On the Living Math Forum group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):
Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.
The equation for a circle is (x-h)2 + (y-k)2 = r2 . Many students see
each equation like this as separate from any other equations/formulas
they know. I try to get them to look at this deeply. From the structure
of it (square plus square equals square), I see
that it's really the Pythagorean Theorem. Why would something for
right triangles show up in the equation of a circle?! (That blew me away a few
years back. I've been teaching math for 30 years, but that question
seemed deep.)
It's because our coordinate system has the two axes perpendicular to
each other. So the distance from the x-coordinate of a point on the
circle to the x-coordinate of the center is measured horizontally and the
similar y distance is measure vertically. You can
build a right triangle from the center to (almost) any point on the
circle. The constant radius is the hypotenuse of that right triangle.
So this equation tells a little story.
So this equation tells a little story.
How does a whole course tell a story?
Algebra is about solving equations and about graphing. We want to see
how real life situations (anything with data that has two components, like time and height) can be
represented with equations and with graphs. In Algebra 1 students learn
to solve simple equations and to graph. And
hopefully they learn how the two skills are connected. There's a bit
more. Systems of equations allow us to model slightly more complex
situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after
factoring) is graphing and recognizing equations of parabolas.
Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.
Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.
The course can tell the tree's story, or any story that can be told
through data, graphs, and/or equations. It tells the story of using math
to help us think quantitatively about problems we care about. (And of
course there are plenty of things we care about
that cannot be quantified.)
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