Trying to find math inside everything else

Potluck Math

I was talking to one of my co-workers about a “Friendsgiving” she is holding, and how the food bill is getting up there as more people are invited. But some of those people are also bringing food – and everyone is worried about having enough.

I realized this is a very common problem with potluck meals. Everyone wants to make sure they have enough food, so the more guests, the more they make. But think about this –

At a 4-person meal, each person makes a dish that feeds 4. (4 servings). So each person then eats 4 servings of food. (Which seems like a normal amount.)

Now it’s a 20 person meal, and each person makes a dish that feeds 20. So now each person eats 20 servings? That seems unlikely – it’s much more likely that people eat 3-6 servings, for 60-120 servings eaten, leaving 80 servings of food left over.

The problem here is that each attendee is treating the problem linearly, when it would better be modeled quadratically. Of course, this is complicated by the one hit dish that every eats a full serving of, and that other dish that no one eats, and everyone wanting to try a little of everything, so figuring out how much to cook can get complicated pretty quickly.

I wanted a lesson to explore the properties of the diagonals of different types of quadrilaterals, but the curriculum map I was following just lead to Khan Academy, and that’s not really my speed. And some scanning through MTBoS resources didn’t find me what I wanted, but chatting out my half-formed ideas with Jasmine in the morning focused the idea into what I did in class today.

I started by having the students draw 6 triangles: 3 scalene, non-right triangles; 1 isosceles non-right triangle; 1 scalene right triangle; and 1 isosceles right triangle. Then we used each of those figures to create a quadrilateral by making some sort of diagonal. Each time, I asked them to identify the quadrilateral and what they noticed about the diagonals.

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First, take one of the scalene triangles and reflect it over one of its sides. Thus we created a kite – which we know because the reflection creates the congruent adjacent sides. Then we can use the properties of isosceles triangles – we know the line of reflection is the median of the isosceles triangles because of the reflection, so it is also the altitude, meaning the diagonals are perpendicular.


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Then, take another scalene triangle and reflect is over the perpendicular bisector of one of the sides. This makes an isosceles trapezoid – we know the top base is parallel to the bottom base because they are both perpendicular to the same line, and it’s isosceles because of the reflected side of the triangle. Then we notice the diagonals are also made of a reflected side of the triangle – and so we can conclude that the diagonals of an isosceles trapezoid are congruent.

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For the third one, I asked them to draw a median and then rotate the triangle 180°. The trickiest bit here is to prove that this is a parallelogram – previously we had classified the quadrilaterals by their symmetries, so using the symmetry definition we could say any quad with 180° rotational symmetry is a parallelogram. Or we can use the congruent angles to prove the sides are parallel. Once we did that, we saw that, because we used the median, that the intersection of the diagonals is the midpoint of both – and thus the diagonals bisect each other.

I then tasked them to figure out how to make a rhombus, rectangle, and square out of the remaining triangles using the triangles. Because we proved the facts about the diagonals of the parent figures, we could then determine the properties of the diagonals of the child figures.

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I think it went pretty well – the students performed the transformations and easily saw the connections between the diagonals. Tomorrow I think we’ll do something about whether or not those diagonal properties are reversible – if every quad with perpendicular diagonals is a kite, for example.

Crossing the Transverse

Oh my god, I haven’t blogged since August! This has been a hell of a year, let me tell you. But maybe I’ll tell you in another post, because this one is about the new game I made in my Geometry class. (My first non-Algebra game!)

So the game is called Crossing the Transverse. The goal of the game (pedagogically) is to help identify the pairs of angles formed by lines cut by a transversal, even in the most complex of diagrams. The goal of the game (play-wise) is to capture your enemy’s flagship.

Here’s the gameboard:

Crossing the Transverse Map

I printed out the board in fourths, on four different pieces of card stocked, and taped them together to make a nice quad-fold board. Then I made the fleet of ships out of centimeter cubes I had, by writing in permanent marker on the pieces the letter for each ship.

Quad Fold Board

Here’s the rules.

In the game, each type of ship moves a different way, which makes it feel a lot like chess – trying to lay a trap for the enemy flagship without being captured yourself.  Many of my students really enjoyed it when we played it yesterday. Today, though, to solidify, I followed up with this worksheet where they had to analyze the angles of a diagram much like on the game board. They did pretty well on it, so I’m satisfied!


Crossing the Transverse Rules

Printable Map (Prints on 4 pages)

No Stars Printable Map (If printing the background galaxy is not for you, here’s a more barebones version.)

Zip File with Everything, including Pages, Doc, and GGB files

The Spirit of the Rules

I just read the book One Man Guy by Michael Barakiva. (Pretty good, but has some problems). I wanted to share a scene from the book. (Emphasis mine.)

“Does that mean your absence last Friday, unlike your earlier absences this semester, was unexcused?” Mr. Weedin asked.

“It does,” Alek admitted.

“Mr. Khederian, you clearly have a strong grip on this material, and if you hadn’t cut, I would’ve considered recommending you for the Honor Track next year. But I’m afraid that I can’t go around making exceptions for students, regardless of how bright they appear.” Mr. Weedin’s picked up his paper and continued reading.

His teacher’s resolution almost made Alek give up. But he knew how important this was for his parents. And, he had to admit, for himself as well.

“Mr. Weedin, don’t you think failing me in a class when you think I’m capable of delivering Honor Track material is counterproductive?” Alek cleared his throat. ” ‘Let us once lose our oaths to find ourselves, / Or else we lose ourselves to keep our oaths.’ ”

“Is that Shakespeare?” Mr. Weedin asked, intrigued.

“Yeah, it’s from Love’s Labour’s Lost. I just wrote an essay comparing and contrasting that play to Romeo and Juliet in English, and that quote really stuck in my head.”

“Why?” Mr. Weeding leaned back and slid his glasses down so he could peer at Alek unobstructed.

“I guess I feel like we spend so much time trying to keep the promises we make, or the rules we set up, but it’s also important to look at those promises and rules and make sure they’re actually doing what we want them to do, and not the other way around.”

“Well, Mr. Khederian, you make a persuasive case.” Mr. Weedin tapped his pencil against his desk three times. “I’m not going to make it easy for you. For the remainder of this class, I’m going to double your homework load. If you complete it all satisfactorily, then I will reduce the penalty from failing to dropping your grade one full letter. So the highest grade you could receive would be a B.”

Alek had to stop himself from hugging Mr. Weeding. “Thank you, Mr. Weedin, thank you so, so much. I promise that I’ll do my best.”

“What is your best, I wonder?”

“I don’t know, Mr. Weeding, but I’m looking forward to finding out.”

“Me too, Alek.”

This was a major theme of the book and one I appreciated (it’s reminds me of Fiddler on the Roof in a way, especially the climax of the book). This isn’t the most moving scene but as this is ostensibly a teaching blog, I thought I would share it.

This seems especially relevant given the two articles I saw earlier today – one about the student who passed because of admin pressure even though she “deserved to fail” and the other about the student who passed because her teacher felt she should, despite her parents thinking she “deserved to fail.”

It’s interesting on its own to contrast the two articles. But now look at it through the lens of the quotation above. What does it mean to pass or fail a student, and why do we do it?  What is the goal of the grades that we give? Often teachers set rules in their classrooms, or grading policies, and stick to them rigidly, thinking that is what is right. But it is easy to lose sight of why we made these rules in the first place – because we want our students to be the best they can be. Most of the time those rules will help that happen – but sometimes they don’t, and so we need to be willing to change when that occurs. It is the spirit of the law that matters, so try not to get lost in the letter.

Fighting for the Center

At the Math Games morning session at Twitter Math Camp 15, we’ve been created curricular games that hit on some topics that there aren’t really good games for. I came up with the idea for this one, and worked on refining it with the help of Paula Torres (@lohstorres1) and John Golden (@mathhombre).

This game is about measures of central tendency (and range for good measure). Not only do students have to determine all of those over and over as they play the game, but they can see how changing the data set changes the values, especially as the size of the data set increases or decreases. It seems really good because it drives the need to make those calculations.

All you need is two decks of cards. The game is designed as a two-player game, but it would definitely be best done as two pairs playing against each other, so they can talk to each other about their strategies and calculations. We also recommend having students keep a running tally of the values.

Last year, before Twitter Math Camp, I was packing and trying to figure out which games to bring with me for the game night we were having before the conference started. I basically had three attributes I was considering: how big the game was, how good it was, and how many players could play it. I wanted to minimize the first one while maximizing the latter two.

So I tried to come up with a bunch of formulas for figuring it out, but nothing was quite working out. (I used BoardGameGeek ratings for “how good it was.”) At first I tried doing {\frac{r \cdot p}{v}}, but it was putting some games that just weren’t very good as top choices. The problem was that the volume was having too big of an effect – games could come in thousands of cubic centimeters of volume, but max at around 8 for rating and 12 for players. (I had to use to look up the dimensions because I wanted to use centimeters.)

So then I tried cube rooting the volume, or doing an exponential functions like {\frac{p \cdot e^{r}}{v}}%s=2, or finding the geometric mean of the three numbers, but still nothing came out right.

I was basically using three games as test cases: Dominion, which is one of the best games I own but it really big; Pixel Tactics, which is one of the smallest but is only 2 players; and The Resistance, which is small-ish, really good, and can go to 10 players. I figured that any good method should tell me to leave the first two games at home, but to bring the Resistance. If they didn’t, it wasn’t right.

Eventually, after doing some research, I determined that a common technique used in psychology when comparing variables of different ranges of values is called standardizing the variables. Basically, for each attribute, I would find the mean and standard deviation. Then, for each game, I would subtract its value from the mean and divide by the standard deviation to get a standardized value. Then I just needed to add up the three standardized values and the ones with the highest score would win. And, as predicted, The Resistance came out on top.

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Whiteboard Desks

After TMC14, I heard a lot of talk about Vertical Non-Permanent Surfaces (though I didn’t go to that session myself). After reading Alex Overwijk’s post about it, I wanted to use the idea in my classroom, but getting vertical boards up seemed challenging considering how long everything takes in school. The researched showed the horizontal non-permanent surfaces were the second best thing, so I decided to take matters into my own hands. I went to the hardware store, bought some whiteboard paint, and got to work.

I put the paper down to prevent drips but, of course, dripped anyway.

Overall the desks have been amazing. The students love getting the markers and working with them, especially because they hate committing things to paper when they might be wrong. (Yes, kids do doodle/play tic tac toe/etc on the desks, too, but I think that’s no different that what they’d do on paper.) Another benefit is how easy it is for me to interact with the students when I go around. Instead of having to write something on the student’s paper or notebook, which always felt intrusive, I can jot something quickly down on the table itself, leaving it to the student to work it into their own thinking. It’s worked great for tutoring (so I don’t have to get up and go to the board). The paint might not last as long as it could when the room is used by other classes who don’t know what’s up (our night school in particular, I’d say), but it’ll last the year, at least, and I’m more than happy to repaint them before next year.

The only picture of student work on desks I had that wasn’t blurry as hell. I need to work on my photography.



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