Chord Star: Another Geometry-Function-Modeling Thing

ImageLast time I wrote about a super-simple geometry situation and how we could turn it into an activity that connected it to linear functions. What does it take to turn something from geometry into a function? This is an interesting question; in my explorations here I’ve found it helpful to look for relationships. And what I mean by that is, where do you have two quantities (in geometry, often distances, but it could be angles or areas or…) where one varies when you change the other.

So one strategy is, think of some theorem or principle, and see if you can find the relationship. To that end, remember teaching geometry and that cool theorem where if you have two chords that cross, the products are the same? That’s where this comes from. Oddly, it took a while to figure out what to plot against what to get a revealing function, but here we go.

Make a circle. Pick a point not near the center, but not too close to the circle itself. Draw a chord through that point. Measure the two segments. Call them L_1 and L_2. Or even x and y. Record the data. Continue reading Chord Star: Another Geometry-Function-Modeling Thing


Isosceles EGADs: Functions, Geometry, and Modeling

ImageIn trying to come up with more activities for EGADs (Exploring (or maybe Enriching) Geometry and Algebra though Data), the following dropped into my lap. Because it’s so simple and so interesting, I’d better write it down…

Everybody get a sheet of paper and draw an isosceles triangle. Try to make your triangle big enough to kinda fill the page, but also try to make it different from those around you. Make your triangle pretty carefully, but don’t measure and don’t use a straightedge.

Individuals can do this too, but I’m writing this as if it’s a class activity. The idea is to get a wide variety of shapes. It is not vital that these just be sketched, but (a) I think that makes the data more interesting, (b) it opens the possibility to drawing more carefully later, and (c) it’s much faster.

Measure the base angles and the vertex angle, and write them on the page.

If you need to introduce vocabulary, do it here. By the way, we don’t assume that these students know that the base angles should be the same. Also, we all know that measuring angles is hard, right?

We’re going to plot the measurements from the whole class. So write your angle measurements on the board.

You may need to help organize this. Will we plot both base angles? Up to you. If so, consider having each kid make two entries in the T-table or whatever.

Now make a graph. Put vertex angle on the horizontal axis and base angle on the vertical. Think about the range of values before you make your axes!

You may want to discuss what goes on which axis. Without having done this with kids, I bet most of us think of the vertex angle as the independent variable and base angle as the dependent. I, at least, think of the vertex angle as the defining angle in an isosceles triangle. This also has the happy consequence of requiring a change of axes in order to get the coolest version of the formula.

At any rate, the graph should look linear. Address outliers (probably due to bad measurement).

Draw the line you think best approximates the data. Find its equation.

Be ready to present your data and line, and explain as much as you can about the line. In particular, why does it have that slope and intercept?

In the spirit of SERP “Poster problems” this could be a poster-plus-gallery-walk event.

Continue reading Isosceles EGADs: Functions, Geometry, and Modeling

Why (most) word problems are not modeling

…and why somebody might try to convince you they are.

It’s even in the Core Standards. This is taken out of context—but not very far:

A model can be very simple, such as writing total cost as a product of unit price and number bought… (Common Core, p 72)


Okay, I could make a case for it, but I won’t.

I’m becoming more convinced that the real hallmark of modeling is simplification  (see this post for more).

Modeling is not simply using math on real-world problems, though that is a Good Thing; you can model to help with pure math as well. And I bet we could find good real-world problems that don’t involve modeling.

But back to simplification. The key element (I believe this afternoon anyway) is taking something and using math in a way that makes it simpler, less complicated than the thing itself. We model to make things tractable. We can handle the model even when the thing it represents it too complicated. If it’s a good model, it captures the essence of what we’re looking at; and exactly what that means may depend on the specific context.

  • We might model a hexnut as a hexagonal prism with a cylindrical hole, and use that geometrical model to find a volume. We avoid the threads and the easing on the corners: they’re too complicated—but we hope our model captures the essence of the hexnut.
  • We might model some messy data as a line or a curve. We can’t make a reasonable prediction from the mess, but we can with a function: just plug in a value and calculate.
  • We might explore the behavior of a system of linked differential equations by creating a numerical model, a system of difference equations we can evaluate on a computer. It’s conceptually simpler (for the computer at least), so we sacrifice some precision for tractability.
  • We might even take all the complexity of Americans and do a Census. When we do, we create a data model: the structure for the information we will collect. We have only approximately captured the people’s information (this is the Census, right, not the NSA). We hope our data has the essence that we need to know—but there is a huge amount of detail that we have ignored. Like the threads, like the deviations in the scatter plot, like the inaccuracies in the numerical model.

What does this have to do with word problems?

Suppose we ask, if Eduardo buys four cans of orange juice for $2.49 a can, how much does he pay altogether?

There is math here, no question. We can argue whether it’s real life.

But it doesn’t involve simplification. All of the information is present. There is no model, and no need for one.

Modeling: Looking for definitions

Modeling is at the center of what I love about math and math education, so I’m thrilled that the Core Standards highlight modeling and that it figures in our latest drafty California framework.

But I’m worried about definition creep. I’m worried that, in two years, when they’re trying to come up with modeling curriculum, people in schools doing the hard day-to-day work will be tempted to say that practically anything is modeling and come up with plausible rationalizations. That, in turn, will dilute the importance of including modeling in policy documents, and result in students who can’t model.

To forestall this, it’s important to know what modeling is and isn’t. So it’s with some embarrassment that I, modeling maven and aficionado, have trouble drawing the lines. So consider this a first step in clarifying these questions for myself:

  • What is modeling?
  • What isn’t modeling?
  • How much do we care whether we can come up with a definition?

Let’s start with the Framework (Modeling appendix, April 2013 review draft, lines 13–14):

Put simply, mathematical modeling is the process of using mathematical tools and methods to ask and answer questions about real world situations (Abrams, 2012).

Of course they go on at length, but the key is a connection to the real world. Here is another definition that I have used recently:

A model is an abstract, simplified, and idealized representation of a real object, a system of relations, or an evolutionary process, within a description of reality. (Henry, 2001, p. 151; quoted in Chaput et al., 2008)

Here, the key ingredients are abstraction and simplification.

Another distinction worth noting is that the first is a definition of modeling—a process—and the second defines a model—a representation. I have no clue whether that matters much.

What do the Core Standards themselves say? First of all, the document identifies modeling as one of eight Mathematical Practices, a great list I have mentioned before. Here is the one called Model with Mathematics, and it’s worth quoting in its entirety:

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

This is great, but it’s easy to imagine this lofty overarching idea getting lost when you’re designing a curriculum—or an assessment—and you have a chart of content to fill in. Fortunately, the Core Standards promote Modeling to the level of a content standard at high school. What do they say? Here’s a quote I find chilling:

Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol.

That is, you should find it everywhere, so we won’t list very many actual skills and goals. One could see this as a good thing: we’re celebrating the ubiquity of modeling. But I’m less sanguine about our ability to keep “overarching ideas” in mind, especially as we design assessments. Continue reading Modeling: Looking for definitions

Wave Slicing and Remainders: a cool way to find the period of a periodic function

Last month, in Falmouth High School in Maine, some Honors Physics students were estimating the period of a mass hanging from a spring. They used InquirySpace/Data Games software and Vernier motion sensors, and got data that looks like this (Reading is in meters; Time in seconds):

oscillating data

To do their investigations, they needed the period of this wave.

  • Some students found the peak of one wave, and subtracted its time from the peak of the next wave. This is the most straightforward and obvious. But if you do that, your period will always be a multiple of the time between points, in this case, 0.05 seconds. (This is part of what must have happened in the previous post.)
  • Some students—sometimes with prodding—would take the time difference across several waves, and divide by the number of periods. It’s not obvious to students that this technique gives a more precise measurement for the period. It’s interesting to think about how we know that this is so; for example, if you use five periods, it’s now possible to get any multiple of 0.01 seconds; but does that mean it’s actually more precise? (Yes.) This technique also gives students a chance to be off by one: do you count the peaks? No. You have to count the spaces between the peaks. (Getting students to explain why is illuminating.)
  • We could imagine trying to fit a sinusoid (and some students would, but it’s hard) or using a Fourier Transform (which is a black box for most students).

But this post is about an alternative to all of these techniques—one that uses all the data and gives a much more precise result than the first two.

You can read about this is excruciating detail in a paper I wrote. And I made one particularly careful group of students this (awkward and quickly-made) video describing the technique. So I will be brief here. Continue reading Wave Slicing and Remainders: a cool way to find the period of a periodic function

Modeling a Spiral, and enjoying Desmos

spiral_20_thumbnailAt a recent meeting, I got to tell people about an old, non-finished book, EGADs (Enriching Geometry and Algebra through Data). The idea of the book is that there are geometrical constructions that have relationships under them—usually a relationship about length—that you can model using a symbolic formula.

Like that spiral. How does the length of the “spokes” of this spiral depend on the spoke number?

This post has two purposes:

  1. To get you to try the spiral example.
  2. To show how you can use the Desmos graphing calculator to do the graphing and calculation.

The draft of the book (link above) is free for now, but it occurred to me that you could do at least one activity (integrates trig, geometry, data, exponential functions) easily using Desmos’s cool new technology. Read on!

The activity: Spiral 20

Here’s what you do:

  1. Download the PDF with the one-page handout. Print it out.
  2. Measure the long legs of the triangles as suggested on the handout.
  3. Go to the Desmos graphing calculator.
  4. Enter the data.
    • To do that you need a table. Click the “<” button under the little panel, upper left, where you will eventually enter a function.
    • desmos_makeTableClick “table.” A table appears.
    • Enter the data. You can change the variable names if you want.
  5. Enter a function in that top panel. Try to match the data!
    • If you leave parameters in the formula, it will ask if you want sliders. I love sliders. You will too.
    • You can type values in, even if you have sliders.
    • You can change the range of sliders by clicking the limits at the slider ends.
  6. Play with the sliders and all the other features of this site.

Here is a graph of mine from Desmos, with only a little of the data:

Blocks and Shadows

Photo of blocks with their shadow
Four blocks and their shadow. I set them on graph paper just to make this shot.

Dan Meyer’s post today is lovely as usual, and mentions the tree/shadow problem (we math teachers make right triangles to help us figure things out because the “tree-ness and shadow-ness don’t matter”).

And that reminded me of a problem I gave teachers long ago in SEQUALS-land that (a) worked really well to get at what I was after and (b) could turn into a great modeling activity that could fit in to that first-year course my fellow revolutionaries and I are gradually getting serious about.

Here’s the idea: we want to be able to predict the length of the shadow of a pile of blocks. So we’re going to make piles of blocks and measure the shadows, which will lead us to make a graph, find a function, etc. etc.

The sneaky part is that we’re doing this in a classroom, so to make good shadows we bring in a floor lamp and turn the class lights off.

I will let you noble readers figure out why this messes things up in a really delicious way. Two delicious ways, actually. I’ll give away the second:

end of the shadow
The end of the shadow, closer up. Really: how long is it?

Of course we have all done height/shadow problems. But have you tried to measure a shadow lately? You have to make a lot of interesting decisions to measure a shadow; and a shadow from a pile of blocks made from a floor lamp exaggerates the problems, such as where do you measure from—the middle of the stack? The base on the shadow side? Where? And where do you measure to—where the fuzzy part of the shadow begins? Where it ends? And why is it fuzzy anyway?

This is why I love measurement as a strand so much. We always think of it as the weakling among content areas at the secondary level; it doesn’t have the intellectual heft of algebra or functions. But if you look closely (and go beyond the words in the standards) it’s a thing of beauty and (since we’re referencing Dan Meyer) perplexity. I did a chapel talk at Asilomar many Sundays ago in which I said that measurement was invented, inexact, and indirect. I still think that’s true, although as alliterative slogans go it’s hard to remember.

So: try this at home. Use Fathom if you have it. Come up with a function that models the shadow lengths. But don’t just figure it out like a math teacher—get the lamp, stack the blocks, and measure.