## Trying to find math inside everything else

### The Integral Struggle

I had an extra day to fill for one of my sections of calculus, thanks to the PSAT, so I set about thinking up a game I could play. I gotta tell you, as much as there is a paucity of good content-related math games out there, it’s extra so as you move up the years in high school. I can still find a good amount of algebra and geometry games online, but Algebra 2? Precalc? Calc? Fuhgeddaboutit. Well, I’m teaching both Pre-Calc and Calc this year, so I guess I’ll just have to make them myself. (I brainstormed two more during said PSAT, so those might happen soon enough.)

We just covered function transformations and how they affected integrals, so this game hits on that topic. (Yeah, we’re doing integrals first.) Here’s how it works: there are 3 functions/graphs, each of which has a total area of 0 on the interval [–10,10]. One team is Team Positive, and the other is Team Negative. Teams take turns placing numbers from –9 to 9 that transform the function and therefore transform the area. Most importantly, they also place the numbers in the limits of integration, so they can just look at a specific part of the graph.

With three functions, it becomes a matter of best of 3 – if the final integral evaluates to a positive number, Team Positive gets a point, and vice versa. If it’s a tie at the end of the game, because one or more of the integrals evaluated to 0 or were undefined, then redo those specific integrals. (I discussed with the students whether it would be better to have it be the total value of all 3 integrals instead of best of 3, but I think that makes the vertical stretch too powerful.)

That’s it! Let me know what you think. Materials below.

### Parallel to a Parabola?

A while back, I was working on a lesson about average rate of change and wondered the following question: “Could you use the word ‘parallel ‘to describe two non-linear functions that have the same rate of change/don’t intersect?”

Jonathan’s response, though, made me think about what it actually means to be parallel. Often when you ask students, they will respond “two lines that never intersect,” which I usually push back against because 1) how do you know they never ever intersect? and b) skew lines never intersect, either. So when I explain parallel lines, I use the fact that they have the same slope/go in the same direction as the actual definition, which has the consequence of never intersecting. So I looked it up on Wikipedia.

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

1. Every point on line m is located at exactly the same minimum distance from line l (equidistant lines).
2. Line m is on the same plane as line l but does not intersect l (even assuming that lines extend to infinity in either direction).
3. Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are congruent. (This is equivalent to Euclid‘s parallel postulate.)

I don’t think statement 3 was particularly useful to me, but the idea of being equidistant was interesting. A vertically shifted parabola is not equidistant from the original – though they never touch, the distance between them gets smaller and smaller.

So that raised the next question – how do I actually measure the distance between two parabolas at a given point? I asked my boyfriend and he responded, “Well, you definitely need calculus….” And who better to swoop in and help with that than Sam Shah.

So now that I know how to find the minimum distance between two functions, all I need to do is find a function that whose minimum distance to my original function is constant, and then I should have something that you could call parallel.

I made a little Desmos graphs with sliders, to help me visualize the process (click to access):

So I have the equation of the perpendicular line

$y = \frac{-1}{f'(a)}(x-a)+f(a)$

But that wasn’t really helping me see what the parallel function would actually look like. So then I turned to Geogebra. I needed to make a point on the perpendicular line that was a certain distance away from the function, say a distance of 1. So to figure out the coordinates of that point (x,y), I just used the distance formula, plugging in y from above.:

$\sqrt{(x-a)^2+([\frac{-1}{f'(a)}(x-a)+f(a)] - f(a))^2} = 1$

That gave me the coordinates of the point that is a distance of 1 away from f(x) at a:

$(a + \frac{f'(a)}{\sqrt{1+(f'(a))^2}},\frac{-1}{\sqrt{1+(f'(a))^2}}+f(a))$

So I made that point in Geogebra and activated the trace, which gave me this:

Lastly, I thought, well, what exactly is this function that I’ve traced? It looks kinda quartic, but that can’t be, because any quartic like this would intersect the parabola, right? So I tried to write the function for it, using parametric equations. Using $f(t) = t^2$, I made the parametric equation $(t + \frac{2t}{\sqrt{1+4t^2}},\frac{-1}{\sqrt{1+4t^2}}+t^2)$.

I tried to plug that into Wolfram-Alpha to get the closed form, but it was a mess, so I still don’t really know what the closed form would look like. But who says a parametric form isn’t a solution?

### Intentions Change Approach (DragonBox 2 vs DragonBox 1)

So since I first had my students play DragonBox last year, We Want to Know came out with a sequel, DragonBox 2. They are now branded as 5+ and 12+, as the original DragonBox is intended to introduce the idea of algebra and solving equations to someone unfamiliar with it, while DragonBox 2 is meant to deepen the equation-solving toolbox of someone already familiar with solving equations, allowing them to deal with more complex equations.

I was trying to decide which one to use with my class this year. It seemed like DragonBox2 would be better at first glance, because I teach high schoolers: we have seen basic equations, and now we need to kick it up a notch. But I wound up going with DragonBox 1, saving the sequel for a handful of students who blazed through it and were advanced. I know I made the right choice because of situations like I tweeted about:

There were several students who could solve the first level (one of the hardest in the game), but not the second, which came later. This showed me that there was something about the structure of an equation that wasn’t getting through and that we needed to work on it.

In DragonBox 1, you only really have four abilities: you can combine inverses into 0, you can divide a card by itself to get 1, you can add a card from the deck to the game (one on each side), and you can attach a card from the deck to another (multiplication/division), as long as you do it to every card in the level. In DragonBox 2, you can do new things like flip a card from one side to the other, divide a night version by a day version (leaving negative 1), combine like terms, factor out common terms, and treat complex expressions as single units to multiply/divide by.

Those are all good things to do, and someone proficient in algebra should be able to do those things. But I backed away from using it in class because it lacked the why. At the end of the first DragonBox lesson, I compile the notes students took while playing to make a comprehensive list of rules and abilities you have in the game. The one student who played DragonBox2 insisted that, in the game, you can slide a card from one side to the other. No matter how much I pressed him, he didn’t see that the card wasn’t sliding over, it was flipping/inverting.

And that’s what I was afraid of by using DragonBox2. These tools are important, but they have to be earned by understanding them. DragonBox2 gives them to you by completing previous levels, not necessarily by understanding how. At the least, in DragonBox 1, because you are stuck with the basics, you have to grapple with where the solutions come from. They can’t magically appear.

So while DragonBox2 is rated as 12+, I wouldn’t give it to any student who didn’t already have a firm grasp on the concept of equality. Maybe post-Algebra 1. Or at least not until much later in the year.

### Slow Rollout

This year has been weird so far. In the past the first week with actual students has never been a full week, usually just 1 or 2 days. So we’ll have some intro days, do intro stuff, and then head full steam into math class the next week. Last year September was so disjointed because of the Jewish holidays that we couldn’t even really get started.

This year, we started with a full week, and have 5 weeks straight of 5-day weeks before the first day off. So because we didn’t have weird intro days and odd days around holidays, I didn’t have a day introducing my class and systems, and instead went straight into math. I also have a lot more systems and routines now then I did in the past. So what I’ve wound up doing was introducing basically one new overarching idea or routine each class.

First class, Habits of Mind survey, then we did the Broken Calculator. (I’ve decided to loosely follow Geoff Krall’s PBL curriculum.) Next class, I introduce my new grading system (hope it works!) and then had to give them a stupid baseline assessment the city demanded. Next time, we set up our Interactive Notebooks, then did the Mullet Ratio. Today, I handed out the rubrics I’m going to be use to grade them (more on that next post), as well as introducing them to Estimation 180, and then we finished with day 2 of the Mullet Ratio. So every class has been a little routine, a little math. But I kind like it. We’ve been building up how the class works, layering it on. By the end of the month, we should be full steam ahead.

Twitter Math Camp has come and gone, and once again it was truly amazing. The energy of all these other exuberant math teachers just recharges my batteries and gets me ready to go again. (Ironically I go on vacation in exactly one week, but I think this will be a productive week!)

I don’t feel that I learned as much at #TMC13 as I did at #TMC12, but that makes sense to me. Before last year I was only at the edge of the #MTBoS. I had only discovered Dan the summer before and was only following a handful of people by the time TMC12 came around. But after that, I dove in with full force, and absorbed so much great teaching. So this year, when TMC13 came about, I was more up to date and had less to learn.

What I did notice instead was that TMC13 was much more collaborative in nature. Last year, there was a focus on sharing things we knew, and exploring new math (the Exeter problems) together. That was still present this year, but so many sessions I went to focused on creating things together. I look forward to many of those projects coming to fruition (and have a lot of work to do on my half to make that happen).

It makes me wonder at the direction TMC will take next year. I have no idea, and that’s exciting.

### 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.