A Picture is Worth…..Everything.

I spent about an hour yesterday playing with geometry tiles.  I was:

  1.  Fully engaged and enjoying myself.
  2. Experiencing OMG, WTF, and AHA moments.  (AKA actively learning.)

The inspiration for my mathematical play time was this photo from  Sara Vanderwerf’s recent post  on Stand and Talks, a bold routine she uses to increase the number of students talking about math.   Really great stuff and worth your time to read it through.

IMG_2094But back to the photo, which is her example of using a Stand and Talk to introduce a new idea, in this case, the Pythagorean Theroem.   My eyes perk up (is that a thing?) because I have been pondering about this very topic.  Specifically, wondering about what activities/tasks would place students in the active role of sense-making.

The first thing that popped into my head was….wait, trapezoids?  (Thank you, Sara!) Then I started some noticin’ and wonderin’ from my teacher-y perspective and also imagining/guessing what students might say.  The more I noticed and wondered, the more I realized how FULL OF MATH IDEAS this one photo is.

I find this exciting.  Really, really exciting.  Can you imagine starting here, with this one little image, asking students what they notice and wonder, and then letting them run with their curiosity?  Isn’t it joyful to know in your heart of hearts that this is so much BETTER than, “Hey kids, here’s a little something called the PT, copy it in your journal/onto your pre-made “doodle” notes, plus copy these three examples of how to get the right answer, and do this worksheet”? How superior this single photo and all its potential are to “I do, we do, you do” and /or anything else that keeps students in the passive role?

It didn’t take me long to drag out my tub of tiles and start messing around with them.  TOTALLY what I would want my students to ask beg to do.

The questions I decided to guide my play were:

First, can a proportional, scaled-up version of each polygon tile (that I have) be built with only that particular tile?
If so, why? Is there a pattern?
If not, why not?

Second, can any three similar polygons be used to make any type of triangular “hole”? Why or why not? Which ones make right triangles? Why? Is there a pattern??

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I started with squares.  Don’t know why, maybe because that’s what is already familiar to me.  No surprises, the areas are the square numbers, 1, 4, 9, 16…..

So I moved on to the equilateral triangles, wondering/kinda expecting that I will make triangular numbers.  Imagine my surprise IMG_2119(my OMG moment) to find SQUARE NUMBERS AGAIN!  Holy shit!  NOT what I was expecting!  Now I feel compelled to keep exploring.  COMPELLED.  (Imagine your students feeling compelled!!)

I quickly confirmed that the next in the series takes 16 triangles, and moved on to IMG_2125the rhombus.  YES, the areas are square numbers AGAIN!  Which gets me reconsidering my understanding of square numbers and forming conjectureish questions:  Can square numbers be made out of any polygon, not just squares? These are all regular; what if they are irregular?

I’ve been reading Tracy Zager’s fabulous book Becoming the Math Teacher You Wish You Had.   She starts Chapter 7, Mathematicians Ask Questions, with a quote from Peter Hilton: “Computation involves going from a question to an answer.  Mathematics involves going from an answer to a question.”  Brilliant distinction.  Looks like I’m living Chapter 7.  Know I want this for my students!

Remember, one photo.

I finally get to the trapezoids, which are more challenging to tile and require some analysis.   I managed to construct one, but the number of tiles it took (7) was a red flag for me, so I  examined it closely to verify if it was proportional or not.  It wasn’t, so I kept playing, keeping theseIMG_2126 two beliefs in my mind: a proportional trapezoid was possible, and it could be made out of 4 tiles.  And lo and behold….

TA DA!  “Square” Numbers!  (Whew!)

 

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By now, I have even more questions.  By now, I have reviewed and applied my prior understanding of proportionality and similar figures, without ever touching a worksheet.  By now, I’ve taken risks, persevered, reasoned, looked for patterns, used vocabulary, formed conjectures, and justified.  All from looking at one picture and allowing myself both time and pleasure to play and be curious….and I haven’t even begun exploring that thing with the triangular “hole”.   Yet.

Finally, the hexagons.  And….What?!?  You can’t even make a similar hex without using some non-hex friends!  WTF!

New question!  WHY NOT?

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One.

Little.

Picture.

Random Acts of Mathness

I finally found Fawn Nguyen’s  current blog! SUH-WEET. I feel like I’ve struck MTBoS gold…again. I’ve decided to read it from her first post forward.

This is not going to be about what an freakishly amazing and intelligent teacher she is, or about how reading her blog makes me waver between feeling incredibly inspired and feeling incredibly inferior. Nope, none of that.

On 11/6/14, she shared  this problem that she had given her students to wrestle with. The problem intrigued me, so I thought I’d try it; I even averted my eyes from the photos of student work so that I would not be influenced. (Want to do the problem, too?  Spoiler alert!  More photos ahead!  Avert eyes!)  In the few minutes I had before leaving for an appointment, I made a sketch and threw down some variables for the missing lengths.

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I noticed that the two right triangles were similar and the requested area was a trapezoid. I wondered if it would be possible to express the variables in terms of one of them; I can easily express “a” as “10 – b”. Could I do the same for “x”?

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I looked at the similar triangles first.

Nah. Still two variables after I simplify. What about working with the triangle with the known area?

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Also a bust. My time was almost up, anyway, so I quickly fiddled with the area formula for the trapezoid, which does not use x, but does still have a second variable, A.

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As I copy my work neatly for this post, I am VERY AWARE that by this point I already had everything I needed. Between my time interruption and my laser-like focus on and blind attachment to my original strategy, I missed that tidbit.

When I again had a few minutes the next day, I started over with a fresh sketch, this time keeping all three variables as I worked.  I revisited the similar triangles and the known area to record everything I knew to be true. I felt these were key. Using the proportions and the ax = 48 fact, I guessed that 6 and 8 were reasonable options to try for “a” since the options for a + b = 10 were limited. Confident, I tested them out….

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…and arrived at a lovely WTF moment. Maybe I need to let go of my assumption that the measurements are whole numbers? Apparently 5(9.6) = 48, but that’s not helpful in my proportion.

I looked at my proportion set up again, this time circling both cross products, and…AHA!

image(Finally, right?) BOTH products = 48. Duh.  So b has to be 3. (Remember, I missed this previously… If 16b = 10x – bx and 48 = 10x – bx, then 16b = 48.  So what.  Now I have two ways so solve this problem!) This is probably the first time I have ever put cross products to good use. (I refrain from introducing this memorized-process-to-get-the-right-answer to students because I rather they’d focus on what’s-the-relationship? instead.)

Figured out what x was (just for kicks..and to validate my proportion) and subbed in 3 for b in my trapezoid area equation to find the solution. Ta DA!

To celebrate, I watched the video Fawn linked at the end of her post, in which Mike Lawler also uses similarity  to find the solution, but via a different path, which is cool to see.  Check out also Mike’s student’s area solution, as well as a completely different area solution by Mike that employs a system.  Holy Multiple Pathways, Batman!

This task is such a poster child for rich problems.  I shared it with my entrepreneurial daughter. She doodled around a bit, dusting off had some long-idle knowledge, determined the exact question she needed to ask me, solved it, cheered, and then demanded another one. EXACTLY what I want for all my students.