Tag Archives: graphs

Another great question from College Algebra

Here is something cool that happened in College Algebra today. We were doing a short thing to summarize our domain and range work before moving on.

A student asked, Is the only way to find range to make a graph?

This stopped me in my tracks. I had not really thought about the knowledge I draw on when identifying the range of a function, and the question cut to the heart of the matter.

My gut instinct answer was yes. But I wanted to explore that a little. I concocted a silly function to do so. \sqrt[3]{x^{5}+x^{2}}+x^{2}-sin(x). I wanted to say that I would need to graph that to know its range.

But the longer I looked at it, the more clear it was that I knew a lot about this silly thing without graphing it. The x^{2} term dominates, for instance, in the long run, so I know it goes to infinity on both sides of the y-axis. I could see that 0 is in both the domain and the range.

But I wasn’t 100% sure whether there were any negative values for the function.

Later in the day, this got me thinking about end behavior. This is why we teach that end behavior silliness, right? It’s not about end behavior, it’s about knowing what values can come out of a function, and having a basis for knowing this.

I am brainstorming here. The point is that the student question showed a sign of her learning, and it pushed me to rethink something too. Win-win.

graph.1

Another cool thing happened, too. We were comparing y= x^{2} and y=2^{x}, looking for sameness and difference. I had to push to get domain and range on the table.

We agreed that the two functions have the same domain—all real numbers. We were split on whether they have the same range.

But not for the reason I expected. Not at all.

A student argued that The only time when they are the same is when x=2. Therefore they do not have the same range.

My students found this argument compelling.

Ignore the second intersection point in the left half-plane. Focus on the essence of the argument.

Do these functions have the same range? is interpreted as Do these functions intersect?

graph.2

 

That seems like a useful insight into the mind of a College Algebra student.

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College Algebra with Christopher

If you assume that Double Stuf Oreos are doubly stuffed (which may turn out to be false), then using the Nutrition Facts labels, you can write the following system of equations, where x represents the number of calories in one wafer and y represents the number of calories in a single layer of stuf.

\begin{cases} 6x+3y=160\\4x+4y=140\end{cases}

The first line represents the caloric content of a single serving (three cookies) of Regular Oreos; the second line represents the caloric content of a single serving (two cookies) of Double Stuf Oreos.

Getting to this system represents some effort on the part of me and my College Algebra students. They are wont to represent their work arithmetically; my job is to help them to transition this arithmetic problem solving to algebraic generality. It is work that I love, but it is hard work.

We had gotten ourselves there, and we had discussed the importance of being very clear about the meanings of our variables when I presented the following graph in class yesterday.

The basic questions in front of us were, What does the blue line represent? What does the red line represent? What is the meaning of this graph?

We had a number of false starts and hesitations. After a few minutes of this, a student pointed our attention to the slope of the red line.

Student: The slope is –2.

Me: Why is it negative?

Student: Because it goes down.

Student: Because if you count the squares over and the squares down, and write rise over run, it’s negative 2, which means the line goes down.

Me: Right. But what does that have to do with Oreos?

A few moments of contemplative silence from 44 college students.

Student: There are twice as many wafers as stufs in the regular Oreos, and the red line represents the regular Oreos.

Me: Right. But why negative?

A few more moments of contemplative silence from the group. This is not a routine they are familiar with, but they are working hard to acculturate themselves to these new expectations.

Me: OK. Let’s do this. Write your answer to this question in your notes.”What is the meaning of x on this graph?”

I allow a few moments for this to occur.

Me: Raise your hand if you wrote that “x represents wafers”.

About 80% of hands go up. I contemplate this. Then…

Student: Isn’t it “number of calories in one wafer”?

Now we have something to work with!

TURD: Common Core graphic

I can’t quite decide whether this is a Truly Unfortunate Representation of Data. Help me out here.

The following is from an Educational Researcher article on the alignment between the math and English/language arts standards of various states and those of the Common Core State Standards (about which, more here).

The graphic (and several others like it) comes with the following disclaimer:

When reading these graphs, the representation of content emphasis is accurate at each column- by-row intersection, but the smoothing between rows and between columns is not meaningful because the data are nominal. (p. 107)

What this means is this, Because the data is categorical, we could really have put them in any order we like. As a consequence, any patterns (any patterns!) we see within each graph are simply artifacts of the order we chose. This smacks of TURD to me. But I stand ready to be convinced. Any takers?

Reference

Porter, A., McMaken, J., Hwang, J. & Yang, R. (2011). Common Core standards: The new U.S. intended curriculum. Educational researcher, 40, 103—116.

Finally! A use for the point-slope form of linear equations!

I’m a slope-intercept man, myself.

y=mx+b, baby.

That’s what you really need in life, right? The slope is the rate, the y-intercept is the starting value. What more information could be necessary?

But the other day I needed the point-slope form. Here’s why.

The textbook I am using for College Algebra introduces difference quotients at the end of a section on operations on functions. It doesn’t flow and I don’t get why it’s there, but it is. So I want to give my students the best possible chance of making sense of the topic.

So I decided to cook up a dynamic graphical demonstration using the software Fathom. I have written about other uses of Fathom elsewhere, notably in an article about Fathom’s power for investigating experimental probabilities. In this case, I was using it to graph a parabola and a secant line to the parabola so that I could move one of the two intersection points back and forth while leaving the other point fixed. I arbitrarily chose (1,1) for the fixed point. The video below is a crummy little screen capture, but it gives the flavor of what I wanted to do.

The function itself (the parabola) was easy enough to plot, and I could use the slider to move the intersection point back and forth on the curve. But how was I going to define that line?

The  slider works by giving a value to a variable called V1. So I wanted the line that goes through (1,1) and (V1, V1^2). (NOTE: by V1^2, I mean “the square of V1”- I’ll figure out how to put exponents in text another time)

Slope intercept form assumes that I know the slope (which is going to change as the point moves) and that I know y-intercept (which is also going to change as the point moves). Indeed figuring out the y-intercept seemed like it was going to be a hassle.

Enter the point-slope form, which is:

point-slope form

Given the two points named above, the slope is:

equation.2

Interestingly, this slope simplifies:

equation.6

So the equation for the line is:

equation.4

But I need to solve for y in order to tell Fathom how to plot the line, so a bit of solving yields this:

equation.5.2

And this is good enough for what I needed to do. It’s the equation that underlies the green line in the demo video above. But by working a little bit more, I can get it into slope-intercept form:

equation.7

I don’t have time to try, but I sense that this simple equation would have been harder to derive by starting with the y-intercept form.