# Tag Archives: college algebra

## 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.

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?

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

## College algebra teachers! Please try this and report back!

My fellow teachers of College Algebra. I want to talk to you today about domain and range. For now, let’s leave aside all the analogies, vending machines, notation and ants crawling on graphs. Let’s get to the heart of the matter with an assessment/instructional task.

This is in the spirit of Eric Mazur, but it is low-technology. It will take 10 minutes. Then you’ll need to report your results back to me. We can talk about what those results mean.

Here is what you do.

Get yourself a College Algebra class that has studying (or even one that is still studying) domain and range.

Then get yourself some index cards in two colors. We used yellow and pink. You may use whatever you have on hand. Make clear that yellow means yes, and that pink means no.

Also make clear that they will raise their cards in unison and on the count of 3. This is to prevent conforming to the majority and will result in more honest representations of your students’ understanding.

Practice this routine on some non-mathematical questions. Some that will be universal nos, some universal yeses, some that are mixed.

Now have your students consider the function $y=x^{2}$, where x is interpreted as the input, and y as the output in the usual way. You are going to ask a series of questions about the range of this function.

Ask, Is 4 in the range of this function?

You should get near universal yellow. Ask someone to state the case and make a note of their argument on the board.

Ask, Is –2 in the range of this function?

My bet is that you’ll have a lot of pinks, but several yellows. Those yellows probably need clarification that we are asking about range, not domain. But don’t assume it. Ask the pinks to state their case. Ask the yellows to refute or question. Never say anything a [student] can say.

Likely, the yellows will argue that there is no number that, when squared, gives -2. They mean real number; I see no need to make a big deal out of this. But make a note of the argument. Perhaps writing something like this on the board:

$($ nothing $)^{2}=-2$

You will probably need to revisit this later on.

Ask, Is $\frac{1}{4}$ in the range of this function?

This is probably all yellow and mostly unproblematic.

Now the fun begins.

Ask, Is π in the range of this function?

If you don’t get a good mix of yellow and pink here, I will eat my hat. And those pink people? They are going to tell you that there is nothing that—when squared—gives π.

Have them talk it out in pairs or threes. Then have them show cards again. And then have the pinks state their case. Nine times out of ten, it’s going to be that there is no number that can be squared to get π.

My fellow College Algebra teachers, I am not interested in your theoretical arguments about what a fabulous job you/your textbook/your online homework platform are doing at teaching domain and range. If you wish to claim that your students will not show pink for π here, the burden of proof on you is high.

Notice with your students the very important difference between:

$($nothing $)^{2}=\pi$

and

$($nothing I can think of $)^{2}=\pi$

Someone will point out that $\sqrt{\pi}$ is a number, and that when you square it you get π. Highlight that contribution and estimate the value of this number.

Ask, Is 0 in the range of this function?

Probably mostly yellow, but worth asking to make sure.

Ask, Is infinity in the range of this function?

Seriously. If you don’t get, like, 80% yellow here then I do not understand your school’s placement system

Reinforce that infinity is not a number. Connect it to the notation and wrap up with one more.

AskIs 12 in the range of this function?

You should get nearly all yellow here. Get back to your regularly scheduled classroom activities.

## Update

There is lots of unexpected pushback in the comments on the value of teaching domain, range and functions in a College Algebra course. I had previously thought these to be de rigueur topics in such a course. I suspect sampling bias here.

Gregory Taylor mostly reproduced my results in his own course and had some lovely mathematical conversations along the way. You should go read his account.

## Questions as evidence of learning

I have argued that learning is having new questions to ask.

Here are a few questions that have surfaced in the early weeks of the semester. These are all student questions in College Algebra.

(1) Can it still be a variable if it only has one value?

This was asked by a student as we were sorting out whether $y=2$ counts as a function, and whether it counts as a one-to-one function.

(2) How do you solve $x=|y|$ for $y$?

This was asked by a student as were considering the relationships among functionsinverses and inverse functions.

(3) Is the inverse of a circle an inside-out circle?

See, we were using a set of equations, considering x as the domain and y as the range. We were asking whether each equation—so viewed—is a function and whether it is one-to-one.

Then we were switching domain and range (i.e. swapping x and y) and asking the same questions about this new equation. Bonus question was to solve each of the new equations for y.

One of our equations was $x^{2}+y^{2}=1$. Swap $x$ and $y$ and get back the same thing. Thus, a circle (as a relation) is its own inverse. Which fact I had never considered.

But my purpose here is to check in on the progress I am making in fostering and noticing student questions as evidence of learning.

## Sameness in College Algebra

Two years ago, I began using unit as an organizing theme in the math content course for future elementary teachers. That led to many adventures, including a TED-Ed video and new ways of talking to my colleagues about fractions, decimals and place value.

That work continues, but it has become part of my instructional practice; one of my habits of mind.

This year, I am thinking about sameness, and about helping my students to notice and pay attention to sameness. The formal name is equivalence, but I am not so worried about the vocabulary and formal definitions here.

I am concerned with helping students understand something about how mathematics views and uses sameness.

It is awkward at first, as any new teaching moves are. But it got us some good stuff recently.

We are studying functions. Our grounding metaphor for functions is vending machines. We discussed the following collection of vending machines the other day.

1.  This is my favorite vending machine of all time. The banana vending machine. It dispenses only bananas. It is like the constant function. More on this below.
2. There are two ways to get the Pocari Sweat in a can. Two inputs, same output. That’s OK. It’s not one-to-one, but it’s a function.
3. You put in a quarter, you turn the knob. Sometimes you get a die. Sometimes you get a top. Sometimes you get a ball. This is not a vending machine, really. Same input gets you different outputs. That’s a problem in the vending machine world, and in the world of functions.
4. The battery vending machine is one-to-one. Each battery type has its own button to push.
5. Put a dollar into this one, get a dollar out. Put in five dollars, get out five dollars. The output has the same value as the input. This is the identity function.

We discussed these in class one day. Then we opened the next class session by having students brainstorm with their partners specific functions with the traits exemplified by the vending machines. We divided up responsibilities for recording these functions on the classroom whiteboards.

Here is what our boards looked like after the large group (45 students) discussion. (Click to make legible.)

In order:

1. Lots of good stuff here. x=2 is not a function because, as a vending machine, it would take your money and not put anything out. All input, no output. The idea that we can write y=5 as y=5+0x was important. More importantly, this led a student to ask* about y=5, “Can it be a variable if it’s always the same value?”
2. Our example the previous day had been absolute value. They weren’t ready to venture much beyond this. As a class, they struggled to identify two x-values that would generate the same y-value. We need to work on that. But I have mentioned that this is College Algebra, right? Students have placed here, or worked their way here through developmental math. Either way, the idea of producing example points to demonstrate properties of a function has not been schooled into them yet. I’m on it.
3. Again, +/– square root was the prior day’s example. I love +/– x as an extension of the technique. Love that. And square root of x is not right. We’ll come back to that. Having a permanent record of the difference will be helpful.
4. Wow. Just wow. That was our example from the previous day. Not even a y=x+3 in the bunch! Work to do here.
5. Now we’re having fun. I love the $y=\frac{x}{1}$. Same function, different notation. I finished off our work by asking whether $y=\frac{x^{2}}{x}$ is the same as $y=x$.

Which (finally!) brings us back to sameness.

My students are highly accustomed to writing $\frac{x^{2}}{x}=x$. But they are not accustomed to thinking about what this means. Because when $x=0$, that equation is not true. The question then becomes, In what sense are these the same?

And that points us to the very heart of the discipline.

In mathematics, we decompose things according to their attributes, and we focus on one (or two, or…) of these attributes at a time, disregarding all of the others. Formally, when we write $\frac{x^{2}}{x}=x$, we mean “These two expressions are the same for all but a finite number of values of x.” We don’t say that, of course, but that is the essence of the equal sign here.

We returned to the sameness question with this video.

Are the two outputs the same? How? Are they different in any way? How? Again, mathematical sameness requires us to specify the precise ways in which two objects are alike.

We will return to machine number 3 above in class shortly. If you just want to get “a cheap plastic toy” out of the machine, then you get that every time. It’s a function. If you want to get “a top” out of the machine, then you get something different every time. Is it a function? Depends on what you mean by “same”.

Much more work to do. I’ll keep you posted.

*I recently argued that learning is having new questions to ask. This student was learning about what variable means, and had a question to ask that she maybe could not have articulated before this.

## 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!

## Some lightly edited thoughts about teaching probability

Michael Pershan boldly posted How I Teach Probability recently. I have struggled mightily with the teaching of probability over the years, so I took his video as an invitation to share and discuss.

The following is a copy/paste of an email I sent him (with, perhaps, some light editing).

I have been working on a longer, better edited version of the basic idea that I lay out here. But I’m working on tons of other stuff, too, so I cannot promise that the other one will ever see the light of day.

So here you are. Enjoy.

An important bit of honesty first: I am crap at teaching probability. Many reasons, mostly having to do with the ephemerality and abstraction of the topic, in contrast to the cold, hard demonstrable reality of (say) fractions.

With that said, I seem to have been making progress with an approach that has one important extended feature to the sort of work you show in your video.

Your video has (1) a scenario, (2) guesses, (3) small group data collection, and (4) large group data collection.

To that I add (2.5) explicit discussion of students’ probability models that inform their guesses, and then (5) some related follow-up activities.

After collecting guesses, I ask someone in class to describe WHY they said what they said. My job, then, is to press for details and to capture what they are thinking as carefully and accurately as possible on the board.

Once I have that, I ask for someone to describe a different way of thinking about it. Even if slightly different, we get it on the board. My writing isn’t abaout me dispensing  wisdom, it is about making a permanent record of a student’s model in enough detail that we will be able to test it later. For anything even moderately complicated (such as rolling two dice and considering their sum), I am disappointed if we don’t get at least four different models.

Now the data collection doesn’t just tell us who guessed closest; it can rule out at least some of our models.

As an example, in your scenario I would expect something like this:

Model 1: Zero multiples of three and and one multiple of three are equally likely, so I’ll bet on either one. There are only four possibilities: 0, 1, 2 and 3 multiples of three, so the probability of each of these outcomes is 1/4.

Model 2: There is more than one way to get 1 multiple of 3. Our model should account for that. There are three ways to get 1 multiple of 3 (on Die1, Die2 or Die3), three ways to get 2 multiples of 3, and only 1 way each to get 0 or 3 multiples of 3. That’s eight possibilities, so the probability of getting 1 multiple of 3 is 3/8, while 0 multiples is 1/8.

Model 3: Each die can come up either “Mo3” or “NotMo3”. Getting all “NotMo3” has probability 1/8 (1/2*1/2*1/2), while getting one “Mo3” and 2 “NotMo3” is also 1/8, but there are 3 ways to do it, so it’s 3/8 altogether.

Et cetera

As many different ways of thinking about it as my students have, I will dutifully record. I encourage argument, as each new argument suggests a new model, which we can test.

Now before we roll those dice, we set up a way to test these models. In my experience, I have to devise that test. My students are not sophisticated enough to think this way yet.

In your example, and with these models, I might identify an important difference to be that Model 1 predicts equal numbers of 0 multiples of 3 and 1 multiple of 3, while Models 2 and 3 predict three times as many 1s as 0s. Hopefully we also have a model that predicts something in between.

So now as we roll, we are not just looking for which happens more often, but to relative frequency. Because the model that better predicts the relative frequency that actually happens has got to be the better model.

Much more to say, examples to offer, etc.

I’m working on a paper that will be submitted for publication. It mostly records and expands on my NCTM presentation in Denver a while back.

Along the way, I decided to a bit of looking into the claims I have made about the standard view of what it means. I did it the lazy way. Google search on “Ready for Calculus”.

See for yourself. (Click on the images to be taken to original sources on the web.)

### Exhibit D

I did find one that had a different nature

The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions… Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets… You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of function underlies everything that calculus considers.

This was nice, and if you’re at all interested in this topic, you should go read the essay in its entirety. (Don’t worry, it’s short.)

### Conclusion

And now let’s all imagine how a community college developmental math program that took Exhibit D more seriously than Exhibits A—C would be different from the present-day state of affairs.