## Trying to find math inside everything else

### Trig without Trig

Over the summer I made a Donors Choose page in the hope of getting some clinometers, so that we can go out into the world and use them to calculate the heights of objects like trees and buildings. And we did!
I created the Clinometer Lab as an introduction to Trigonometry. As such, prior to the lab, they had not seen any trig at all. So I started off with this video as the homework from the night before.

In class, we talked about how, when the angle is the same, the ratio of the height and shadow is the same. And how, long ago, mathematicians made huge lists of all of these ratios.
So if they told me any angle, I could tell them the ratio, and then they could just set up the proportion and solve.

So we left the building and went to the park, armed with clinometers, measuring tape, calculators, and INBs, so do some calculations. I had the students get the angles from their eyes to the height of a tree, from two different spots, and the distances, so they could set up a proportion and discover the height of the tree. (When they finished, they did a nearby building. If they finished that too, the extension problem was to switch it up: given the height of a famous building, the Metlife Tower, that they could see from the park, and determine how far away they were.)

It went really well, and I think they got the idea. The fact that they could choose which tree to measure and were given free range of the park, and could choose where to stand to look at the top, but always had to come back to me to get their ratio, worked seamlessly. I could check in on them easily and keep them on the right path. (Especially with my co-teacher there to keep them all wrangled, or the AP who observed/helped chaperone the classes without my co-teacher.)

The next class, I revealed to them that I was not using that crazy chart to give them their ratio, but rather they can just get it themselves from the calculator, which basically internalized the list. Then they got it, that is was just ratios and proportions, not some crazy function thing.

Trig as a topic did not go great in my class, but that is the fault of my follow-up lesson, where I tried to squeeze in too much and didn’t do any practice, not that fault of this lesson, which still sticks in their minds. Later in the year we were doing a project about utilizing unused space, so we picked empty lots but couldn’t get in. So in order to figure out how big they were, we used clinometers, and trig. Now that’s a real world application.

### The Lab

Clinometer Lab Instructions

Clinometer Lab Sheet

### The Archimedes Lab

I love this lab, but I’m sure it can be improved, so I’d love to hear feedback in the comments.

When I was student teaching at Banana Kelly, they had a combined math/science program that often had many lab activities, both math labs and science labs. But I noticed the Systems of Equations unit didn’t really have a lab, so I set out to make one. I created the Archimedes lab because of that.

The story of Archimedes and the Crown of Syracuse says that he was able to determine that the crown was not pure gold by using mass and volume. We start by watching a short video about the story. (I had assigned the video as homework, but no one watched it. While my video homework watching rates aren’t stellar, they’ve never been this bad. I blame the fact that it’s been a while: the last video was in December.)

Then I tell them that we can do Archimedes one better. It’s been over 2000 years since he lived and we have algebra and precise tools on our side: we can figure out exactly how much gold was stolen, not just if some was.

To demonstrate this, we built several Mystery Blocks out of two different materials. Back at BK, we had lots of different materials in the science closet to use. Because the students would know the type of material (brass, aluminum, wood, etc) but not the size and shape, they would have to use density to solve the problem. (But I had seeded the story throughout the whole unit.) The density kit I requested last year never arrived, though, so I had to scrounge around for something, and I found the perfect thing: Cuisinaire Rods.

I noticed that, despite what I expected, the longer Cuisinaire rods are much less dense that the small single cubes. So I used some amount of cubes and some amount of rods to make the mystery blocks. Then I simply give the kids one cube, one rod, the block, a scale, and a ruler, and ask them to figure out how many if each kind are in the block.

This lab was much more procedural when I first made it, but I’ve embraced the problem-solving classroom since then. And while my students were unsure of what to do, I feel like we’ve built up to this kind of unstructured problem, and most of them took to it well.

When they thought they had a solution, I would press them on it. “Why do you think it’s 5 yellow blocks and 5 cubes?” If they responded that that matches up in size to the block, I would point out any others do also, like 4 yellow and 10 cubes. I would implore them to use the tools available to them. (We also talked about mass, volume, and density in the warm-up.) If they told me some combination gave them the right mass, I would point out that their volume would be off. If they told me no combination gave them the right volume AND right mass, I would ask why. Most would identify the tape/foil/wrapping paper as the culprit.

Most importantly, if they gave me a solution that was well supported, even if incorrect, I told them there was only one way to find out if they were right: unwrap the mystery block and see. The excitement at each table when they got to that point was palpable. One student said it felt like Christmas. If they were right, they were ecstatic. If they were not, they tried to figure out where they went wrong.

When I first made this lab, I put it at the end of the Systems unit, thinking they would need those tools. Now I have the confidence to put it at the beginning, and let them figure it out. No one used equations explicitly to solve the problem, but they definitely grappled with the idea that two separate constraints need to both be satisfied in order to solve a problem, which is an important understanding for systems.

This post is getting a little long, so I’ll probably split off into another posts with problems I had with the lesson, things that went well, and a call for suggestions. I just wanted to get this out there. It’s been a long time since I’ve blogged, because I’ve had a lot going on in my life. So I want to get back to it.

### The Lab Sheet

The Archimedes Lab

Accompanying Graph

### Fish Populations and Proportions

One of the labs I did back at Banana Kelly was a fish population estimation lab. You may have seen something like it before elsewhere. The idea is to explore proportions and the mark and recapture technique of population estimation.

The gist is this: students have “lakes” filled with “fish” (boxes filled with lima beans). They use a sampling tool to collect a sample of fish and tag them all with stickers. Then they release the fish, mix them up, re-sample, and use proportions to determine the population of the lake. They do it a few times and average, then they count the actual population to see how close they were.

But I was at a BBQ the week before I did this lesson, and I was talking to my friend Rachel, who is a marine biologist. I mentioned the lab, and we talked about what they use tags for. One thing is to track populations over time, so they can determine the changes in populations since each different year has a different tag. I wondered if I could change the lab to include that.

(Rachel also dug up the video that I had students watch the night before. I’ve decide to have a little “flip” in my classroom by having students watch a video before we do a lab and start asking questions, which I can then address in the next class.)
So I thought about how I could change it. It actually took a lot of thinking, jotting things down on the white board, consulting with the living environment teacher to make sure I was on the right track. But I extended it, so now they would do at least 5 different calculations in the process, instead of spending all that time on just one proportion.

Now, students do the first part the same as before. Then, a random sample of fish “die” and are removed from the lake and put side, and a bunch of new fish are “born” by taking them from the bag of beans I had. Then when they took a sample of the new lake, they tagged the new fish (not already tagged) with a different color sticker. Now they had data from both years and could figure out the new population, and the difference from the old population.

Not every group got to the extension, but I think it improved the task overall.

### The Materials

Fish Lab Instructions (formatted to fit in an INB)

The Lab Report

### Math Labs

When I student taught at Banana Kelly High School, the 9th grade math and science teachers there used a wonderful curriculum called Thinking Math and Science, which they had been developing for about 10 years. Those classes were integrated with math and science together, and so very often the classes were doing labs. But the labs weren’t just science, they just as often had math labs. And I wanted to bring that idea into my own classroom.

I had decided last year that I wanted to introduce new topics with labs, so the students could explore an idea before getting the mathematical language that does with it. When I sat down over the summer with my co-teacher Sarah, we created a template for our math lab reports, taking the steps of the scientific method and putting a mathematical twist on it. Here’s an example of it, using the first lab we did, Pythagorean Theorem in 3D.

The beginning is much the same, asking the driving question that we want to answer. Then, instead of background research, since I want to work with a low barrier of entry and move up, we have “What do you notice?” (thanks @maxmathforum).

The next step is to construct a hypothesis. This is often still relevant with math, of course, and may go unchanged for some labs. But I thought another way we could look at a hypothesis is an estimate, since both are educated guesses, right? I set it up using Dan Meyer’s suggestion of “too low, too high, actual guess,” which gives us nice bounds, and I think this does it visually as well. Although not completely, since some students haven’t gotten it, so I wonder if I can improve on that. (I have two versions in the file: the arrow one is the one I used, and the dotted line one is a new idea I have, I’ll try it soon.)

Then we do our calculations, which is our experiment, they go hand in hand. And finally we analyze what we did with discussion questions.

I’m don’t think the format/template is perfect, but I think it’s a start.

### Recursive Combinations with Replacement

So I was in my classroom last night with my boyfriend, waiting for his phone to charge before we went to dinner. Since we had some time, we played some of the math games I have in my room. (He’s a math PhD student, so he was all for it.) We played Set, of course, and then played a bit of 24. We idly wondered if it were possible to get 24 with any combination of 4 digits. So I looked at the box, and saw it came with 192 possible configurations. Well, if we determined how many possibilities there were (maybe there were 192), that might give us an idea of the feasibility.

So we tried to calculate how many configurations there were. Shouldn’t be too hard, right? Well, it kinda is, especially when you’re not already familiar with combinations with replacement. So we started using what we did know of combinations, but were stuck because we could use the same number multiple times, which made it trickier. Otherwise it would just be 9 C 4.

So, unsure how to solve, we tried to make a simpler case. What if we only had 2 digits to choose from, not 9? There’s there’s 5 possibilities. (1111, 1112, 1122, 1222, 2222.) And with 3 digits, there’s 15 (1111, 1112, 1113, 1122, 1123, 1133, 1222, 1223, 1233, 1333, 2223, 2233, 2333, 3333). We got a lot of fruitful thinking out of this, finding patterns, but didn’t really get closer to the answer. (Four digits had 35, btw. But we didn’t want to list all the ones for 5 digits and beyond.)

At this point it was time to go to dinner, so we put the whiteboard aside. But that couldn’t stop us thinking and talking about it, which we did as we walked to the restaurant and waited for out table, when we finally had a breakthrough.

Instead of trying to figure out the pattern with fewer digits but the same number of slots, let’s try to iterate up with the same number of digits, but using increasing number of slots. Let me explain, using 4 possible digits.

If we only have 1 number slot on the card, there are only 4 possibilities. (1, 2, 3, 4.) When we increase to 2 slots, we could start by putting a 1 in front of each of those possibilities. (11, 12, 13, 14). But, because order doesn’t matter, we can’t also put 2 in front of everything, because 21 is the same as 22. So we don’t use the one, and get 22, 23, 24. Same logic for 3 gives us 33, 34, and then finally 44.

This gives us a total of 10 possibilities. (4 + 3 + 2 + 1.) Now let’s think about 3 slots. In the same way, we can add a 1 in front of everything we’ve done so far. So for 3 digit possibilities there are 10 that start 1. Since we have to eliminate the four that two-digit configurations that have 1, there are 6 remaining, so that’s how many will start with 2. (3 + 2 + 1). Then three will start with 3. (2 + 1) And 1 will start with 4.

The process here is to add up all of what we had before, chopping off the start, to get the total number of new possibilities. So now, with 3 slots, we have 20 possibilities. (10 + 6 + 3 + 1.) To get for 4 slots, we use the same process: 20 start with 1, 10 start with 2 (6 + 3 + 1), 4 start with 3 (3 + 1), and 1 starts with 4, for a total of 35. Which is what we found before.

(If there were 5 slots, it would be 35 + 15 + 5 + 1, or 56.)

I don’t know of this recursive method of solving for combinations with replacements has been done before. I’m sure it is, but I haven’t found it in a very short google search. If someone knows of it, please let me know. But I wanted to share what I did. You can tell I love math, and so does my boyfriend, because we got completely distracted from a board game by solving a problem. He told me I’d make a good mathematician, because of how I tackled the problem. That may be true.

### 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.

### Is Algebra 2 Necessary?

So, of course, Andrew Hacker’s article “Is Algebra Necessary?” had caused quite the stir, and the obvious answer to that question was “Yes, algebra is necessary.” But the article makes you think if all of what we learn of algebra is necessary. And I think it isn’t, but that comes from thinking about what high school is for.

Do we expect that, when a student gets to college, they can skip the lower levels of Biology because they took bio in high school? No, of course not. (Excepting AP courses, of course.) So what is our goal for learning biology in high school? It’s to provide a general foundation of the subject, that most people should know, and it prepares you for a college level course or major in Biology.

Really, all of what we learn in high school is designed to broaden our horizons, to provide experiences and content we wouldn’t see otherwise, and to provide a baseline of knowledge that we feel everyone should have.

I remember reading from someone, though I don’t recall who, that they had struggled through Algebra 2 and Pre-Calculus, slogging along, and then when they got to Calculus a light turned on. “This was why we’ve been learning everything we’ve done in the past two years! It was all for this!” Even the wikipedia page on Pre-Calc says “…precalculus does not involve calculus, but explores topics that will be applied in calculus.” It’s putting the work before the motivating problem, again.

But now thinking about the normal course sequence for a student that is not advanced: Algebra –> Geometry –> Algebra 2 –> Pre-Calculus –> Graduated from High School, so no Calc! So these students will have two whole years of math without the payoff that shows why we do it.

And as teachers we know that you need to start with the motivating factor, not have it at the end. So why don’t we have calculus first, before those two? If we consider our goal in high school is to spread ideas people might not see otherwise, I think Calculus has a lot of important ideas people should see that would improve their lives. Optimization? The very idea of it can improve how you look at all the problems in your life. Related rates, limits, the idea of changing rates and local rates, the relationships between functions, these are all good ideas to be familiar with.

Can the students learn these things without having done Algebra 2/Pre-Calc? I think so. As Bowman Dickson says, “The hardest part of calculus is algebra.” So what if we taught it in a way that didn’t rely on that? We can get the ideas across without jumping into the nitty-gritty of a lot of it. Save that for AP level classes, or for college calc. What you take in college is more in depth that high school, so it should be the same here.

Now, there would certainly be some stuff from Algebra 2/Pre-Calc that we really need first. But why not have those in Algebra 1? I accidentally taught several things from Alg 2 when I taught Alg 1 my first year, because they seemed like natural extensions of what we were doing, and I didn’t know they weren’t required until I started planning for the next year. But also, consider this. If we made Probability & Statistics one of the main courses of the math sequence, I don’t have to teach it in Algebra 1. I spent about 7 weeks on those topics last year. That’s 7 weeks of Alg 2 content I could fold in, without worrying about reviewing old stuff because we just did it.

So then the new math sequence could be Statistics –> Geometry –> Algebra –> Calculus. (And I think that might fit well with the science sequence of Biology –> Earth Science –> Chemistry –> Physics.)
Thoughts?

A bit ago I got yelled at by a commenter on Kate’s blog who claimed that being always right is why we like math. The problem with that point of view is that, while yes, you can always be right while doing computation, math isn’t just computation. So the other day I was talking with a friend of mine, and that prompted me to post the following tweets:

My friend Phil (@albrecht_letao) responded to the question, and he came up with an answer of $20/hr. When I worked it out with my friend, we came up with$14.25. Does that mean one of us is wrong, since we got different numbers?

No, of course not. What happened is we approached the problems in different ways. Phil only calculated the monetary value: with his amount, my friend would earn the same amount of money she does now. He figured this was an important way to look at it, for paying bills and whatnot. Our calculation came from thinking about how her time is being compensated. Since those 16 hours are being wasted (she has to work them for free; actually, she pays to lose that time), we calculated her “real” hourly rate and used that.

There can be more answers than even these two, depending on what you think is important. But it’s a clear example of a problem, solved using math, with no one right answer. That’s what math is about. I tweeted it thinking maybe it could be a problem worth considering in class, to show that essential idea to students.

What do you think?

P.S. The right answer, of course, came from @calcdave:

So I’ve been working on creating this board game, Totally Radical. (Tagline: Don’t Be a Square.) After some play-testing and adjustments, and bouncing ideas off of other teachers, I’m ready to post about it.

(But first, thanks to my co-teacher Sarah for helping come up with the game, my coworkers Cindy and Jenn and my Tweeps Max, Jami, and Jamie for playtesting.)

The idea behind the game came before I didn’t really have a good application for simplifying radicals. But I’ve been annoyed at how I see math games designed: do some math action and, if you are correct, you then get to do some game action. While this is certainly how some games work (like Trivial Pursuit), it just separates the math from the game and makes the math seem worthless. So I wanted a game where the math action WAS the game action.

You can read the rules of the game right here: Totally Radical Rules. During the game you have a choice of 5 actions: 3 involve actions we take when simplifying (breaking a number into two factors, taking a root and putting it outside the radical symbol, multiplying two terms together) while two are purely game actions (draw a card, play a special “Action” card).

Other touches of note: the factor cards are exactly half the size of the radicand cards, so that students break up “larger” numbers into “smaller” ones.

You can use factor cards on their own or combined into multi-digit numbers, like so:

(the top would be two factors, 2 and 5, and the bottom would be one factor, 25)
The numbers in the radicand cards are not just simple numbers. There’s prime numbers, composite numbers that can’t be simplified, perfect squares, as well as numbers that can be simplified (going all the way up to 250).

So, how can get this game, you may ask? Two ways!

### Make It Yourself

If you want it for free, or are just in that #Made4Math mindset, you can print out the following files on card stock:

Prototype Factor Deck

Cut the cards out and label the backs. Print out the instructions (found here). You’ll also need to make a board: 4 big radical signs (I also recommend cardstock.) That might look something like this:

(I also drew in spots to put the card decks in).

Don’t want to make it or want the awesome one pictured above? Then go for option 2:

I found this great website called The Game Crafter where you can send in artwork, pick out the pieces, etc, and they will print and construct the game for you. So if you click the button below, it should bring you to the shop to buy it.

### Lab –> Lecture –> Assessment

Next year, the weekly schedule at my school is going to be 2 double periods for a particular class (alternating sections on an A/B day schedule) with a single period for every section on Wednesday. Because of the new schedule, I wanted to make a new structure for my class, which is the title of this post: Lab –> Lecture –> Assessment.

There are roughly 30 proper weeks of learning in the year, so I figured I would have 30 Learning Goals to cover, and do one each week. I would introduce each learning goal with a “math lab,” which may be an actual lab (like the popular M&M Lab for exponential growth/decay) or a 3 Act problem or something else that the students can really engage in before getting down to the nitty-gritty and symbolic way mathematicians deal with the problem.

The next double wouldn’t necessarily just be lecture, but it would be the abstraction of what we did the lesson before, including lecturing on technique and practicing what we’ve learned. Then assessment could be any number of things, but will almost certainly involve a targeted quiz.

Seems like a good structure, right? Problem is, while I have a lot of good labs and problems for most of the topics (and will keep improving), not all of them do. Particularly:

1. Radicals – Simplifying & Arithmetic
2. Unit Conversion
3. Solving in Terms Of
4. Box-and-Whisker Plots / Percentiles
5. Scientific Notation
6. Statistics Vocabulary (univariate/bivariate, etc.)

So my major goal this summer will be to develop something for each of those. The rest I can fall back on what I have, even if I don’t come up with something new/better. But these have nothing. My first task/idea is to develop a board game about radicals. That’s still under development. Any other suggestions would be appreciated.