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:

An Unexpected Expected-Value Problem, and What Was Wrong With It

"no permutations" graphicIt’s such a joy when my daughter asks for help with math. It used to happen all the time; it’s rare now. She just started medical school, and had come home for the weekend to get a quiet space for concentrated study.

“Dad, I have a statistics question.” Be still, my heart!

“It’s asking, if you have a random mRNA sequence with 2000 base pairs, how many times do you expect the stop codon AUG to appear? How do you figure that out?”

I got her to explain enough about messenger RNA so that I could picture this random sequence of 2000 characters, each one A, U, G, or C, and remembered from somewhere that a codon was a chunk of three of these.

“I think it’s more of a probability, or combinatoric question than stats…” I said. (I was wrong about that; interval estimates come up later. Read on.)

Continue reading An Unexpected Expected-Value Problem, and What Was Wrong With It

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.

KOLD Curriculum: Killing the Darlings in Math

“Kill your darlings, kill your darlings, even when it breaks your egocentric little scribbler’s heart, kill your darlings”

—Stephen King, On Writing

Fiction writers have heard this advice: “Kill your [little] darlings.” I realized (at a wonderful lunch yesterday with colleagues who are planning a revolution) how this might apply to reinventing math curriculum.

The problem is that it’s really hard to imagine a math course without some of our favorite parts of math. We all have things we despise (one of mine is factoring trinomials) but every one of these things is someone other teacher’s fave, the thing where they suddenly got how cool math was. And if we have that meeting where we decide what to throw out in order to put modeling in, we’ll keep everything. It would be like writing science standards in the 90s.

It’s a values/positions thing. We need to figure out carefully what each darling (ours and the others’) really means and see where the meat of it fits. Maybe we really can get at it with a modeling approach. Maybe it needs to remain the way it is. Or maybe it’s just later in the sequence.

And it’s not a zero-sum game (thanks, Mariel!). Ideally, kids get everything, using the best tools, always appropriately, in the most efficient imaginable sequence. But we will, occasionally, have to Kill Our Little Darlings. KOLD, but necessary.

Just for fun, I list a few of mine. Okay, some are not darlings, but I want to kill them off. So my lesson in collaboration may be letting them live…

  • Factoring trinomials (but recognize squares!)
  • Obscure trig identities (but keep Pythagoras, double angle, etc.)
  • Remainder Theorem
  • Integration by partial fractions

What are yours? And what can’t you give up?

Reflection on Modeling

Capybara. The world’s largest rodent.

I’m writing a paper for a book, and just finished a section whose draft is worth posting. For what it’s worth, I claim here that the book publisher (Springer) will own the copyright and I’m posting this here as fair use and besides, it will get edited.

Here we go:

Modeling activities exist along a continuum of abstraction. This is important because we can choose a level of abstraction appropriate to the students we’re targeting; presumably, a sequence of activities can bring students along that continuum towards abstraction if that is our goal.

As an example, consider this problem:

What are the dimensions of the Queen’s two pet pens?
The Queen wants you to use a total of 100 meters of fence to build a Circular pen for her pet Capybara and a Square pen for her pet Sloth. Because she prizes her pets, she wants the pet pens paved in platinum. Because she is a prudent queen, she wants you to minimize the total area.

Let’s look at approaches to this problem at several stops along this continuum:

a. Each pair of students gets 100 centimeters of string. They cut the string in an arbitrary place, form one piece into a circle and the other into a square, measure the dimensions of the figures, and calculate the areas. Glue or tape these to pieces of paper. The class makes a display of these shapes and their areas, organizes them—perhaps by the sizes of the squares, and draws a conclusion about the approximate dimensions of the minimum-area enclosures.

b. Same as above, but we plot them on a graph. A sketch of the curve through the points helps us figure out the dimensions and the minimum area.

Using Fathom to analyze area data. Sliders control (and display) parameter values. I have suppressed the residual plot, which is essential for getting a good fit.

c. This time we enter the data into dynamic data software, guess that the points fit a parabola, and enter a quadratic in vertex form, adjusting its parameters to fit the data. We see that two of these parameters are the side of the square and the minimum area.

d. Instead of making the shapes with string, we draw them on paper. Any of the three previous schemes apply here; and an individual or a small group can more easily make several different sets of enclosures. Here, however, the students need to ensure that the total perimeter is constant—the string no longer enforces the constraint. Note that we are still using specific dimensions.

e. We use dynamic geometry software to enforce the constraint; we drag a point along a segment to indicate where to divide the fence. We instruct the software to draw the enclosures and calculate the area. (In 2014, Dan Meyer did a number on a related problem and made two terrific dynamic geometry widgets, Act One and Act Two.)

f. We make a diagram, but use a variable for the length of a side. Using that, we write expressions for the areas of the figures and plot their sum as a function of the side length. We read the minimum off the graph.

g. As above, but we use algebraic techniques (including completing the square) to convert the expression to vertex form, from which we read the exact solutions. In this version, we might not even have plotted the function.

h. As above, but we avoid some messy algebra by using calculus.

Now let’s comment on these different versions.

Continue reading Reflection on Modeling