**Opening Pandora’s Trig Box – Unit Circle Socratic Seminar**

This was my most exciting project this semester. I have implemented both formal and informal
Socratic Seminars in the past, but this time I decided to go out on a limb by
using it to introduce a new concept. In
fact, I used it to introduce one of the most important concepts in trigonometry –
the Unit Circle.

One of my past mentors in college always said I was a “take
the bull by the horns” type of person, and that personality in my surely came
through on this one. For weeks before I
oscillated between the “are you nuts Tara?” and the “I am so excited about
this” thoughts on a daily basis; during the school day sometimes on an hourly
basis.

I have taught the Unit Circle at least a couple of times before,
but it was always flat, and the beauty of this concept deserved much more. This time around, I was determined to be as
creative and student-driven with it as I could be. It was a huge risk, I was playing with their
trig foundation, but I was committed to giving them an experience learning it
that was fun, exploratory, and long lasting.
It worked, it was an amazing experience for me to create this activity
and then watch as they put smaller ideas together with oohs and aahs throughout
the seminar; to watch them derive and form the single most important concept
for use and application in trigonometry.
Did my colleagues raise their eyebrows at me when I talked about what I
was going to do – of course! Just as
they had done when I began Socratic Seminars at my old school. That just fueled my intentions as always!

**The Journey to Pandora’s Box**

I hyped it up – of course I did, I am a math nut that
way. I kept telling them we would be
opening Pandora’s trig box for key concepts that contribute to a very important trig foundation.
They kept arguing that Pandora’s box ended up containing evil. I got cheesy with it telling them that some
evil is necessary and can transition to beauty if you can get past the
uncomfortable mystery of it. Some
students had friends in Accelerated Pre-Calculus, one level above honors
pre-calculus, and they had already heard about the Unit Circle and that it was
painful. They would ask when we were
going to learn it, and I told them they were going to discover it and develop
it themselves. They looked at me in
shock, I told them I wanted them to find the beauty and patterns in it, so
their experience was not painful. I told
them to trust me, and that they did.

The ideas were swimming around in my head, I knew I wanted
them to take parts and put them together like a puzzle. I started developing the activities I would
give them to draw out their knowledge and start making connections. The following are pictures and explanations
of each.

**“Angle Order”**– This is where I integrated the ideas behind Clothesline math, but without the string. They way in which they received the assignment made it too hard to get a string with cards going – you will see as this story develops. Still, I had them order radian measure from least to greatest in and around the 0˚, 90˚, 180˚, and 360˚ values, which were the only angles they knew the radian measures to at this point without conversion. I did not tell them to place the radian angles they already knew, the intent was for them to use that as a basis on their own. I would later find out that many did just that, and that certainly was evident in the Socratic discussion.

**“Radian Reference Angles”**– This had 2 intentions, to bring back the idea of reference angles and give more practice with them, and then also to allow students a visual follow-up to the order of radian angles around the unit circle. I could not even hazard a proper estimate of how many times the term “reference angle” was used in Socratic discussion when it had been a less than popular topic when introduced before radians and the unit circle.

**A Scattering of Previous Ideas**– These 4 cards contained ideas that would enhance work with reference angles, and “terminal points” was an early on concept that I had not given much air time to in the days leading up to the seminar. It was the concept they needed for the coordinate values on the circle, but that was for them to put together. They did just that during the seminar, but probably not with the prep assignment when prompted. That was OK, that was just what I wanted them to do – define it in their own words, and then apply it to break away into unit circle values.

**Which One Does Not Belong?**– I could not resist! There is a beautiful visual pattern that develops as the Unit Circle develops, and I wanted that to be somewhere in their minds before the seminar without knowing why it was in their minds. That was the discussion point right before the seminar formally started – there were many different choices among them for which one did not belong to the subject matter at hand, and they were all arguing their points. This is the entire nature of what “Which One Does Not Belong” is intended for. There are no wrong answers rather an answer that may “fit” better in a specific situation or parameter. This was one of the most fun pieces I put into the mix!

**“What is the Connection Here?”**– We had reviewed their past right triangle trigonometry in the week before the seminar. We had also talked about quadrants and the positivity and negativity of x and y in the coordinate plane as well as axes coordinate points. They had not seen the circle on the plane yet in our discussions, and I had not used a hypotenuse of 1 intentionally in the right triangle trig review. We had discussed and used Pythagorean theorem both in exact and approximate values, used the trig ratios of sine, cosine, and tangent only in exact values, but not with hypotenuse 1. I saved that for this connection; I wanted to see if they could start to integrate the idea of hypotenuse of 1 can connect into circle of radius 1.

**Picking their Brains: “Points to Ponder”**- This was a guide sheet to go along with the other activities I developed and prompt thought and connections of those ideas. I reinforced A LOT when they were given this activity that I wanted them to explore with their thoughts and own individual answers and that at this stage of the game, there were no wrong answers; only their own understanding was required and also sacred.

The files for the above activities can be downloaded here: Pandora's Trig Box Activities

**The Implementation of and Opening Pandora’s Box.**

First of all, I wanted their pre-seminar assignment of my
activities to be different than they had ever experienced in math. I also wanted it to be distributed in a way
that reflected the suspense of the Pandora concept. For a few weeks, I had my eye on the treasure
box in one of our workroom suites that was covered in old-world maps. I did not have to convince the secretary in
that suite very much to let me borrow it – she said I could have it. Although I told her I would return it, it
still sits in my room by my desk because the whole project was a true treasure
in our classroom, and I am not ready to let go of it yet!

The two weeks before the seminar, I began prepping the
activities. I copied them onto color
copy paper, and double copied many of them onto on side to make the papers they
would work with smaller pieces of paper.
I then folded each activity individually in different ways, the “Which
One Does Not Belong” activity was in the form of a scroll, and a couple of the
activities were folded into the ways we used to fold notes that we passed back
in forth in junior high back in my day.
The funny thing with that was they thought it was some sort of exquisite
origami paper folding art rather than what I knew it to be. Perhaps it was, the generational gap between
me and them in the age of cell phones for communication kept me from realizing
that they would not see it the same way as my memories did. Such a cool and unexpected element of the
experience!

I placed each folded activity into a sandwich bag, and I then placed all the bags (90 in all) into the treasure chest I scored from the copy room. Was that a lot of work-yes it was. Was it worth it – a million times over when watching their reactions to it. I brought the treasure chest to school, and the seminar was scheduled for Monday, October 16

^{th}, so I gave them their seminar prep-work homework in sandwich bags on the Friday before.
Seminar prep - homework bags for students and tucking them into Pandora's Trig Box:

Students opening Pandora's Trig Box and finding the clues for the seminar prep work:

**The Unit Circle Seminar**

The day of the seminar arrived. I had assigned roles to all students; they
did not know before that day if they would be in the discussion circle, the writeboard writer's circle, or the observe and record circle. Because they participate in Socratic Seminars
in their ELA and Social Studies courses, they knew what each role entailed, and
I also provided a 10-minute Q & A session on the structure of a math
Socratic the previous Friday before opening Pandora’s Box. I assigned students into roles based of data
that I had collected from the previous quiz, and characteristics in my students
that I had discovered thus far into the year.

Classroom set -up for seminar:

**Data from Previous Quiz:**I gave a Likert Scale response question on the previous quiz (not for points on the quiz), which asked them to rate from 1 – 5 any previous knowledge they had of the unit circle. The leaders of the discussion group were chosen from those reporting a 5 rating on the question, and the discussion and writing groups were chosen in a mixture of a few 3-4 ratings and more 1-2 ratings. I wanted some knowledge to be in each group to keep discussion transitioning, but I also wanted a fair number of students who were in pure discovery and connection mode.

**Student Characteristics:**I put students who were shyer in the writer’s circle, so they could still contribute their ideas and knowledge via whiteboard, and so they would not be so stressed by the pressure of speaking in the spotlight of the discussion group. As with any classroom, there are very eager and verbal students that like to answer all the questions if they could, and for this activity, they were assigned the role of observe and record. It is true that the leader of the discussion group has the role of managing the discussion to avoid participants monopolizing the discussion, but I did not want to place that stress on them the first time we had a seminar in our classroom. It is not that I do not value the eager students’ knowledge, but I knew that they would share their knowledge in their , and know how to effectively transcribe the contributions of others.

**Resources used during the seminar were as follows:**

- A projected image of a Unit Circle Template on the whiteboard
- Each student had a paper copy of the template to reference at their desks and their bag of seminar-prep homework; nothing else was allowed on their desks.
- The discussion group (inner circle) and
whiteboard writing group (2
^{nd}-ring circle) were not allowed pencils that could stifle group discussion. - The third ring circle was comprised of students assigned to observe and record the contributions of the discussion group. Each member of this circle was given a blank unit circle template, allowed to use pencils or pens, and was assigned one member of the discussion group to follow.
- My role was to record the contributions of the whiteboard writing circle, and provide prompts if needed.

A more detailed explanation of each of the roles in the
seminar along with the evaluation tools I used can be found on a previous blog
I wrote: Socratic Seminars in the Math Classroom - Why Not?

**Summary**

As I stated in the beginning of this blog, I was so nervous
to try this and knew there was a risk of it not serving the purpose
intended. I opened the seminar by projecting a blank unit circle
template onto the white board and saying “There are 64 Important Facts to Know Within
this Image, You May Now Begin the Journey to Finding Them!”

My fears were quickly alleviated within the first few
minutes of the first seminar and transformed into amazement. The interpretations of the prep-work I gave them
came flowing out of them as they worked to put all the elements of the unit
circle together. There were two students
assigned to scribe the information onto a unit circle template, and they found
challenge if keeping up with scribing the flow of ideas coming out of
discussion. There was so much use and
proficientcy of vocabulary that I did not realize they had in their
discussions. Below are a few of my
favorite quotes from students:

- “What are we supposed to fill in for the
coordinate points? Oh wait, are
*those*the terminal points?” - “I think we get the terminal points by referencing the angles to form triangles and find the sides”
- “I thought ordering the angles in radians was easy because I know that sixths are smaller than fourths and fourths are smaller than thirds.” (This was my all-time favorite contribution – evidence of pure number talk!)
- “I think this one does not belong because it has pi only in the circle, and everything else we learned before with degrees”
- “Does this mean the x-value is always the adjacent side and the y-value is always the opposite side?”
- “How does tangent tie into this – is it x over y?”
- “Oh wait – it is this one that does not belong here because it is more complex and what we had to figure out”

I know there are many more, but I just cannot remember them
all, nor could I get them all down on paper as I observed and recorded the process. The worry I had about executing
this topic in a Socratic Seminar was quickly canceled out by the amazing
discovery and connection of previous ideas they had and collective efforts they
displayed in putting together the puzzle that is the Unit Circle. Each of the three periods of students had
their own way of bringing the development of the Unit Circle to
completion. One of the classes
experienced more roadblocks than others needing more prompting from me, but my
prompts were only in the form of further questions. In the end, this class overcame the
roadblocks and got everything tied together minus the idea of tangent
values. Though they did not take it as
far as the other two classes, the ability to push through the challenge and
uncomfortable spots gives them a different type of strength with just as much
value.

Here is a quick video recording of the start of one of the
seminars. The teacher I am mentoring
recorded the whole thing, but it gets cut down in the link-up:

Seminar Video:

The message I wish to share in all of this is to run with
ideas you are truly passionate about no matter how risky they may seem; the
passion will carry you through it and help you to make it work. Do not worry about raised eyebrows and those who question rather, tell them you will let them know. If you
truly believe in it, you will develop an unforgettable experience for your
students with sustainable learning that you will then be able to share with
others!

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