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.

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.

Created with Geogebra

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!"

In the Cups

Our department has a shared e-folder where we can share tasks we develop for use in specific courses. I've used it a lot this summer to support my planning for the intermediate algebra course I'm teaching. The activities there range from routine to inspiring--and both types have been useful!--which reminds of a quote I saw recently: Good teachers borrow, great teachers steal.

So I have been drawing heavily on the excellent activities there. Some tasks I use almost verbatim, and others I modify to suit my own purposes. One of the activities I found was a nice math modeling activity called "In the Cups". It is designed to support students with creating and using linear models to make predictions and solve problems.

The original document presents the following data to students:

 

Number of Cups	Height of stack (in cm)
1	7
2	7.6
3	8.2
4	8.7
5	9.2
7	10.1
8	10.6

The task proceeds to ask students to predict the height of 6-cup, 10-cup, and 25-cup stacks, and eventually invites students to find a model expressing the stack height as a function of the number of cups in the stack.

I noticed that the data in the table do not quite conform to a perfectly linear relationship--notice that the unit rate of change varies from 0.6cm to 0.5cm per cup--which got me thinking. I wondered if the author had introduced some random variation into the stack heights presented in the table in attempt to make the task seem more authentic. Or perhaps he or she had really measured some cups and the data were authentic approximations. Either way, I liked it.

I started measuring a stack of dixie cups I had in my office, then I realized that by doing the measuring myself, I was about to rob my students them of an opportunity to do some critical thinking. So I decided to replace the table of values with a photograph. 

That one picture (of 27 cups) contains all of the information needed to model the relationship, and it invites students to consider questions about the accuracy of the approximation. Replacing the table with the photograph felt like a more authentic way to present the task to my students.

What's with that title anyway?

Let me say a bit about the title of this new blog of mine.

My teaching and scholarship have always centered around striving for understanding, and so I settled on the title "...do we understand" as I reflected on the types of questions I anticipate exploring here. These are the questions that interest me as a teacher, as a researcher, and as a human being.

So I expect I'll be exploring questions like:

• Do we understand the content we are teaching and learning? How? What? Why?
• Do we understand our students' ways of thinking? How? What? Why?
• If not, how can we find out?  

What it means to solve

"Do we understand" was on my mind as I was grading intermediate algebra quizzes over lunch today. In particular, I wondered: do my students really understand what it means to solve an equation or inequality? It was clear enough that they knew how to do it, but do they know what they are accomplishing in the process?

One of the learning targets I have identified for my students reads:

I can explain what it means to solve equations, inequalities, and systems, and I can use this knowledge to check my answers for reasonableness and correctness. 

I'll be honest: I really love that learning target. It was inspired based on one of the Common Core State Standards for Mathematics, specifically, one of the Grade 6 Expressions and Equations standards:

CCSS-6.EE.5 [Students will] understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? ... (emphasis added)

That concept is fundamental to the work we do in algebra. Consider the wide array of situations where the sole directive is "solve". Come to think of it, one of my old doctoral advisors once described "solve" as one of the four universal directives of algebra homework sets. (Can you come up with the other three? Answers appear at the end of the post.)

In the case of "solve", the meaning is universal and boils down to "find the set of [thing(s)] that make this [thing] true". Consider: 

• "Solve 3x+4=7" means "Find the values of x that make the equation true."
• "Solve {3x + y = 4, x - 2y = -1}" means "find the ordered pairs that simultaneously make both equations true."
• "Solve y'(x) - y(x) = 0" means "find the family of functions that make the differential equation true."
• "Solve Ax = λx" means "find the real numbers λ and corresponding vectors x, called the eigenvalues and eigenvectors of A, that make the matrix equation true."
• "Solve Ax = 0" means "describe the set of vectors x, called the null space of A, that make the matrix equation true."