## Trying to find math inside everything else

### The Cold War

In my first year teaching I came up with this activity for working with quadratic-linear systems, based in the Cold War and missile defense. It didn’t work as well as I hoped, mostly because it was too complicated, but I like the core of the idea. Maybe now, with more experience and the brainstorming power of the MTBoS, we can think of a way to make it work. But first, I’ll describe what .i actually did.

Students entered the room to find the desks rearranged – four big group tables, and the room split down the middle by a wall of desks, representing the “Iron Curtain.” Each student was then randomly assigned to one of four groups: US Missile Command, US Missile Defense, USSR Missile Command, and USSR Missile Defense. (Only one student, the son of the Georgian consulate, demanded to be switched from the USSR group to the US side.)

Each student then had two roles – one of the roles was their job on the team. Treasurer, secretary, chief engineer, etc. These roles were public. Their other roles were secret – they were things like Double Agent, Handler, FBI Agent, Innocent.

The idea was that each missile team was trying to build a missile that could hit the other country, while bypassing their missile defense. And the missile defense teams were trying to shoot down the missiles. The missiles were represented by quadratic equations and the missile defense by linear functions. But the best way to find out what the other side was planning is through espionage.

Of course, the thing they’ll probably learn is that the missile defense fails and everyone dies – we all lose the cold war.

Below are the files I made way back when. What are your ideas to make this workable?

### Wits and Wagers and Number Sense

I bought my boyfriend the game Wits and Wagers for Christmas, after seeing it on Tabletop and thinking he would enjoy it. (Feel free to watch the episode of Tabletop for a good example of how it works, though they play the Family edition, and we have the standard version.)

The basic premise of the game is that everyone is asked the same question, which always has a numerical answer (including dates). Everyone secretly writes down their guess, with the goal of being the closest without going over. Then, everyone reveals their answers and they are put in order on the board. At that point, everyone bets on which they think is the best answer. The answer is revealed, and the person who wrote the best answer and everyone who guessed it gets points.

I played it with him and some friends on New Year’s Eve, and saw some interesting results that made me think the game would be a good tool for developing number sense. In fact, one of his friends said that she wasn’t good at the game because she had no idea who was even a good range for an answer. But the game itself provides you with that sense, by consensus.

I know I read once, though I can’t remember where, about how when prompting a class for a guess, most student guesses will fall somewhere in the same order of magnitude as the first guess, even if that first guess is super ridiculous. (Like, for a guess at how tall the Eiffel Tower is, the first person guesses 6 miles. No, that can’t be right, the second person says. More like 4 miles.) The way to avoid this is to have people right down their answers ahead of time, a mechanic built into the game.

Sometimes that leads to interesting situations. One question asked how many episodes of Friends there had been, and this had been our responses:

This almost feels like a Math Mistakes question, where did this person go wrong in their guess? But comparing their sense of answer to the consensus helps us get an idea of what’s right, and what’s misinterpreted. (In this case, the person thought it was asking how many episodes have been shown on TV ever, like in syndication and whatnot, in which her answer then makes a lot of sense. So never dismiss an answer just because it seems so far off the mark. There’s always a reason!)

A lot of questions had a historical bent as well (years), so then can help build a sense of time as well. (As long as a rogue history teacher isn’t sitting nearby shouting out answers even though he isn’t playing the game.)

In the end, I think this game could go along with something like Estimation180 for building number sense, but in a more communal gaming way. If you talk to people about how they chose their numbers, we can get a sense of their mathematical thinking. And that’s worth a lot.

### How Many Representatives Should We Have

Back in 1788, James Madison wrote up 20 proposed articles to amend to the constitution. 12 of those were approved by Congress. The latter 10 were ratified by the states and became the Bill of Rights. The second was ratified over 200 years later and became the 27th Amendment. But Article the First was never ratified. Here’s what it said (corrected):

After the first enumeration required by the first article of the Constitution, there shall be one Representative for every thirty thousand, until the number shall amount to one hundred, after which the proportion shall be so regulated by Congress, that there shall be not less than one hundred Representatives, nor less than one Representative for every forty thousand persons, until the number of Representatives shall amount to two hundred; after which the proportion shall be so regulated by Congress, that there shall not be less than two hundred Representatives, nor less than one Representative for every fifty thousand persons.

Back in 1911, Congress froze its size at 435 members of the House of Representatives, and so the amount of people representative by each representative has grown extraordinarily. (Note that this is before we even had all the states (only 46), so the Reps continued to spread thinner.) The average district size now is about 700,000 people, which is a lot of people and opinions to accurately represent. Of course, if we followed Article the First to the letter, we would now have about 6300 representatives, which seems like a lot.

Source: thirty-thousand.org

But what if the article is a formula, not meant to stop at districts of 50000? The way it is written, it seems like every 100 Representatives would prompt an increase in the size cap of districts by 10000. So how could we model that to determine how many reps we need?

Well, in general, the population divided by the number of people in the average district should give us the number of reps. So if P = U.S. Population, R = number of representatives, and D = max size of district, then $R=\frac{P}{D}$.

To represent Article the First, since 0-100 reps have 30000 each, 100-200 have 40000 each, 200-300 have 50000, etc, it seems like we could say $R=\frac{D-20000}{100}$ to give a rough estimate. (Anyone have anything more precise?) So we can substitute, as well as plugging in 308,745,000 for P (according to the 2010 census), to get

$\frac{D-20000}{100}=\frac{308745000}{D}$, and solving for D gets us approximately 186000 people per district. Plug in for D to get 1660 representatives. (Exact amount varies by the precise district make-up.) That seems quite possible, not even four times as many as we have now.

Follow-up questions to consider:

• Is 700000 people too many to represent? Is 190000? What would be an ideal amount?
• How would representing 30000 people in 1790 be different from representing that many people now? How does technology change how effectively we can represent people?
• How could we accommodate having 1700 representatives? What changes would need to be made?
• What other representative systems could you come up with? How would it work?
• How would having more representatives change our current representation?
• How are representatives apportioned in other countries? What methods do they use for determining the size?

For that last one, I think it’s interesting to just look at the Congressional districts of New York City as an example.

I live in District 12. It’s easy to see that the district is half in Manhattan, in the affluent Upper East Side, and half in Queens, in Astoria/Sunnyside/Long Island City. I think it would be very easy to believe that the desires of the people on the UES don’t always line up with the desires of the Queens constituents. Yet we are represented by just one person. However, with a smaller district, they can be divvied up more logically. All of Astoria has 166,000 people, which is almost a full district, and it would be nice to have a district that is clearly where you live.

Since US History doesn’t usually line up with Algebra, this idea might be hard to implement in math. Though it could work fine in Algebra 2. And it might work even better as a history lesson with a bit of math, instead of a math lesson with a bit of history? I dunno. But I think it can definitely be food for thought for any class.

### Math Needs to Be the Spark

At Twitter Math Camp I gave the following talk. The abstract from the program said:

When planning interdisciplinary projects, math teachers need to take the lead in order to create cohesive and authentic projects, and to ensure that the project doesn’t just become psuedocontext for their math goals. Uses two major interdisciplinary projects developed at my school as examples of how to bring all the subjects together, so math isn’t left out in the cold.

Here’s the talk:

After that I opened to questions. The one that I remember was asked by @JamiDanielle: “How can you get other teachers who might not be on board for these types of projects to join in?” And I think this process is actually how. If you go to a teacher with an idea and just dump on them to figure out how to connect it to their class, it’s not going to end well. It’s easier and less work to just not take part. But if you go to them with an idea already half-formed of how they can implement it, it is much easier to build off of that idea and will make teachers more willing to work together.

### The Projects

High Line Field Guide v5 – This is the High Line field guide project mentioned in the video, and first mentioned in this blog post, “The Start of the New Year.”

Intersession Project Requirements – It would be difficult to post everything we did in the Intersession project, but the overview from the video and this packet of requirements for the product should be useful. Anyone interested in more can ask.

### They Don’t All Go So Well

I’ve known what my next post was going to be about for some time, which is why it’s been so long between posts, as I’ve been putting it off. The failures are less fun to write about, but it’s just as important when your lesson is a bust. Now I have lots of other things I want to write about, so more posts in the next few days.

Shortly after my successful Egyptian Fractions lesson, I wanted to tie a lesson into another ancient society they were learning about, so I decided to teach the Mesopotamian Number System. The idea was that we’d reinforce some ideas about exponents, place value, and scientific notation by working with another base.

The problem: working with another base is hard, especially if you’ve never done it before, and sexagesimal is not a great place to start, even with the boost I got with the fraction lesson. Introducing the idea with binary probably would have worked, but I didn’t have the time to do both and also teach the cuneiform and do the activity. Unfortunately, to save the activity, the basis of the understanding got cut. Which left me with a fun but useless activity.

I used hours:minutes:seconds as an analog to help understand base-60, but because they got that they couldn’t move past it. I gave them numbers to translate and had them carve cuneiform tax tablets (and they learned about Babylonian taxes), but that didn’t work out too well.

And then I didn’t even get nice product to display for too long, because they were too brittle.

As I said. A bust. Or rather, busted.

### Egyptian Fractions

As I stated earlier, I’ve been trying hard this to integrate the other subjects more into my math lessons (and the other teachers are happy to work vice versa, because I’m on a great grade team). This process is made easier by actually having a Special Ed co-teacher for one section, and she specializes in math (and sees every subject, so can comment on all of them). So my first lesson explicitly tying history to math just went off, a lesson on Egyptian Fractions.

My goal for this lesson was really to get some fraction practice in while still learning something new, while also highlighting the “symbol that represents the multiplicative inverse,” , which I’d tie in on the next lesson about exponents (aka an exponent of -1). We worried, though, that the translation process would be too tough while dealing with fractions. That’s when we came up with this:

The Fraction Board has 60 square on it (which will be good reference for when I deal with sexagesimal Mesopotamian numbers soon), so each piece is cut to fit the amount of square that will cover that fraction of the board. To make the boards, I just made a 6×10 table in word as square-like as I could, printed on card stock. Then I cut the pieces out of the extra boards and had slave labor student volunteers color them in for me.

Each fraction have multiple pieces to represent the different ways you can fit them. (For example, 1/2 is 30 square, so I have a 3 x 10 piece and a 5 x 6 piece). But each fraction is also colored the same, because in Egyptian Fractions you can only use one of each unit fraction.

Then I would put up a slide like this on the board:

And the students would have to make that shape on their boards, with no overlapping and only using each color once. For the first one I shared a possible solution:

But I got really excited when the students could come up with multiple different solutions for each problem. And I would increase the difficulty of each one, until I would just get to a fraction with no picture:

And they still nailed it. Eventually I would move away from the boards and show the process of how to do it without the boards. We’d do some simultaneous calculation (using the greedy algorithm or more natural intuition) and checking on the board. Then we’d try with non-sexagesimal fractions. And every time we would translate our answers into hieroglyphics as well. So by the end of the lesson they could work on a worksheet where I just gave a fraction and they gave me hieroglyphics in return. (Not all of them could do this completely, but most could do some of the sheet). I think, overall, it went pretty well.

Egyptian Fraction Slides (Powerpoint)

Egyptian Fractions Slides (pdf)

(WordPress doesn’t seem like it’ll host my slides in their original Keynote form. That’s bothersome.)

### How to Order the Topics

Not much posting recently, but hey, it’s summer. I’ve mostly done vacationing, now, and am really thinking about the new year.

I just finished reading through the first half of Merzbach’s and Boyer’s A History of Mathematics, up until the Renaissance. I took a list of topics associated with different cultures as I read through, as they may lead to some interesting lessons in the upcoming school year. I’m not really sure of the best way to integrate with the Global curriculum, but the 9th Grade Team is meeting tomorrow and I’m hoping I can talk with the history teacher about it. Obviously an ordering by mathematical sense won’t match a chronological historical ordering, or even a topical historical ordering, but I’m sure something will come out of it.

At least, I feel that, if one had to come first, it is better to have the historical context before the math, than vice versa. Here’s the list I made, though there’s not much to it.

 Algebra Tiles Ancient China Counting Rods Trigonometry Ancient India Number Systems Ancient India Lattice Multiplication Ancient India Radicals Ancient India Fractions Egypt Unit Fractions Adding Like Terms Greece As opposed to the Babylonians Geometric Algebra Greece Ratios Greece Euxodus, in Plato’s Academy Trigonometric Ratios Greece Ptolomy, using circles Longitude and Latitude Greece Completing the Square Islamic Empire Number Systems Maya Bases Systems of Equations Medieval China Pascal’s Triangle Medieval China From the Jade Mirror Number Systems Mesopotamia Bases Context Clues Mesopotamia Place Value Exponents Mesopotamia Place Value Fibonnacci Middle Ages Slope Middle Ages Sine and inclined planes Proportions Pythagoreans Music Radicals Pythagoreans The expulsion of Hippasus Types of Numbers Pythagoreans Numerology

### A History of Math

One of my favorite math courses when I was in undergrad was “History of Math.” I thought it was fascinating to see where it all came from and how math was done before we had more modern advancements or with different number systems.

I wanted to bring some of that history into my Algebra class and have often lamented that it doesn’t match up nicely with Global Studies. The students do Geometry in 10th grade, but the Greeks in 9th. They do Algebra in 9th grade, but the Arabs in 10th. So the obvious connections are not really there.

But there are more, I think, such as perhaps having a lesson about Egyptian fractions to improve students fraction skills while they learn about ancient Egypt in global, as well as to introduce the idea of a negative exponent. I couldn’t find my History of Math textbook from when I was in college, as I may have sold it back to the store, so I just bought a new one.  I’ll start looking through it over the summer and see what connections and lessons I can make.