# Tag Archives: range

## 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.

## Experiments with datasets

Go out and collect a modest-sized, discrete dataset. Name lengths of all of the students in your classroom, say, or the number of people in each of their households.

This bar graph is only tangentially relevant, being more of a case-value plot of four different populations. But it breaks up a texty post. So deal with it.

Now play with that data.

If we add one or more new (hypothetical) cases, can we…

1. Increase both the median and the mean?
2. Decrease both the median and the mean?
3. Increase the mean while decreasing the median?
4. Vice versa?
5. Increase the Mean Absolute Deviation (MAD) while decreasing the mean?
6. Vice versa?
7. Decrease both the MAD and the range?
8. Decrease the MAD while increasing the range?
9. Vice versa?

If we delete one or more actual cases, can we…

1. [same list as before]

Thanks to Susan Friel, Connected Mathematics and tons of other creative folks for getting me started with this. Future elementary teachers to tackle this shortly. I’ll report back.