Fun with The Magic Octagon

So, we are wrapping up our unit in Geometry on rigid transformations, which means it is that wonderful time of year when I show the students Dan Meyer’s The Magic Octagon!

Seriously… have you seen this? (Go through it like a student. Pause it to make your first prediction.)

The Magic Octagon from Dan Meyer on Vimeo.

Isn’t that cool? Not sure whether you predicted correctly or not (I did not the first time), but I’ve used this video with 6 or so geometry classes and the results are somewhat predictable. 85-90% of the class guesses 10:00. Most of the remaining voters choose 7ish:00 and get called crazy.

Then they see the answer and they JUST… CAN’T… BELIEVE IT!

“Wait… wait. How does one side go clockwise and the other side go counterclockwise?

“No. Run the video again. What did I miss?

That would be a 10-out-of-10 on the perplexity scale, when, like 85-90% of the class gets the math problem wrong and that suddenly becomes a motivator!

Then as they start trying to figure it out, they start making lots of hand gestures, which is surprisingly helpful to them and

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Then they don’t want you to move on. They want two more minutes to talk about it. Then a classmate starts explaining it. Not all of them get it the first time, but some of the demand to have it explained again.

Then they move on to the second rotation and they feel so confident. You ask for explanations. They give them… quickly. Quickly because they can’t wait to see the answer. And then they did.

And those who got it right cheered! Quite loudly.

Then a boy stopped us and offered a sequel.

“If the front side arrow is pointed at 5:00, would the other arrow point at 5:00, too?”

He turned the tables enough close to the end of the hour that we left with that question unanswered.

 

And I fully expect a couple students to have something to say about it tomorrow.

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Kids, I’ll tell you why you need math…

So I was at Kroger and I was thirsty.

I went to the cooler to pick out a 20 oz. Coke. $1.69. Reasonable. About what you’ll pay most places, at least around these parts.

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But I had some more shopping to do. So, I kept walking. Gathering the items on my list.

Then I saw some 2-liter Cokes. (Remind me. Which is a bigger amount of Coke? 2 Liters? or 20 ounces?)

The 2-Liter Coke was priced at “2/$3.00”. (Which, I’m pretty sure is less than $1.69.)

(Oh! And you don’t have to buy 2 to get that price. Kroger is awesome like that.)
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I get pricing a little bit. I worked concessions at a college football stadium for a while (true story, by the way). I know the shtick. It’s upselling.

“For an extra quarter, you get almost double the pop!”

But this isn’t like that.

You have to buy 4 twenty-ouncers to fill a 2-liter. 4! You include the 10-cent bottle deposit here in the great state of Michigan, and that’s over $7.00!

2-Liter? $1.50

This isn’t even close.

Kids, THAT’S why you learn math.

Okay… I solved it, but how’s that math?

Its on these warm, spring days that some learners tend to start checking out. They recognize our typical reviewing and recirculating attempts to help students recapture some learning before the last tests and exams. But some of them learned the content the first time. Essentially, this time of year can bring with it a lot of down time for our students who learn the fastest.

That’s when I like to pull out my old college textbooks. I spoke before of the power of ungraded bonus problems. If interesting and placed properly, they can provide powerful opportunities for thinking simply for the sake of thinking. I like to give them a window into math that “doesn’t look like math.” After all, most of the content that K-12 mathematics includes has some commonalities that students get used to. They’ve gotten used to “what math looks like.”

Today, four groups got a problem that I adapted from one of my undergrad courses that I took at WMU with Dr. Ping Zhang who, along with Dr. Gary Chartrand co-authored this book which was the textbook for the course I took.

The problem is adapted from Example 1.1 from Chapter 1. It asks the students to create a schedule for 7 committees that share ten people. Now, the purpose of the problem in the book is to give a simple example of how a graph can be used to visualize a complex situation. Often the way a problem is mathematically modeled can change the intensity of the solution process.

 

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Figure 1.1 from Page 1

 

The reason I like to give this problem (and problems detached from the K-12 curriculum in general) is that in May, to curious students, these problems tend to hit the perplexity button just in the right spot. In fact, all the students looked at the handout at first and were unimpressed. Until I asked them to read it… then, they seemed to just want to see what the answer looked like. As one student put it, “It seems really easy at first, then you get into it and it’s actually harder than we thought.”

Graph theory is about creating visual representations. I didn’t want to ruin their experience by pigeon-holing them into trying to represent this they way I knew they could. And given the time, that didn’t stop this group from creating a similar idea. Not with a graph, but they did use crayons.

 

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To hear Kailey explain: “We just gave each person their own color. Then we knew we could have two committees meet at the same time if they didn’t have any colors in common.” That isn’t different thinking, really. Just a different representation.

I enjoy the conversations that come out it. No grade. Just doing math for the sake of thinking about something that’s interesting. It’s especially interesting to students like Katie who said, “Okay… I solved it, but how’s that math?”

 

Reference

Introduction to Graph Theory (2005) Chartrand, G., and Zhang, P., New York: McGraw Hill

(not) Explaining “Half”

“If you can’t explain it to a 6-year-old, you don’t understand it yourself.” – Albert Einstein

 

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“I’ve drank half my water”

 

My daughter isn’t a 6-year-old… yet. I get a couple more months before Dr. Einstein’s quote applies. Regardless, today I found myself at the breakfast table trying to make sense of one of those topics that exists in the intersection of math and common language.

It’s good to make mathematics common place, but the problem with making technically-specific terms commonplace is that it often leads to the usage of the word becoming a bit less technical. ELA teachers will tell you that this has happened with the word “literally.” (For more on “literally”, this post will literally blow your mind.)

In that same spirit, today’s breakfast was abuzz with the word “half”. My daughter is comfortable with the word “half.” She will often take a couple of gulps from her cup leaving ~5 ounces in a 6-ounce cup. Then she’ll say, “I drank half my water.” To her that means that she has consumed a some portion of her water, more than “just a little”, and less than “all” or even “a lot”. It seems the developing spectrum looks something like this:

Water Drinking Spectrum

Water Drinking Spectrum

It’s clear that spectra like this make clear the need for fractions, but that’s a discussion for another time.

Well, today, she asked about what “half” means. Which was a question that I should be qualified to answer, but you wouldn’t know it from how the conversation went.

Her: “Dad, what does ‘half’ mean?”

Me: “Well, it’s like broken into two pieces that are the same size.”

Her: “No, like, I drank half my water”

[See? My explanation didn’t work because I was thinking the two pieces were the water in the cup and the empty space. She only sees water. That’s only one piece.]

Me: “That means that the amount of water that you have already drank is the same as the amount of water you have left.”

Sage: “I don’t get it.”

Me: [holding up clear cylindrical glass that is “about half-full”] “See? This glass has water up to here [I point at the water line], and then the rest of the glass is empty. It’s like there’s the same amount of water and empty.”

Sage: “Um… Maybe we should talk about this when I’m older.” [Sage continues eating oatmeal and the topic changes]

 

Good thing she’s only five. I have a few months to prepare for Dr. Einstein’s test of whether or not I really understand the idea of “half.”

 

Perception and Reality – (Lean not unto thine own understanding…)

In Basic Economics, Thomas Sowell tells a story about a decision made by a New York politician who was attempting to address the homeless problem in New York City. The politician noticed that most of the people who were homeless were also not very wealthy. The politician moved forward with the idea that the apartment rent prices were simply too high for these people to afford a place to stay.

So, he decided to cap the rent prices… and the homeless problem got worse. How could this possibly be?

Well, according to Dr. Sowell, lowering rent prices, while making the apartments more affordable for those in need, did the same for everyone else. The suddenly cheaper rent prices decreased the rates of young folks sharing apartments. Also, people who have several places they call home throughout the year might not have found it reasonable to pay a high rent price to keep a NYC apartment that they might only stay in a few times throughout the year. Lower rent prices made that seem more reasonable.

Evidence also suggested that there was an increase in apartments being condemned. Lowering rent costs meant that landlords found themselves with fewer resources to maintain buildings, repair damages, pay for inspections, etc.

While the decision made the apartments more affordable, it also made them more scarce. There was a disconnect between a decision-maker’s perception of a situation and the reality. That disconnect led to a decision that ended-up being counterproductive.

I may have just done the same thing… maybe.

Perception-Reality

 

Sometimes things make so much sense. If we did this, it would HAVE to produce that. It make so much sense. How could it possibly not work?

This perception was in place among some in my community. It led me to decide to try The 70-70 Trial, which I’ve been at for about 10 weeks now. The perception in place goes like this:

a. Formative assessments prepare students for summative assessments.

b. Students who struggle on formative assessments are more likely to struggle on summative assessments (and the inverse is also true.)

It’s with these two perceptions in mind that we assume that the if we can ensure a student achieves success on each of the formative assessments (regardless of the timeline or the number of tries), we improve their chances of success on the summative assessment.

The 70-70 trial did what it could to ensure that at least 70% of the class achieved 70% or higher proficiency on each formative assessment. (There were four.) This included in-class reteach sessions and offering second (and in some cases third) versions of each assessment. With all of those students making “C-” or better on each formative assessment, how could they possibly struggle on the unit test? That was the perception.

50% of the students scored under 50% on the summative assessment. That was the reality.

Now, I am not an alarmist. I understand that one struggling class in one unit doesn’t discredit an entire education theory. But it sure was perplexing. I’ve never seen a test where, after 8 weeks of instruction on a single unit (Unit 4 from Geometry), half of an entire class unable to successfully complete even half of the unit test.

And when you consider that this class was the one class I had put the most effort into defeating just that kind of struggling, well it seems like the intersection of my perception and the reality wasn’t nearly big enough. I just got a better view.

And I’m having a hard time making sense of what I’m seeing.

Coke vs. Sprite – One Class’ Response to Dan Meyer’s #wcydwt Video

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Almost three years ago, I highlighted the Coke vs. Sprite video that @ddmeyer made. It is a very intriguing question (which glass contains a greater amount of its original pop) without a clear answer.

Today, I let a class give it a go and here’s what they came up with.

First, they chose to model it with integers. Said one student: “I pretended it was a jar with 10,000 marbles.” 2014-04-03 11.07.36

Then they assigned “a dropper-full” to be 100 marbles. So, the first dropper took 100 red marbles and placed them into the jar with the 10,000 green marbles.

Now we assumed the stirring made the mixture homogeneous which meant that there was a consistent 100:1 green-to-red marble ratio in the right jar. So, when we pull another 100 marbles to put back, (with a little bit of rounding) we pull 99 green marbles and a single red marble back into the left jar.

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A little number crunching reveals (at least in this model) that there would be 9901 original-colored marbles and 99 other-colored marbles in each jar.

My original assumption matches their investigation. It seems (at least by this model) that at the end of the video, each glass is containing equal amounts of their original soda.

Another teacher Chris Hunter (@ChrisHunter36) also battled with this video and created an excellent read about his experiences. Check them out.

 

Proof and Consequences: Circular Reasoning

I’m frustrating my students in ways that I don’t want to. I’m not sure exactly what to do about it. In geometry there’s proof. With proof comes a certain logical structure. Once you know this structure, it is terribly difficult to unknow.

Currently we are dealing with similarity, which involves using SSS, SAS, and AA postulates to prove whether or not two triangles are similar.

Suppose I gave this image to a student and asked them to find whether or not triangle FES and triangle GHS were similar.

Similarity4

Let’s suppose the student divides 54 by 24, and also 58.5 by 26. Both times the student gets 2.25 as a solution. The student assumes this is a scale factor and applies it to GH, finding that FE = 45. The student then divides 45 by 20 and gets 2.25 for a third time. That’s three pairs of proportional side lengths and BAM! Similarity proven by SSS.

Except…

Me, the teacher, is there is tell the student that he or she isn’t quite right. (You see the mistake, right?)

The student assumed similarity before it was proven. Then proceeded to use the assumed scale factor to find the missing side length, which ensured that the third quotient was going to be the same as the first two. This is circular reasoning. They are similar because FE = 45. FE = 45 because they are similar. I have seen this play out countless times.

I have addressed it with little success. I can’t seem to make sense to the students why that argument is weak. It sounds like “geometry teacher says we can’t, so do what geometry teacher says.”  (especially when the very next question asks for FE, which is 45… because the triangles ARE similar…). I can’t stand using my authority as a teacher to enforce a math idea that the students are perfectly capable of actually learning.

I’m trying to decide how picky to be with this. I have a hard time allowing that circular reasoning argument to be called correct, although it is clear that the student has learned a lot about similarity, proportionality, and the structure of a proof.

But the more I push the point, the more frustrated I get and the students don’t seem to be getting any significant gains. I just continue to enforce that “math teachers are just picky like that.”

I am hoping for some help on this one. I’ve tried a lot of things, but that thing you like that works really well for you… I haven’t tried that one. Toss it my way. I want to see how well it works.