Category Archives: Transformation

Whoa. Like Whoa. My #TMC14 Transformation

Upon returning from Twitter Math Camp 2014 I felt like this:

I didn’t really know how to process it.  I couldn’t compare my experience to another TMC because this was my first.  As the weeks approached, I felt a little like a the slow crescendo.

RollercoasterClimb

Climbing…..Climbing….

I arrived late with John Stevens, Mrs. Stevens, and Sadie Estrella late the night before the big event.  We had great conversation and got to sleep a few hours after midnight, just a few hours before needing to be up for the big show at Jenks High School.  Around 7am, we start seeing faces.  For some this is a long awaited reunion, and for others this is a first encounter.  No matter the previous experience, everyone seemed to be feeling like this as twitter handles turned into real life:

Diving In

In some ways I didn’t truly understand what was happening around me.  I’ve felt like the outlier in my incessant passion for math and learning.  Our backgrounds were varied, the common ground of interests kept us bouncing from one conversation to the next.

Then we get to the facility.

 

Again. Whoa.

It just kept escalating.  Then start our morning sessions.  This was a pleasant twist on conference workshops.  Being able to meet for a few hours each morning over multiple days allowed us to truly develop a deeper insight into some specific math content, and package that understanding into something we could take home with us.  It’s not really possible to do this at another conference.  TMC keeps a good balance of meeting the masses, while supporting conversations in smaller communities.  This is what makes it so special.  I have heard some worries about TMC losing value as it grows, but as long as we have groups like the morning sessions in which we can have the intimacy and depth of relationship TMC will still maintain its appeal for me.

Then I get to be in a session with Pershan on complex numbers and geometric rotations.  His craft as a teacher is what impressed me most.  He let us work through some material in small groups, and guided us in the classic, “I’m going to pretend like I don’t know where this will end up.”  Then another whoa.

Every night I would fall asleep exhausted from the constant mind blowing experiences.  My awesome roommate, Chris Shore, was able to help me remember this experience.

 

Occasionally I may have had a moment where the rise and fall seemed less impacting relative to other extreme moments at the conference,

Little bumps can still turn your stomach

but the bumps kept coming.

 

Then I return home, work for a couple of days and then off to more conferences.  Only now is TMC truly starting to settle for me.  This conference has transformed me, and only now am I able to process the experience.  As this rollercoaster feeling diminishes, I’m seeing how the conference is having direct effect in my professional and personal life.  Only 350ish days until we do this again.

to be continued…..

(I plan to follow up this post with another soon on how I plan to incorporate TMC into my work and that of my colleagues as well.)

 

 

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Sum it Up, Angle Edition: Part 2

There is a little bit of mystery and magic to these relationships, if you don’t believe me just ask Mr. Vaudrey.  Students trust that a triangle is simple, yet if you asked them to communicate anything beyond the magical balance of 3 angles, and 3 sides, most wouldn’t know what else is true.  Sometimes students see triangles as  snowflakes, each one of them unique.  Little do they know how much all these triangles are alike.

The Hook:

  1. Get quarter sheets of graph paper
  2. Draw your own unique triangle
  3. Color in the angles in each corner
  4. Cut Out the triangle
  5. Tear off each corner
  6. Piece together the puzzle, and what do you see?

Playtime:

Practice:

Describe in your own words what’s happening.

This is HUGE.  Students need time to digest this transformation.  If it feels like the engine is stalling, change gears:

Start practicing lo-tech with some paper examples.

TriangleAngleSumWS

Recursive Reflection

Constantly bring back the triangle with the transforming corners.  Have the students take some

TriangleAngleSumsColoring

Then go back again to the applet.  Get the class to a point where students are articulating what is happening in the triangle.  Have them say it in multiple ways.  These angle relationships have patterns and consistencies, but often get lost in the multiple perspectives (what about this corner, or that one, or the inside, or the out, or what does a parallel line have anything to do with it).  If students can transform a triangle and its angles, then adding in the relationships with parallel lines is only a half step away.

Don’t forget to check out the other gems over at Transformulas.

One Formula, to Rule them All

Two-dimensional area starts and ends in pretty much the same place, with base and height.  Kids in elementary school calculate space by counting grids.  Calculus classrooms do the same thing (on a more complex level of course), but through the short cut of integration.  Somewhere in the middle, with geometry and the like, it gets complicated and students lose the conceptual understanding.

How do we get from the simple to the complex

How do we get from the simple to the complex

Here’s what we did instead.

Start with a few applets:

  1. Rectangle vs. Parallelogram
  2. Triangles
  3. Kite / Rhombus
  4. Trapezoid

Then we document our thoughts.  Some people call this notes.:

Students watch in amazement as if this were a magical experience.  Audible comments of “wow” and “that’s cool” are common.

So then we conclude that there really is just one way to calculate 2 dimensional straight line areas:

base times height (and sometimes half)

Sum it Up, Angle Edition: Part 1

360 Wheel

Somewhere, I don’t know where for sure, a student learned that one common way to measure a full rotation is 360o.  I would even say this generation is more aware of this concept with the popularity of sports that involve rotation(s).  I could mention the number 1080 in a class and many students would have at least a minimal intuition that it had something to do with revolutions/rotations.  Imagine you are that student and start from there:

The Basics (prior knowledge)

  1. Full Turn =  360o.
  2. Half Turn = 180o (call it straight angle)
  3. Quarter Turn = 90o (call it a right angle)

Number Sense (and decomposition)

“How many ways do you think you can fill in these boxes?”

90 = [?] + [?]
90 = [?] + 41
90 = 29 + [?]
180 = [?] + [?]
180 = [89] + [?]
180 = [?] + 27
180 = [?] + [?] + [?]
180 = 90 + 80 + [?]
180 = 70 + [?] + [?]
 Supplementary Supplementary 3 piece Complementary

Playtime

After 1-3 minutes of getting out their Chromebooks, loading the applet and playing, I give the students a specific angle measure to move to, like angle BAE = 40o.  Next a simple task: find all the measures.

“Mr Butler, I don’t know what to do?”  “Mr. Butler, but there’s not enough information.”  and more of this kind of thing for about 30 seconds.

Then one student elbows the other, “ow, why’d you do that?!”

“Check this out, I can find this one.”

“Oh oh, I can find the next one look here it’s a quarter turn, we just need the other piece and go backwards like we did a few minutes ago”

10 minutes later we share.  And there are soooooo many different methods used, a good amount of them correct.  In the end it was easy to check though.  The total angles made a full revolution, the one measure everyone knew coming in.  For those that were incorrect we talk about how the error took place, (was it arithmetic? or reasoning?)

We repeat this process with two or three more different starting measures for the angle controlled by the slider (let the students choose the angle for at least one of those).

Vocabulary

Oddly enough, vocabulary seems to stick better once the students have a context to put it in.  These quarter turns, that make a CCCCorner are call CCCComplementary, these half turns, that make a SSSSStraight line (or Straight angle) are called SSSSSupplementary.  Somewhere in this process students start to notice the relationship in angles across from each other at an intersection are equal.  We draw the criss-cross X and get the term Vertical.  At this point we are only about 3/4 through 1 class period.

Finale

Move the angle to _____ and  prove to me (with addition or subtraction) which angles are:

  • Complementary
  • Supplementary
  • Vertical

(There’s two for each)

Mr. Butler adds, “Hey class, oh and by the way, we just reviewed Algebra and solving equations.”

“You’re sneaky Mr. Butler”

Then the bell rings…

… to be continued…

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