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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:

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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.

 

 

 

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Now for Some Real User Power #MTBoS30 3 of 30

I enjoy math, thoroughly.  I also enjoy design and technology.  Recently I’ve had a wildfire like experience in processing and learning material through Geogebra and Desmos.  I’ve been learning and experimenting with these much faster than I could possibly archive organize the material.  I look forward to the upcoming extended break at summer to truly polish this material.  Currently I’m feeling more like

Greetings Programs

This little presentation I had tonight was a breath of fresh air.  It was a mixed crowd and we had a great time.  About half mathies  in the room, and half techies.  It wasn’t a large crowd, so we had a casual, yet productive time.  At around 2/3 of the way through the time allotted, one of those in the room inquired about the coding behind some of the applets.  Others seconded the question, so then we transitioned from math to tech.

Technology geeks, myself included, often dive into the code and lose some of the social part of the experience.  Working on the backend of a program experience too often is a lonely one.  Talking about this experience in a real life, social platform was great for myself and them.  I was able to reflect and process on my wildfire experience of learning, and from what it seems they were about to start their own versions of something similar.

On the other end of the screen, it all looks so easy.

Geogebra and Desmos both use clean user interfaces that allow for wide audience.  Knowing some of the coding and design on behind the scenes still has it’s place though.  Recently education has seen growth in the art of coding.  It has become more accessible with drag and drops like scratch, tutorials and screencasts shared freely online, and organized movements from large institutions like Khan Academy.

Now I would like to throw another idea into the mix of developing a coding mindset.  For those of you that know me, you could probably guess that I’m thinking of Geogebra (and Desmos as well).  Tonight we talked about the object oriented code experience it offers, and it’s simplicity with design and interaction.  To toggle a picture or make an object move, all you need is a slider control and some checkboxes.  I’m not sure where this can go from here, but I like it.

If you haven’t already you should check out these online interactive tools.  And when you do, look at them as tools for tech and coding, not just math.

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PS: bonus points if you get the geeky references.

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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