mathhombre.blogspot.com/2010/12/triangle-puzzle.html
Document your work, and be sure to include a reflection.

Schema: I decided to take a look at your 7 triangle mosaic – nice work! This would be difficult for me to create on the computer, so I am very impressed. When first thinking about a lesson in regards to the mosaic, I could only consider it being a fun puzzle. With our exposure in class to different workshops regarding the original mosaic, I began to think about the properties each triangle in your mosaic possessed. You have:
  • -Two right triangles
  • -One isosceles triangles
  • -One right isosceles triangle
  • -One equilateral triangle
  • -Two scalene triangles (One acute and one obtuse)
Fabulous! You have one example of everything.

Focus: With my class, I would want to explore why these triangles fit together the way they do. Assuming the students have not yet learned about triangles, this could be used as an introduction. Let’s say I have this class for 60 min. Here’s how my day would go.

Lesson: Properties of Triangles

Introduction: (5 min) Talk about puzzles!
  • What kinds of puzzles do students like to do?
  • What makes a puzzle puzzling?
  • What are some mathematical properties of puzzles?

Introduce Mosaic

Mosaic Play and Record: (15 min) Allow students to play with the pieces and try to create the mosaic. As they do this, I would like them to document their actions:
  1. What they tried
  2. What pieces worked together?
  3. What didn’t work together?
  4. Qualities they notice about the triangles
**If students solve the mosaic, they should focus on:
  1. Is there another way to solve it?
  2. Why do some types of triangles fit together and others don’t?
Discussion: (20 min) I would ask all students to pull apart their mosaics and separate the individual triangles. Then I would ask them if they saw any similarities between any of the triangles?

*As this is happening I will write up student ideas on the board. If need be, they may come up to the board and illustrate their thinking.

Assuming they already know terminology for a line, angle, point etc. I will have students pull out the rulers and protractors to assist them in drawing more comparisons. Once we have a pool of properties, we can begin to group the triangles accordingly. Once we are able to do this accordingly by the deduced properties, I will write the names of the triangles on the board (but not yet with their corresponding group). Instead I will ask students what they think goes with each.

Properties: (With any luck, we get some or all of the following, although I’m sure I’ll get some other interesting thoughts!)
  • 3 equal sides
  • 2 equal sides
  • No equal sides
  • 3 equal angles
  • 2 equal angles
  • No equal angles
  • Right angle
  • Obtuse angles
  • Acute angles
Hopefully, they will see comparisons between the word “Equilateral” and the same angle and side measures, “Right” and the triangles with 90 degree, or right angles, “Scalene” and the triangles that depend on their individual scale/measure, although “Isosceles” doesn’t work too well, but it can be the odd guy out.

I will want to pay special attention to that sneaky little purple “Right Isosceles Triangle.” This guy is important because he shows that two properties can hold for one triangle. Maybe we could explore which properties can hold together and which ones don’t (As I’m writing this these ideas are just kind of coming…)
  • A scalene can be a right triangle. Why? Because one angle may be 90 degrees, the other two differing, and all sides of different lengths. A scalene cannot be isosceles or equilateral because it goes against the definition of scalene.
  • An isosceles triangle can also be right, but can an equilateral triangle also be isosceles? No, the definition of isosceles is EXACTLY two sides of equal length. Although it can be either acute or obtuse depending on the size of the angles
  • A right triangle can then be isosceles or scalene. It cannot be equilateral because one angle must be 90 degrees, thus going against the fact that all angles in an equilateral triangle must be 60 degrees.
Teacher Question: So then, are triangles actually right triangles? Or does the word “right” just classify a specific type of isosceles or scalene triangle? A right triangle cannot exist outside of one of the two classifications.

Sorry for my tangent. The above discussion over the “right isosceles triangle” may be something for another day! My hope would be to get to the last part of my lesson…

Discovery: (15 min) The students would then need to reassemble the mosaic (I will show them the put together puzzle if they need it). With their protractors and rulers I would like them to work on:

Measuring the divided angles in the corners of the square. What is the sum of these angles? What type of triangles have an angle like this?

Measuring the divided angles along a straight line within a puzzle. What is the sum of these angles? What do they notice about all of these sums along a straight line? How does this compare to the sum of the angles within a triangle?

Lastly, I would like them to paste their mosaic together on a piece of paper and write out the angle measures, side lengths, and classification for each triangle. Students should make a note of anything else they notice.

Reflection: (last 5 min of class) What is one realization that surprised them today? Can they put anything they’ve seen into another context? How might it relate to something else?

MY Reflection: Wow, This was wonderful. I had the initial idea for the lesson because I thought it was so cool how the angle measures across a straight line will add up to 180 degrees. A simple concept, but it helped a lot of things make more sense when I recognized it. I think a lot of times we have this subconscious knowledge that we utilize everyday but don’t fully recognize. Once I started planning out how I would eventually get to a measuring activity, ideas just lead into one another, making this a lot longer lesson that I intended. There is no way I would get through the discovery part in 15 min! For me, this order of events seemed to make the concept clear. Perhaps it should be a day and a half sort of lesson?

Do you have any feedback for Jill or I about the lesson?  What would you try?

Photo credits: Jill Beauchamp, quinn.anya and bjornmeansbear @ Flickr