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Circles Part 2: Angle Relationships


Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments.  This post will look at using visual information from central angles and inscribed angles.

Pi Charts

Students sometimes lack intuition for the measure of something.  Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises.  Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.

Non-Central Angles

At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees.  Built with this intuition in mind, we look at a triangle created by inscribed angles.

The next day we get to see the formula that collapses 3 ideas down to 1.

Dynamic Angles in…

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Interior Angles with Triangles (and Circles) #MTBoS30 9 of 30

The foundation:

The are only a few givens in geometry, building on those we derive many other patterns theorems.  One theorem I often prefer to focus on visually is the sum of the interior angles of a convex polygon.  There’s different ways to approach the process, and most of them refer to creating a fan of triangles inside the polygon:


Early on in most geometry studies, we learn that the sum of the interior angles of a triangle is equal to 180, building off of this we can use the above animation to calculate the sum of the convex hexagon: (180)(4)=720.

The Detour

There’s often the student who wants to divide the hexagon up differently.  The student wants to draw lines criss crossing all over the shape.



The Brakes:

Too often I would have totally passed up this opportunity and said, “That’s not how we do it, so it can’t work.”

Taking The Long Way isn’t an Error:

Thankfully, this year we saw something different.  There were still triangles inside, and with a few more lines added, the picture had only triangles inside.


Once we have all triangles, we can just count those and multiply by 180.



(16)(180)=2880.  That can’t be right, can it?

From here we get a better definition for what is an interior angle of a hexagon.  It’s an angle inside the hexagon, but still attached to the edge of the shape.  Somehow we need to get rid of the angles that are inside the hexagon but not touching the edge of the shape, like these red circles:



There are 6 of these circle sets of angles, each of them having a value of 360 degrees.  We just need to discount those (aka subtract).


This kind of process also builds a more flexible understanding of how to decompose a geometric shape in multiple ways.  I encourage all math teachers out there to try this next time a student chooses to slice it up however he/she wants.

By the way, thanks to @justinaion  for inspiring this with his post earlier today.

Circles, Triangles and Kites oh my part 2 #MTBoS30 7 of 30

It’s Thursday.  Yesterday we talked about circles, chords, and kites.  Today we asked a very similar question:


Here’s the applet in action: (click the animation to open the applet)

Circle Inscribed Find Area of Triangle


This was similar to Wednesday.  We asked the same type of questions.  We saw similar relationships.  But it stood out enough for a unique post because a student enlightened me with  an observation I didn’t articulate.  When I repeated yesterday’s question, “What shapes do you see inside the triangle?” one student almost immediately replied, “Is that a kite?”  I had to look at it myself.  “Yeah, wow, that makes this conversation easy.”  My original plan was:

  1. Focus on the Right angles/right triangles
  2. Question if certain segments were congruent
  3. Look at the reflection or congruence theorem that helps confirm the congruence
  4. then finish off with some color coding.

Instead this students recognizes the 3 kites, then refers to her knowledge of the symmetry in kites.  Congruence, simplified.

To help students in transcribing the diagram onto paper to start doing some hand calculations we took a tip from a student in the first class:

  1.  Lay Chromebook on its back
  2. Increase brightness of screen, turn off some/all light in class
  3. Trace

Student Tracing on Screen

This was another good introduction and discussion with segments in circles.  We of course spent the last 1/3 or so of class practicing and becoming fluent with the skill with problems like:

Perimeter of Polygon with Tangents



At the end of the week we took a short quiz.  2 questions, same as these.  Today/tomorrow I’m going to try something new with how I grade them (thanks to some inspiration from Michael Fenton, Michael Pershan, and Ashli Black).  More on that later this weekend.

In the meantime, let’s keep rounding out this circle thing and see what other shenanigans we can come up with.




Circles, Triangles and Kites oh my part 1 #MTBoS30 6 of 30

In geometry we had some fun with kite like shapes for the last couple of days.  On Wednesday we asked:


See it in Action (click image to open)

Circle Chords and Kite


I asked the students to play with the applet.  I prompted them to ask for more information.  Some noticed the sample questions below the applet and asked those to start the discussion.  The key question had to do with decomposing the shape into other shapes that we had more familiar tools to work with.  The students were quick to see the right triangles.  Once we were able to identify that, the next question was, “How does that help me?”  Getting students to connect and then apply relationships they know into a seemingly new context is a constant challenge, but they are getting better.  I reminded them to recognize what are we trying to find.  Once they narrowed in on the task of finding a length, and the length was a part of a right triangle, some started to see it, “Pythag Thyrem.” (this seems to be a tongue twister for the general high school math population).  “Okay, how do you mean?” was my reply.  Some didn’t see the given values for the hypotenuse that was also a radius.

Circles seem to be a pain for many geometry teachers, and I feel that it’s because so many people approach circles as a never ending list of formulas.  We need to find ways to simplify the overall question and give students an opportunity to fill in the structure(s) needed to respond to the question.  I know there are some awesome activities out there dealing with volume and area with round objects.  Here I’m trying to put together a series of interactive questions the see the overlaps and relationships with circles, segments and angles.



What’s your Angle? (or Arc) #MTBoS30 5 of 30

Today Geometry started with a question:


I was purposely vague, and no student went so far as to talk arc length (as opposed to arc measure in degrees).  I told them to give their best guess but back it up with process and reasoning.  This is what they came up with:

How big is the Arc

Students we able to trace the process of splitting into portions or multiplying into larger parts.  They also tried using the context of percents without being prompted to do so.  The first class didn’t automatically use units in degrees so I prompted, “If I told you the measure of this arc was 22.5 degrees, how do you think I came to that.”  Students were able to articulate my process by relating it to their own.  The second class had a few students actually use the process of putting the measures into degrees of a circle.  Thanks to a recent tweet by NCTM illuminations:

I was able to quickly put together a worksheet of “spinners” as a follow up the this discussion.We practiced these in the same fashion.  I made sure the students included the calculation of the arc in degrees.

I could have made this worksheet much longer, including measures in terms of percent, number, or even with angles.  I wanted to keep it simple for now.  Tomorrow we go further with Algebraic expressions, and start the discussion chords, area of slice of pizza and the length of string cheese. Now I want some pizza.



Limited Area, Maximum Space


Simple questions with not so simple answers have been growing in the classroom.  I’ve seen multiple takes on the classic “which glass has more” from Piaget, to Meyer, Timon Piccini, and I’m sure plenty of others.  I figured we could also do the same type of thing lo tech style.


You have to use half of a standard size sheet of paper.  What is the largest cylinder you can make ignoring the top and bottom lids (bases).

Cylinders with Constant Lateral Area

In other words, paper = lateral area.

There are at least 2 ways of splitting a sheet of paper.  Many teachers refer to them in similar foods:

versus ,  versuschicken_veggie_burrito_50, or whatever else the students can come up with.  I started this prompt with my first class and quickly had to apply some planned scaffolding.

Mr. Butler we don’t know the radius?  Can we use a ruler?  What is the height?  and of course…I don’t get it.

We rebooted and brainstormed what we did know about what we had:

  • The base is a circle
  • The wall (lateral area) rolls out to be a rectangle
  • Area of a Circle = Pi*r^2
  • Circumference of a Circle = 2*Pi*r
After the students got this out there was still a connection missing.  There was no mention of dimensions of a piece of paper. I asked if they would like to know how big a sheet of paper was.  Students were quick to realize that this would help with calculating the height, so they pleaded with an almost sarcastic whine, “Please!?”  After telling them standard american sheets of paper are 8.5 x 11 inches, they still struggled with making the connection to the radius.  I showed the students one example for finding the radius using the base edge as a circumference and working backwards.  By now half the period was over and not much progress had been made in answering the question (save for the on visual approximations without measure).
After one period of work we were able to ask a question, make some attempts at a reasonable answer, and reach absolutely no conclusions.
Enter the next class.
Modifications are a necessary part of teaching and learning.  Fit teachers truly survive through refinement and adaptation.  For this activity I narrowed our resources to be more specific with using only graph paper.  This removed the required knowledge and/or tools for measuring edges of a rectangular piece of paper.  Students could simply count.  Some students were wanting clarification on what the units actually were, “Mr. Butler are these little squares the same as centimeters?  Or something else?”  I replied with, “Let’s call them units for our activity,” and eventually whispered, “Most graph paper sheets we use are split into quarter inches.”   I made a mental note to include varying sizes/scale of grid squares for future attempts at this question.  Students could see that the geometrical relationships are not dependent on actual units as long as one is consistent.
The questions in this class were more closely targeted at the actual question of which cylinder was bigger.
  • I can use the grid to measure the height but how do I find the radius?
  • Do I even need to find the radius?  Can I answer the question with just the height and the size of the paper?
  • Can I use the grid to measure the radius?
    • Mr. Butler, I can just stand up the cylinder like this and look down.  I can see the radius across the inside.  Can I just use this to measure it? (He was referring to the diameter, but I got the point, more importantly this caused me to determine the point of the whole activity:
Am I trying to
  1. increase fluency and familiarity with the formulas for surface area and volume of a sphere?        …or…
  2. practice using a tool (grid paper) appropriately and apply problem solving?
So I had a little micro conference with the opposing arguments on my shoulders:
 Procedural Thinking  Problem Solving
Practice with the formulas.
The abstract concept will apply to any situation.
Students can’t use grid paper on the standardized test
Encourage and support practical application.
Using tools appropriately is a universal skill.
Life beyond high school will depend more on tools than abstract concepts.
From experience in the classroom I chose the most classical teaching strategy I know: stall with a question, “How ’bout you try it an tell me what you find.”  I didn’t want other tables to just settle solely with the approximation of the radius with the grid paper, so I continued to prod each group as I circulated for alternative methods.  After 10 or so minutes of struggle I could see the class needed a bone thrown in its direction.
Alright, let’s try to see if we can calculate one of these cylinders using something we know about the lateral area.  I displayed an animation we looked at when first expoloring a conceptual understanding of surface area of a cylinder.
T: So which edge helps us with the height?
S: The height. (said with a tone implying, “Duh”).
*Side note: This question was pretty elementary, with more of an intention to rule out one edge and focus on the other.  There’s probably a better way to set that up with answering my own question.
T: Awesome, anything worth mentioning about the other edge?
Crickets.  Wait time.   Wait for it.
S: “The diameter?”
T: Thank you, but I’m not sure, let’s see.  So if this rectangle edge is the diameter, then it should be the same rolled up and laid flat.  What do you think?
S: Oh, no, not really.
Crickets. Wait time. Wait for it.   Wait…
S: whispers, “I’m not sure, I think so”
T: “Go ahead, it’s better to try something than just sit and wait.”
S: mumbles, “circumference?”
T: hand up to ear, “Say again?”
S: Circumference.  (Stated, slightly more confident).
T: Let’s see, what do we know about circuference…
From here with the help of a few different students we’re able to breakdown the measure of the radius after knowing the circumference.  Then calculate the volume with said radius.  This is one of at least 4 calculations needed for the process.   I leave the work up at the board and recirculate constantly repeating the actual question, “Which one is bigger?”
More than half the class reaches a conclusion in the next 5 minutes.  They determine out of the four that the widest cylinder is in fact the largest.  For a group that reached this result significantly faster than the others I challenged them to answer if a cylinder half as tall as the widest cylinder and again using the same lateral area would be larger or small.  A simple construction and then comparison of fitting the taller of the 2 cylinders into the half as tall cylinder twice with room to spare answered the question with simple visual reasoning:
Double Up Cylinders
I was much more satisfied with the second experience, and now as I’m writing this I have an even better question to help the students generalize the reasoning as well as increase number sense with the calculation:
What matters more in the volume of a cylinder?  The radius or the height?
Is it the same for Surface Area?  
The necessary argument for these would call for the students to use examples from their activity and articulate a conclusion with at least a minimally satisfying level of reasoning.  Maybe next time I’ll include a state for putting together the data into a graph to help students see relationship.
Why does this even matter?  Ask yourself about packaging.  Why does it make sense to use cylinders?  Why would one want to optimze volume?  Why might materials be restricted?  I’m writing this as I just finished a beverage that was served in a glass that makes me consider the intent of its design.  Did I really get the most out of my drink?


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…

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