Monthly Archives: March 2014

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?



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.


Recursive Reflection

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


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


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).


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.


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…

Pizza Pi

The hook

The students walk in on Friday to wrap up a roller coaster like week (more posts on that to come). The students heard the music and saw a simple question:

We had been covering a unit on deeper extensions into 2-dimensional area and perimeter.  This question fit right in with calculating the the area of a sector in a circle (also known to the layman as slices of pizza).

The students saw the question and almost felt lost.  They didn’t even realize it was a question because it wasn’t specific enough.  A common response was, “Do you mean which was is bigger?”  I replied, “I don’t know, do you want more or less pizza?”


Students then commenced with table talk and less than 10 minutes later were ready with their responses.  Before looking at calculations and actual numbers, we took a short poll from all the tables.  For each class it the split was nearly even.  So we settled the argument with the calculations.

The results were not exactly close.  1 slice had about 10 square inches while the other had 15.   I noticed that we could make the decision harder by using a combination of slices to create an equal proportion.  2 of the 15 square inch slices, and 3 of the 10 gave two nearly equal groups.  Now the extension.

I had created a #Geogebra applet before school that was originally intended to help visualize the comparison of the two pizzas.  This allowed the students to interact with the question.  The visualization of numbers is often a challenge, on both sides of the learning experience.  These visualizations can be hard to create in a static snapshot, much less something that is interactive and dynamic.  Organizations like Geogebra and Desmos have all but eliminated those obstacles.


For homework the students weren’t asked to complete 20 problems practicing this exercise.  Instead the students were asked to create 2 more Would You Rather combinations with the following parameters:

  1. Each pizza needs to have a different diameter
  2. The difference in the areas must be less than 5 square inches.
  3. Show process/work for your calculations.


Not every lesson can be awesome like this one.  But the format of activity in my classroom has changed significantly with dynamic programs like Geogebra and Desmos.  It’s transforming into something much better.

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