## Trying to find math inside everything else

### Slope

At Twitter Math Camp, Karim Kai Ani and I debated for a bit on what slope really means, and how best to teach it. Since slope is the upcoming topic for this week, I thought it would be good to reflect back on our arguments.

Karim argued that slope should always have units, and that removing the units created a contextless concept that made it difficult for students to grasp. I argued that, while that is true and units are useful in many cases, the concept of slope as a unitless ratio is an important concept, digging deep into what it means to be a ratio, so that a line with a slope of 2 could be 2 miles up, 1 mile over, or 2 cm up, 1 cm over, it didn’t matter. The differences are exemplified in two of our lessons: my “Steepest Stairs and Wacky Measurements” (soon to be updated) and his “iCost.”

(c) Mathalicious 2011

I mentioned this debate at dinner last night to my boyfriend, who is a math PhD candidate. He said what we were talking about reminded him of the difference between a rate and a ratio. He said that a ratio was a “quotient of quantities of the same unit” and a rate was a “quotient of quantities of differing units.” Further clarification was that a ratio’s units had to be the same dimension, while a rates did not.

So then, really, the question becomes, is slope a rate or a ratio?

It’s both. Karim argued for rate but that’s really just the algebraic or calculus-based definition of slope. My argument for ratio was a geometric one. Both are important, and are related, which is why they go by the same name.

But I wonder if it would be easier if the concepts had a different word. What if we only used “grade” or “gradient” for the geometric definition, and slope for the algebraic one? Or slope for the geometric, and just rate for the algebraic? The problem is they are so intertwined. For which there is only one person to blame.

Damn you Descartes!

### Today’s Roles: IT Department, Programmer, Lecturer, Assessor, Tutor, Co-ordinator….

5:30 – Alarm goes off. That is not happening.

6:20-7:00 – Wake up, shower, pack up, go. I decide to take the subway to school today, instead of biking, because I have too much to do to lose those 40 minutes.

7:00-7:07 – Walk to subway. Catch up on Twitter while walking.

7:07-7:40 – Subway to work. My train still isn’t running to my job because of Sandy, so I have to transfer. While I’m riding, I grade math labs. (Despite grading for several hours over the weekend, I didn’t finish.) I don’t finish by the time we arrive.

7:40-7:50 – Walk to work plus breakfast.

7:50-8:55 – Enter my classroom to discover 1) it’s a sauna, and 2) that the wi-fi is down in my classroom (and only my room). This is awesome, because I have a computer based lesson today. Also, the person in charge of the laptop cart doesn’t get in until later. Luckily, I am technologically proficient, so I spent this time creating an ad-hoc network and setting it up on the ancient Dell laptops (after tracking down the AP to unlock the tech room) so the students could get and give files. I also spent some of this time inputting the grading I did into the gradebook.

8:55-9:46 – Start of contracted time. Embassy class, which is our special version of advisory. We finally have a curriculum to follow, so I need to modify for my students.

9:46-11:30 – Math class, students brought in survey results to analyze. So I pass out laptops and walk them through the analysis excel file I made yesterday. Minor tech problems, so most of my time is spent fixing those and running around teaching the quirks of excel while the students do data entry and create conclusions. I explain the requirements for their project, stop a student from hacking into one of the computers, and general maintenance. We also have problems getting the files back to me, because they don’t follow directions.

11:30-11:50 – Putting away the laptops and making sure all files are saved.

11:50-12:20 – Run and get lunch, while planning the next lesson with my co-teacher. When we get back, we meet with a third teacher about two students who need resource room (since I’m the programmer, and can change it.)

12:20-1:20 – Student lunch period, so some kids come up to my classroom to work on their projects. I continue setting up laptops (since my afternoon class is larger). At 12:45, the wi-fi returns, so I switch the computers back to that instead of the ad-hoc network.

1:20-2:00 – First opportunity to use the bathroom. I run to the programming office to change a schedule. Then I go back to grading, or, more accurately, data entry.

2:05-3:45 – Another math class, this one with 4 languages spoken and no ESL support. The tech problems seem even worse at first, but balance out in the end. Unlike my morning class, which is very industrious, several pairs in this class did not come prepared and needed to do alternate work/catch up work. End of contracted time.

3:45-3:55 – I bring some of my students to the Teacher Work Room to make copies of their surveys for them, so they can catch up on their project. I get a cookie from a co-worker.

3:55-4:15 – Break and decompress, including short chats with coworkers in the hall.

4:15-4:45 – Back on the grading grind.

4:45-5:30 – I plan with my co-teacher on Thursday’s lesson, which we won’t have time to do tomorrow because of other meetings. (I’ll have a Math for America meeting in the evening.) We adjusted my Lying with Statistics Stations because they were confusing and ill-timed last year, opting this year for a looser flow. I finish grading while I do this.

5:30-5:35 – I write an e-mail to a parent because her son is way behind on the project.

5:45-6:20 – Time to head home. I grab a hot dog on the way. True to my pledge to not bring work home (even though I broke it over the weekend), I play my 3DS on the subway ride home. I almost fall asleep on the train, and my game freezes, losing all progress.

6:20-6:35 – I stop at the supermarket on the way home, to get some stuff for dinner and breakfast.

6:45-7 – I forgot the shallot. So I change my plans, because I don’t want to go back out. It’s a tough decision, I seriously thought about it for 5 minutes because I wanted the shallot but was so tired. I make more bachelor-y food.

7-7:45 – Watch Daily Show/Colbert while catching up on tweets/blogs.

8-9 – Leisure Time

9-9:20 – I try to go into the Global Math Department meeting about homework, but the audio is too messed up, so I duck out early. Now I’m going to lay down and read the news/play with my DS probably until around 11, when I’ll hit the sack.

### 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