Art & Geometry

I recently was fortunate enough to attend the NCTM conference in Boston and while I was there, I checked out a really cool art museum in Providence, RI.  I took these two pictures while I was there because I thought that they could connect to what we are learning about in our polygons unit.

When showing the pictures to my students, I projected each one and had them brainstorm some questions they were curious to know about.  Of course, I encouraged them to be mathematically based questions, but I did still get, "What color is it?"  Check out some of the questions that they came up with:

How many different squares do you see in the picture?

What is the scale factor used if you dilate the center square to the outer square?  What about from the outer square to the center square?

How many lines are there?

I asked my students to explore 2 different questions.  First, I said, "If I asked you to determine the number of 'spokes' in the picture, how might you do it?"  Groups brainstormed ideas for how they might tackle the problem.

The general response was that students would break the picture down into smaller parts, count the number of spokes in that section and then multiply by the total number of those parts.

Once we had shared some ideas of how you might determine the number of spokes, I asked them to actually figure it out.  They came up with 128 spokes.

The second question I had students explore was what would be the degree measure of the angle created by two adjacent spokes?  

This was a little trickier, not because of the calculation they used (360/128), but they had a hard time defending their use of 360.  Was it the Interior Angle Sum Theorem?  The Exterior Angle Sum Theorem?  Or the fact that there are 360 degrees around the center point where the vertex of the angle is?

This was the second picture I shared with them.  I really like this because it's deceiving.  Is it a hexagon?  An octagon?  Neither...more of a hybrid of the two to form a heptagon.  I find the inscribed circle to be really fun, too!

The question that I posed to the students was to find the measure of each of the seven interior angles. Lots of students found the sum of the interior angles (900 degrees) and then divided by 7, but soon realized that that would not work because not all angles are equal in measure.  So, they revised and were able to make a lot of progress.

I find questions/problems like these to be so much more interesting than the way that I've taught this unit in the past.  I feel like polygons have so much potential for interesting explorations, and I was happy to find some of those in art!

What mathematical questions do you have about these pieces? 

Breaking Bad...Definitions

I posted a picture on Twitter the other day (see below) that included the 8 SMPs in "Kid Speak".  Ashli Black (@Mythagon) pushed back on #6, saying that precision is more than just numerical accuracy, but also precision of language.  I could not agree more!

I have found that necessitating the precision of language is fairly easy in my classroom because my students do a ton of talking and writing.  But I think the major difference is that they share their writing and verbal thoughts with their partners and the whole class, so I'm not the only audience member, which seems to be more important to the kids.  (Remember, these are teenagers.  They care very much about what their peers think.)

We were getting into our unit on polygons and I wanted to try out an activity that my colleague Dr. Patrick Callahan had shown me a few years ago.  I was actually attending a PD session at UC Davis where he was presenting and he asked all of us (a room full of math educators) to write down our definition of the word "polygon".  I've adapted the activity slightly and I used it with my students just the other day.

We started with this handout:

The instructions read:

"The purpose of this activity is to work toward a mathematically viable definition of the term “polygon”.  To begin, write a first draft of your definition.  Then, exchange papers with a peer.  When you read your peer’s definition, consider if you can “break” it.  To break a definition, draw a picture that does satisfy their definition, but is not a polygon.  We will have a mathematically precise definition of polygon when we cannot draw a picture to break that definition.

            For example, if we were trying to define the term “vegetable”, my first draft might be: A vegetable is a food.  When I exchange papers with my partner, they might break my definition by writing the word “chicken”.  This breaks my definition because it does satisfy my proposed definition, but it is not a vegetable."

I should say that I was essentially looking for the following elements in an "unbreakable" definition of polygon:

• Closed
• 2-Dimensional
• Straight sides
• At least 3 sides/More than 2 sides
• Sides that intersect only at endpoints

I asked the students to begin writing down their initial ideas of what a polygon is.  Many got right to it, others stalled to wait and see what others wrote, and some got flustered, panicked, and asked me "What if I have no idea?"  I reassured them that we were not going for the right answer at this stage and I just wanted them to put down ANYTHING that they knew about the word.  This seemed to help a bit.

I walked around and saw a variety of first drafts:

polygon - a shape

A polygon is a shape or a figure

A closed shape with more than 2 sides

A polygon is a closed shape with no curved edges.

A shape with more than one side

A shape with 6 sides

 As papers started to move around for definitions to be broken, I found that this part was really challenging for some students.  "How am I supposed to break their definition if I don't even know what a polygon is?"

I grabbed a sample paper (see below) and put it under the document camera.  The first draft read: "polygon - a shape".  I asked the students how we might break this definition of polygon and one student said, "You could draw a circle.  Because even though it is a shape, it is not a polygon."  This started a lot of conversation at each table and students were brainstorming ways that they could break the definition on the paper in front of them.

(as mentioned above)

Here are some ways that students broke the first drafts:

This student was calling out the fact that sides could not be curved

This student was addressing the fact that a polygon could actually have less than 5 sides

Now, not all students really got the way that the breaking process was working.  Here are some examples of pictures that did not actually break the proposed definition:

"A polygon is a shape that has multiple sides and is not round" is not actually broken by a circle

"A polygon is a geometric figure".  And so is the shape drawn in the box.

It was at this stage that I realized that my description of "breaking" a definition was not precise enough.  In my vegetable example, I had gone in the direction of not being specific enough in my definition, or as Dr. Callahan would say, I've "included too much".  I hadn't considered the other direction of not including enough.  In the example above where the student broke the definition by drawing a trapezoid, the original definition had left out figures with 3 and 4 sides, which had excluded too much.  This was an opportunity to revise my own definition of "breaking".  Obviously I always love a great opportunity to model what I am asking my students to do!

Also at this point, students got their papers back and started revising.  You can see that the precision of language was called out without me really having to say too much.  There was a sense of competition that students had with one another because they wanted to come up with an "unbreakable" definition before anyone else did.  Take a look at their second drafts:

(Please note that not all of these are great, but you can see that students were attending to the precision of their language, regardless of whether or not the picture actually broke their previous draft.  Many students relied on the examples show to the whole class under the document camera or other papers that they saw to revise their definitions.)

From this point on, I didn't need to do a lot of modeling to the whole class, they were switching papers back and forth on their own.  Students would call me over if they thought they had an unbreakable definition and I might tell them that there was still a way to break it, or I might pose it to the whole class to decide.  Could they break it?

Here are a couple examples of further revisions:

Also, after a few rounds of revision, most students were good with their definitions including closed figure, straight sides, and at least 3 sides.  Now, it was time for me to push them on the piece that could describe the requirement of sides only intersecting at endpoints.

I drew these images on the board:

They knew that the first picture was unacceptable because it wasn't closed.  They were fine with the middle image.  But I told them that I could break a lot of their existing definitions with the third picture.

A lot of students said, "Sides can't intersect!!!" would be their revision.  This is when I pushed back with, "Well, you can't have both.  You can't say that it's a closed figure AND that sides don't intersect."

Silence.

I asked them to brainstorm with their group how they might describe the difference between the middle picture and the one on the right.  What made one acceptable, but not the other?  Check out what they came up with:

She told me later that she meant "vertex" not "vertices"

This student actually came up with this language before I addressed it with the class

This is one of my top performing students.  He had most of it down from the beginning.

This last definition is one that the group came up with together

You can see this student's process and her final product below

See above

I loved this idea of the vertical angles connection

At the end of the day, of course, I was wanting my students to have a solid understanding of what the term "polygon" means, but this activity allowed for so much more than that.  I know that I mentioned the importance of the precision of language, but another layer of this is the process of determining if something "belongs" or not.  Furthermore, if I think it should belong, but my definition doesn't allow for it, I need to revise my definition.  Or, if I don't think it belongs and my definition does allow for it, again, revision is needed.  Dr. Callahan really made this clear for me when I debriefed this activity with him.  It was something that I had not really considered, but was so powerful for students.

And just to wrap it all up, I had to share Dr. Callahan's creative side...

You're welcome.

The Mathy Murk: An Impact Survey

Hello Colleagues!

I'm wanting to gather some information about what impact my blog has on teaching practices.  If you have 4 minutes, I would appreciate hearing from you!  Thank you in advance for your feedback!

Please choose ONE of the versions below.  You only need to answer once, as the questions are exactly the same.  The only reason I created a Survey Monkey version is because I know that Google is blocked in some parts of the world. :)

Thanks!

Google Forms version: http://bit.ly/mathymurksurvey

Survey Monkey version: https://www.surveymonkey.com/s/59WVQJJ

One Hot Topic: Law of Sines & Cosines

When I work with high school math teachers in my consulting life, I am always surprised by how often the topic of Law of Sines and Law of Cosines gets brought up.  Now, this may seem random, but it is often in the context of what content standards we must teach students, yet we don't have time for.  However, with the adoption of Common Core State Standards, you may notice that these are actually now "Plus Standards", which means that they will not actually be assessed.  So, my take away from this feeling of panic from teachers, is that they are finding the transition  to Common Core difficult, they are unsure of what they still need to teach from the previous standards, and what is it exactly that they can now take off of their plates.

http://www.corestandards.org/Math/Content/HSG/SRT/

This always sparks great discussion because the bigger conversation, in my opinion, is: "Is it really that important that we teach students a formula to memorize?"  Law of Sines and Cosines are not the only ideas or concepts that fall into this category, and I was guilty for many years of spending my classroom time lecturing students on topics that involved some nasty formula to memorize.  However, I have made a huge effort to change this culture in my classroom.

If you haven't seen Diana Laufenberg's TED Talk, you need to!  (Find it here!)  I love the way she talks about the role of school changing as technology becomes more available to our students and they no longer need to rely on their teacher for all the information; they can find it elsewhere.  I am not suggesting that there is no longer a place for the teacher in a child's education, it just looks different than it used to.

So, I tried something a little different this year.  My students had just returned from a week long spring break and before the vacation we had finished up Right Triangle Trigonometry.  So, as a review, I drew this picture on the board and asked them to find all of the missing parts:

The context was: you have a 12 ft ladder that is propped up against the side of your house and the bottom of the ladder is 5 ft away from your house.

Students proficiently used their trig tables that we had generated together (I blogged about that HERE) to accurately solve for the missing pieces.  All spirits were high and I was pleased to see that students comfortably fell back into the swing of school after a week away.

Then I threw them a curve ball...

Same situation, you've still got a 12 ft ladder that is 5 ft away from the base of the building, however, instead of this building being your house, it's the Leaning Tower of Pisa.  Students realized immediately that this was no longer a right triangle and they were curious to know how they might solve it.  "Can we use trig?  Can we use the Pythagorean Theorem?"  I told them that the angle in the bottom left was 60 degrees and that it was indeed NOT a right triangle, so our "traditional" methods for solving were unavailable.

I gave them a quick pep talk that essentially summarized Diana's TED Talk and let them know that my purpose for giving them this problem was to see what they could discover and research on their own.  I did not want them to come back with the values of the missing side and angles, instead, I wanted them to explore possible methods for how they might find the answers.  Also, I told them that I made up this problem, so don't bother Googling it directly.

My hopes were for them to think about:

• What makes this problem different/the same as the first one?  
• What question should I Google/ask?
• How do I decide if what I've found is valid?
• Is this method that I've found really applicable/appropriate?

I'll admit, I had no idea how the process would go and was mentally preparing myself for this to be a total flop.  But when they came back to class, my mind was blown...

When students returned to class, I reminded them that I didn't expect them to have found the answers for the missing values.  I wanted them to come to class with an idea of how to solve for the missing values and have a discussion about these ideas with their peers.

Within seconds of discussion I was hearing students say, "I found this thing called the Law of Sines that I think we can use here."  Another student said, "I found that, too!"  One student said, "I found the Law of Sines, but I also found the Law of Cosines and I think that is the one we should use based on the information given."

Are you frickin' kidding me?!?!

Now, not all students had this information.  Some hadn't looked at it at all, others used the Pythagorean theorem and right triangle trig properties, but there were a good handful of students who were on the right path and had hit the jackpot!  I took a quick lap around the room and was able to capture some images of what students had written down on their papers.  Check it out:

Trying to use some of our more "traditional" methods for solving non-right triangles

I almost fell over when I saw this written down.  Imagine how I felt when she started explaining it to her peers.

Note: when I came by to take the picture, she said, "Don't mind the raspberry juice that I spilled on my paper."

Perhaps she's onto something :)

This student admitted that she had found the formula on Wolfram Alpha and got stuck

I feel like the question mark says something about her understanding of the concept,

but she was able to plug in values correctly

Admittedly, students really were unsure about what the Law of Sines and Law of Cosines was all about, but with a little nudging, I was able to get them plugging things in, refining their research, and persevering in their problem solving.  It was fun to bounce back and forth between groups of students to help push them further in the problem or answer some questions that had arisen since my last visit. It was also exciting to hear them argue with each other and ask such thoughtful questions of their peers.

But the most beautiful part was that students were sincerely interested in finding the values.  The opportunity for them to take ownership of their learning and for them to discover what was important played a major role in their willingness to try.

My big ah-ha moment here was that students really are capable of finding things on their own, but it is still important for the teacher to be involved.  My students still needed me to answer specific questions and to help explain things along the way, but THEY were a huge part of that conversation and process, which was magical and something that I hope they will remember.  I know I certainly will.

I told them that my goal was for them to take away some bigger skills from this activity than just remembering a formula.  I admitted to them that when I was a math major in college, it was essential that I knew how to work with others and ask appropriate questions when solving a problem.  Without these skills, I would have sunk like a stone in many of my upper division math classes!

After about an hour of working on this problem, I told them that they could move on to the next activity that I had planned for the day.  The only groups that moved on were those who had found the missing side and angles of the triangle.  Other groups persevered and continued working.  I brought to their attention that they had just spent an hour working on a challenging math problem and for them to reflect on this feeling of wanting to finish.  I asked them to contrast this experience with approaching a challenging problem in their math classes of the past.  Many students nodded and smiled; they knew that this was something special.

One student said to me, "Yeah, I've never spent this much time on one single math problem and actually enjoyed it.  What have you done to my brain, Ms. Balli?"

I see it as this: I've helped them learn how to use their own brain and not rely on mine.  Isn't that what we want for all of our students?

International Consulting

I had the incredible opportunity to provide some math education professional development for an amazing group of K-12 teachers in China last week!  Having lived in Asia for most of my life, I felt right at home, surrounded by my international people. :) 

My colleague, Patrick Callahan, and I presented on Common Core Mathematics (an overview), the implications for international schools & students, mathematical modeling, as well as some teaching strategies that can help increase communication and critical thinking among students.

It was an incredible trip for many reasons, but one of my favorite parts was getting to read about the PD experience through the eyes of one of the teacher participants.  Evan Weinberg is a high school math teacher in Hangzhou, China who blogged about his take-aways from the day here.  Make sure to check out all the cool things that he's blogging about!  He's an extremely passionate and innovative educator who has a ton of great ideas to share!  His blog is: http://evanweinberg.com

I hope that this will be the first of many international consulting experiences that I can be a part of.  The teachers were fabulous, and of course getting to see the world while doing what you love is a true gift!  I can't wait to go back!