This post is based on a student comment that caught my eye while grading. The data below shows the sodium content (in mg) for 23 brands of "regular" Peanut Butter.
The student wrote one of the following two statements in defense of choosing the preferred measure of center.
1) The data are balanced on each side of the median.
2) The data are balanced on each side of the mean.
Which statement do you think is more correct? In what sense might someone think the other one is correct, too?
Ready... GO!!
Showing posts with label procedural understanding. Show all posts
Showing posts with label procedural understanding. Show all posts
Tuesday, February 11, 2014
Tuesday, June 25, 2013
What (else) it means to solve
In class yesterday, I had an opportunity to reinforce what it means to solve equations and inequalities (a connection I've discussed in a previous post). My students and I were working on a task that required that we verify that x = 2 and x = 0 are the two solutions of the quadratic equation -2x^2 + 4x + 1 = 1. How can we do that? A chorus of replies: "plug them in and check."
"Right," I said, "We need to see if we have found both (all) of the values that make the equation true."
Learner #1 raised his hand and asked, "Doesn't it also mean finding the points where the parabola intersects the line y = 1?" I know from a previous conversation (described in this previous post) that this was how he learned to solve quadratics in high school: enter the parabola into Y1, enter the other expression into Y2, and use the CALC -> INTERSECT to solve.
So I was happy to affirm the connection.
A short time later, in the same lesson, we were exploring the function f(x) = (x-3)^2 + 5. We had determined that because the parabola opened upward and had its vertex at (3,5), so there must not be any solutions to the quadratic equation f(x) = 1. Furthermore, we agreed, the equation f(x) = 5 would have just one solution (at the vertex) and f(x) = k for any k > 5 would have two real solutions.
Then Learner #2 offered: "So if we ever find that there's no solution, we can conclude that the curves do not touch, right?" It was a nice extension of Learner #1's remark.
But it depends a bit on what you mean by "no solution," doesn't it? The situation is exactly analogous to the 1st grader who asks the question, "Teacher, what's 1 minus 3?" to which the teacher response helpfully, "Honey, you can't do 1 minus 3." I like to imagine a precocious 1st grader reply: "Yeah, maybe you can't, but I can! It's negative 2!"
"Right," I said, "We need to see if we have found both (all) of the values that make the equation true."
Learner #1 raised his hand and asked, "Doesn't it also mean finding the points where the parabola intersects the line y = 1?" I know from a previous conversation (described in this previous post) that this was how he learned to solve quadratics in high school: enter the parabola into Y1, enter the other expression into Y2, and use the CALC -> INTERSECT to solve.
.png) |
| Created with Geogebra |
So I was happy to affirm the connection.
A short time later, in the same lesson, we were exploring the function f(x) = (x-3)^2 + 5. We had determined that because the parabola opened upward and had its vertex at (3,5), so there must not be any solutions to the quadratic equation f(x) = 1. Furthermore, we agreed, the equation f(x) = 5 would have just one solution (at the vertex) and f(x) = k for any k > 5 would have two real solutions.
.gif) |
| Created with Geogebra |
But it depends a bit on what you mean by "no solution," doesn't it? The situation is exactly analogous to the 1st grader who asks the question, "Teacher, what's 1 minus 3?" to which the teacher response helpfully, "Honey, you can't do 1 minus 3." I like to imagine a precocious 1st grader reply: "Yeah, maybe you can't, but I can! It's negative 2!"
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