Category Archives: Geometry
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.
NonCentral 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.
http://tube.geogebra.org/material/simple/id/115590
The next day we get to see the formula that collapses 3 ideas down to 1.
Dynamic Angles in…
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2 is better than 1 (sometimes)
I’m a fan of visual context over simple memorization of formulas. Another pet peeve of mine is the requirement that denominators be rationalized without reason. A good conversation on rationalizing the denominator takes place over at “Why am I teaching this?”. One place I’ve see rationalized denominators lose their context is in a unit circle. I get that it’s a standard. I get that the angle radian measure makes most sense with a radius equal to one.
A friend of mine, Jen Silverman, makes some great protractors that help with this radian angle measure as well. I think my frustration comes from the idea that the unit circle is a beautiful overlay that simplifies so many interactive relationships between the trigonometry ratios. Too often students never come to this realization and instead resort to tricks.
Here’s another post using the same trick.
I used to try to get students to think of the Unit Circle key segments as just a small set of lengths that interact in different ways.
Some applets to help in relating the unit circle to the cartesian form of basic trigonometric graphs:
While others simply go back to memorizing the rationalized denominator forms.
Still, I feel the abstraction away from Trigonometric Ratios loses the relationships within the right triangles created by these points along the unit circle.
So my proposal is: Introduce Circle Trigonometry with a radius of 2. This would double all the segment lengths. Trigonometry ratios would relate more directly, and students could redraw and label the pieces more intuitively [it’s pretty hard to imagine what Sqrt(2)/2 looks like compared to Sqrt(3)/3)]. I haven’t built the applet for this, yet. Expect it to show up in a future revision of this post soon. If you’ve ventured this way before, or you’re interested in encouraging/discouraging me from doing so, please comment below.
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).
28806(360)=28802160=720.
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)
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:
 Focus on the Right angles/right triangles
 Question if certain segments were congruent
 Look at the reflection or congruence theorem that helps confirm the congruence
 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:
 Lay Chromebook on its back
 Increase brightness of screen, turn off some/all light in class
 Trace
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:
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)
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:
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:
Attn all teachers! Thx for all the hard work you do day in & day out! Let Dynamic Paper make ur life a little easier! http://t.co/Q5OlDu8hgp
— NCTM Illuminations (@NCTMIllum) May 5, 2014
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.
Hook:
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).
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 , versus, 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
 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:
 increase fluency and familiarity with the formulas for surface area and volume of a sphere? …or…
 practice using a tool (grid paper) appropriately and apply problem solving?
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. 
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:
 Get quarter sheets of graph paper
 Draw your own unique triangle
 Color in the angles in each corner
 Cut Out the triangle
 Tear off each corner
 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 lotech 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
Twodimensional 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.
Here’s what we did instead.
Start with a few applets:
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
Somewhere, I don’t know where for sure, a student learned that one common way to measure a full rotation is 360^{o}. 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)
 Full Turn = 360^{o}.
 Half Turn = 180^{o} (call it straight angle)
 Quarter Turn = 90^{o} (call it a right angle)
Number Sense (and decomposition)
“How many ways do you think you can fill in these boxes?”



Playtime
After 13 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 = 40^{o}. 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 crisscross 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…