## Trying to find math inside everything else

### Slopes and Lattices Game

Okay, here’s a game I came up with off the cuff today. It kinda worked, but I guess if other people tried it and gave feedback, that’d be swell.

Players: 2 (or 2 teams), each with two colors

Board: A 10×10 grid.

The game is played in two phases. In the first phase, each team takes turns placing points on the grid, until each team has placed 5 points. The origin always is claimed as a neutral point. Every point has to be on a lattice point. (In the example below, I was blue and my student was yellow.)

In the second phase, on their turn, each player may place a new lattice point and form a line with one of their original 5 points. If that line then passes through one (or more!) of the opponent’s original 5 points, those points are stricken. If one player can strike out all of the other player’s points first, they win. (If not, then whoever strikes out the most.)

There is one caveats to round 2 – when a line is drawn, determine the slope of that line and write it below. That slope can’t be used again.

After playing the first time, it became clear that much of the game came down to placing the points. If you could place one of your points so it was collinear with two of your opponents, you can strike them both with a single line. (But this only works if there is space for a 4th, alternate color point in phase 2 to form the line.) You also want to place your points defensively, with weird slopes that don’t pass through a lot of lattice points, to keep them safe. The second player definitely has an advantage when placing points, but the first player has an advantage when drawing lines, so I’m hoping those balance out.

Thoughts?

### Hiking and Slope

Last summer, I went on vacation out west to see some National Parks (Yellowstone, Glacier, Craters of the Moon). At Craters of the Moon, all the trails had these lovely signs talking about how steep they were – since one of us hikers had a bad knee, we needed to make sure the trails weren’t too tough. What’s interesting is that they didn’t just talk about the average grade, which many hiking books do (as we learned to our chagrin in Glacier), but also the maximum and minimum. I feel like this is a good opportunity to talk about average rate of change versus instantaneous.

Later on in the trip, we had a discussion about what it means if a trail is twice as steep as another one. If I told you the the next trail is twice as steep as this one, what would you expect? What would it feel like? Then we also talked about whether we were doubling the slope or double the angle. That distinction is tricky because, for angles less than 10°, which are the most common, the difference between doubling the slope or doubling the angle (up to 20°) is less than 1% extra grade.

Anyway, there’s a lot of data here, so I pose to you: what could you do with this?

### Walking in San Francisco

I had this same thought the last time I was here, but I feel it could be fruitful. Sometimes there is a route someolace that is “shorter,” laterally, but there are tons of up and down hills between. So is that way really shorter? I feel like you can do something with this: use the Pythagorean theorem to determine how far you are actually walking, determine different walking speeds on different inclines, and then get a topographical map and determine the speed and length of different routes, then check what Google Maps says.

For an example of what I just experienced: because we didn’t really know where we were going, we wound up walking up the giant hill up Powell St and then down the hill on California St, but if we had gone down Sutter first and then up Grant, it would have been much flatter.

### Steepest Stairs Redux

Last year I made a lesson about determining the steepest stairs, using pictures my co-teacher and I took and based on an idea from Dan Meyer. It took about a period, and was mostly teacher-led. But after arguments and deep thinking about slope, I wanted to go into the lesson deeper, so I turned it into a lab.

I started the same way, throwing up the (new and improved) opening slide and asking which they thought was the steepest and which was the shallowest.

I really like this new improved one because I took a picture of the toy staircase from the board game 13 Dead End Drive (middle left). Last time, there were overall agreement on the shallowest (the Holiday Market) while there was disagreement on steepest. This time, because the toy was tiny (if not shallow), we had some disagreement there, which really let us tease out some definitions of “steeper” and “shallower.”

Once we had definitions of steeper (which usually came out to something like “closer to vertical” or “at a bigger angle”), I handed out the pictures on a sheet of paper and asked them to develop a method for determine which was steeper, or the steepest. I mentioned coming up with some sort of “steepness grade” (because I thought it would be amusing to throw the word “grade” in there).

So I let them struggle, and come up with what information they had to ask me for, which I would then provide. If I had to do it again, I would also have pictures of the width of each stair, as a distracter, because some kids asked for it. Interestingly, some also asked for the angle, because of our prior experience in the year with the clinometers. I told them I didn’t have the clinometer with me at the time. One kid called me on it, because she knew I had a clinometer app on my iPad. So I told her (truthfully) some of the pictures were taken last year, before I had it.

So I had them come up with their own measures. If they tried to base it off of only height or only depth, I deflected with examples of really tall, really shallow stairs, or really short, really steep stairs.

By the end of the classes, students usually came up with one of three different measures: slope, the inverse of slope (depth over height), and grade (that is, slope as a percentage).

So they had to then reason as to why they might prefer height/depth to depth/over. (Their logic: it seems more natural to have bigger numbers be steeper stairs, rather than the other way around.) And so it was that point that I told them this “steepness” grade that they developed was often called “slope” by mathematicians.

At which point, I got a big “Ohhhhhhhhhhh.” Which always makes it worthwhile.

### The Materials

Stairs – Portrait

Stairs – Landscape

Steepest Stairs Lab

### 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!

### Steepest Stairs and Wacky Measurements

After reading Dan Meyer’s post mentioning a Steepest/Shallowest stairs contest, I decided to go for it. But Dan had them do it for homework after they knew what slope was. I decided that I thought steepness of stairs would be a great way to introduce the concept, and then we can have the contest after. So Ms. Barnett (my co-teacher) and I went around the area and took lots of pictures of stairs we can find. Then I put them up as a warm-up and asked them which were steepest and which were shallowest.

In every class, there was near-universal agreement on which stairs were the shallowest (the top-right), but lots of different votes for the steepest. So then I asked them, “How can you know? What does it mean to be steep?” I got a lot of good, intuitive answers from that (My favorite was that something is steeper when it is closer to being vertical). I asked them what they needed to know to find out which was steeper, and they said we should measure it.

But what exactly should we measure? That took a little cajoling and probing, until we eventually decided on the height of the step and how deep it was. So I gave it to them:

Alright, now we have these measurements, what can we do with them? I lead them on a discussion on how best to use these numbers (a ratio), and we looked at another example. This is a pretty clear example (1/2), but not all of them are. So we used our estimating skills.

And my personal favorite…

(They really asked if I had 11 cell phones. I guess my Photoshop skills are better than I thought.)

The best part of these pictures is that they so naturally prompted them to question the units of measurement. “That one is cell phones, but the other one is hands. How can we compare them?” And so it’s natural to talk about slope as a ratio with no units. I didn’t have to artificially insert it. I even had a picture of a curved slide at the end, so we could theorize about the steepness of that.

Finally at the end I mentioned the contest. Unfortunately, I’m afraid Ms. Barnett and I did too well finding stairs. I’ve had students say they’ve been looking, and some say they found some (but don’t have pictures yet, though they have one more week). I hope someone can knock us off our thrones:

## SLIDES:

Steep Stairs (PDF)

Steep Stairs (PPT)