Pam  0:00  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:10  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:17  
We know that algorithms are amazing human achievements, but, ya'll, they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. I made it.

Kim  0:32  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:39  
We invite you to join us to make math more figure-out-able. Alright, so, Kim, I'm a little excited today because we're going to do something fun.

Kim  0:47  
Am I excited? 

Pam  0:49  
I think so. Hold your hats. Okay, ya'll, if this is one episode that you maybe don't do while you're driving, this may be one. In fact, pull up a graphing calculator. You could just... 

Kim  1:01  
All the

Kim  1:02  
people in the car right now are so annoyed because they have the podcast

Kim  1:06  
ready. 

Pam  1:06  
Yeah, sorry. But, you know... Yeah, pull up a Desmos. Grab your TI. Whatever you've got handy. I think we're going to be in Desmos today for some reasons. Here you go. Ready, Kim? 

Kim  1:19  
Yeah.

Pam  1:20  
Okay, I'm just going to ask you if you've ever thought about the equation 2x minus 5 equals 4. If you've ever thought about that as the graph of two equations and that equal sign is where they intersect?

Kim  1:39  
Um, I don't think so.

Pam  1:43  
Okay.

Kim  1:43  
Like picturing the two lines?

Pam  1:45  
Well, so if I had the line the line y equals 2x minus 5.

Kim  1:50  
Yeah. 

Pam  1:51  
What do you picture?

Kim  1:52  
So, first, I picture y equals x.

Pam  1:55  
Okay.

Kim  1:55  
And I think about it's twice as steep.

Pam  2:00  
Mmhm. 

Kim  2:00  
And then it shifted down.

Kim  2:01  
5. 

Pam  2:02  
Nice. And then do you have any idea where the line y equals 4 is? 

Kim  2:07  
At 4. 

Pam  2:08  
And is it horizontal or vertical? 

Kim  2:12  
It's horizontal. 

Pam  2:13  
Okay, so horizontal line at y equals 4. So, if I've got that 2x minus 5. 

Kim  2:17  
Yeah.

Pam  2:18  
It's sort of steep and it shifted down. Will it intersect the line, the horizontal line, y equals 4.

Kim  2:25  
Well, yeah, it'll intersect because you said they were equivalent.

Pam  2:30  
It could be a question. It could be, are they equivalent?

Kim  2:32  
But... Oh, well, didn't you say they were? 

Pam  2:34  
Yeah, that's a little tricky.

Pam  2:36  
Okay.

Kim  2:37  
So, how many times can two lines intersect? I guess would be a question.

Pam  2:42  
Lines? 

Kim  2:42  
Yeah, like we have two lines. We have 2x minus 5, and we have y equals 4. How many times can they intersect? 

Pam and Kim  2:50  
Once.

Kim  2:51  
Or none.

Pam  2:52  
Or none.

Pam  2:54  
Or what if they were parallel? 

Kim  2:57  
If they're parallel, none.

Pam  2:58  
Then none. So, we could conceivably have equations that could be the same equation. They could represent the same line. That means they would have infinite solutions. We could have equations that intersect once. That means they have one solution. And we could have lines that never intersect. Which means they have no solutions.

Kim  3:17  
Yeah.

Pam  3:17  
You agree?

Kim  3:17  
Yeah. 

Pam  3:18  
Okay, so I'm just going to ask you to type in y equals 2x minus 5. And then I'm going to agree that the way that you described it is what I see on my screen right now. 

Kim  3:30  
Okay.

Pam  3:31  
Okay, and then could you graph a second line, y equals 4. And then I'm kind of curious. Do they, on your screen, intersect in one place like you kind of thought they

Pam  3:41  
would?

Kim  3:42  
I think my...

Kim  3:44  
Hang on.

Pam  3:45  
Do you want tell us what you're looking at?

Pam  3:47  
Or... 

Kim  3:47  
So, I put the first line in and then in a second

Kim  3:53  
entry...

Pam  3:53  
Y equals.

Kim  3:54  
...the 4. 

Pam  3:55  
Not just 4, but y equals 4. 

Kim  3:57  
Why didn't I have to do that for the first one? 

Pam  4:00  
Well, that's a Desmos question. 

Kim  4:02  
Okay.

Pam  4:05  
Okay, so...

Kim  4:05  
Okay.

Pam  4:06  
...do you now have a horizontal y equals 4 intersecting a... What do you call that? Skiwampus. What... Diagonal! There we go. A diagonal. 

Kim  4:16  
Yeah.

Pam  4:16  
Okay, and can you tell about where? Now, when I say "where", this is tricky. Do I mean the x value or do I mean the y value? In this case, I mean the x value. And most often mathematicians when they say, "Where is the function doing something?" they mean the x value. So, can you tell the x value of that intersection point? 

Kim  4:40  
If I zoom in, I can. Hang on. 

Pam  4:41  
Well, you can zoom in, but you can also just put your cursor on top of it and click on the intersection point. It should pop up. 

Kim  4:47  
Lovely. 

Pam  4:48  
Did it pop up? 

Kim  4:50  
Yeah. 

Pam  4:50  
Okay, so is the x value 4.5? 

Kim  4:52  
Yeah. 

Pam  4:53  
So, question. Is 2 times 4.5... What is 2 times 4.5? 

Kim  4:59  
9.

Pam  5:00  
9. So, 9 minus 5, is that 4? 

Pam and Kim  5:02  
Yes. 

Pam  5:03  
So, that makes sense that that is the solution to that equation, yes?

Kim  5:07  
Yes. 

Pam  5:07  
Okay, so often we might look at the equation. And I wish I was writing something. Where did my pen go? Okay, there it is. If we were writing the equation 2x minus 5 equals 4, we might say to ourselves, "Well, something minus 5 equals 4. That something has to be 9. Would you agree?

Kim  5:28  
Mmhm. 

Pam  5:28  
Okay, so the "something" is 2x. So, now we've kind of got the equation 2x equals 9. So, I want to graph that. So, on a third. Keep the two that you have. Keep those in there. But on a third line, will you type in y equals 2x? And then on the next line, will you type y equals 9. 

Kim  5:47  
Okay. 

Pam  5:49  
Now, if you're on the home screen, you're not going to see that y equals 9. It's going to be too high. So...

Kim  5:57  
Oh, I see it. 

Pam  5:58  
Can you drag the whole thing down? 

Kim  5:59  
Yeah. I think I already done that because I see it. 

Pam  6:02  
Oh, okay. Where does then... At least for me, it's green. But where does the green y equals 2x intersect the purple y equals 9? (unclear). 

Kim  6:11  
They're different for me, but... 

Pam  6:12  
Sorry?

Kim  6:14  
They're different colors for me, but they intersect at 4.5.

Pam  6:19  
Hey, so x is still 4.5. 

Kim  6:21  
Mmhm.

Pam  6:21  
Yeah, so that's interesting. In other words, we can create an equivalent set of equations that has the same solution. And we can create the equivalent set of equations that has the same intersection. At least x value. What are you thinking?

Kim  6:38  
Well, so when you asked me if I had ever thought about the two lines being equivalent, I don't think that I ever thought about those as two independent lines. But what I have thought about a lot is the equivalency between 2x minus 5 being 4, and that being equivalent to 2x equals 9. 

Pam  7:05  
Mmm, nice.

Kim  7:06  
So, like this doesn't surprise me to see it's just another pair of equivalent lines.

Pam  7:12  
So, I'm going to be careful about equivalent lines. It's another pair of lines that have the same x coordinate for the intersection.

Pam  7:25  
No? 

Kim  7:25  
Well, I think there's one pair of parallel lines and a second pair of parallel lines.

Pam  7:31  
Ah, okay, okay,

Pam  7:33  
okay.

Kim  7:33  
So, yeah. Not the first ones you gave me to graph, and then the second ones being parallel. I mean like a and c being parallel and b and d being

Kim  7:42  
parallel.

Pam  7:44  
Gotcha, gotcha. Yeah. Okay, because we shifted them both. 

Kim  7:48  
Yeah.

Pam  7:48  
Yeah, nice. 

Kim  7:49  
Okay.

Pam  7:50  
I like that. 

Kim  7:50  
Alright. 

Pam  7:51  
Cool, cool. Okay, so another thing that we could think about would be to graph the line x equals 4.5. So, you actually have to type in x equals 4.5. And maybe I should have said, listeners, what will an x line look like? Like, we had y equals something was a horizontal line. Now, we're typing x equals 4.5. That should be a... And indeed, it is a vertical line.  

Kim  8:14  
Mmhm, yeah. 

Pam  8:15  
And then does that vertical line go through those two intersection points that we were looking at?

Kim  8:19  
Yeah, mmhm.

Pam  8:19  
Yeah. So, then we could create other sets of... Well, we call these systems of equations because they have the same intersection point. 

Kim  8:27  
Mmhm.

Pam  8:27  
Cool. Okay, so the place where I wanted to take this was... We kind of already said a little bit. Where if I have a line and another line, they could not intersect at all if they were parallel. They could intersect once. Or they could the equation could be the same. It could be a different way of writing the same line, and then we could have infinite solutions. What if I had a quadratic? So, now I'm going to invite you to get rid of all those. We're going to go to base camp. All gone. 

Kim  9:04  
Yep. 

Pam  9:04  
This time I want you to type in y equals x^2. So, that's just the parent function of a quadratic parabola.

Kim  9:14  
How do I make it a square? (unclear). 

Pam  9:16  
Oh, caret.

Kim  9:17  
Oh, I see it.

Pam  9:18  
Caret. And well, or there. Yeah, you get on the keyboard or you can just do caret too. Okay, so y equals x^2. And I'm going to ask where does it hit the x-axis?

Kim  9:29  
0. 

Pam  9:29  
And that's pretty obvious it hits that one point. So, what if I wanted to think about where a parabola hits the x-axis, but I'm going to move that parabola around. How many different solutions might there be with a parabola? (unclear)

Kim  9:47  
(unclear) 

Pam  9:47  
Mmhm, go ahead. 

Kim  9:49  
If it's shifted horizontally, we'll have either 0... Horizontally, still 1. But if it's shifted vertically, then it could have 0 or it could have 2.

Pam  10:02  
So, like... And I'm watching you in the video. So, you shifted it up, and you said it could have 0...

Kim  10:07  
Mmhm.

Pam  10:08  
..x-intercepts. And if you shift it down, it could have 2.

Kim  10:10  
Right.

Pam  10:11  
So, let's go ahead and shift it down. So, will you type... Keep that same one.

Kim  10:15  
Yeah.

Pam  10:15  
And just change it to x^2. So, just add on the end of it minus 4. You might have to...

Kim  10:21  
Keep the same one? 

Pam  10:23  
Yeah, keep the same one. 

Kim  10:24  
Okay. 

Pam  10:24  
And then did we shift it down? 

Kim  10:24  
Yep. 

Pam  10:24  
And sure enough, there's two x-intercepts.

Kim  10:27  
Mmhm. 

Pam  10:27  
Okay, cool. So, a quadratic could have... Oh, and we could shift it up. We could do plus 4.

Kim  10:34  
Mmhm. 

Pam  10:34  
And we would have no x-intercepts.

Kim  10:37  
Right.

Pam  10:37  
It could have one x-intercept we already saw. And we could shift it down, and it can have two x-intercepts.

Pam  10:43  
In the one that you have right there, what are the two x-intercepts? Can you tell in your graph?

Kim  10:47  
Yeah. -1, 1.

Pam  10:48  
Kim's squinting. We're both at that age where we have to...

Kim  10:51  
I'm having a hard time with eyes.

Pam  10:52  
We need... Yeah, both of us. So, did you say -2 and 2? 

Kim  10:57  
-1 and 1. 

Pam  10:57  
For x^2 minus 4? 

Kim  11:01  
Oh, I think said minus 1. 

Pam  11:03  
Oh, oops. 

Kim  11:04  
So, -2 and 2. 

Pam  11:05  
There we go. Okay, cool. So, then I'm just going to wonder what about a cubic function? So, before you type anything in, listeners, everybody, do we know the behavior of a cubic function? And then do we know enough about it that we could start to reason how many x-intercepts might a cubic have? What are the all the possibilities? And then I'll let you play, Kim. We can get rid of the quadratic. So, you just want to do x. And I just hit the shift 6 key, and then cubed. That's a way to do that.

Kim  11:44  
I think it's going to have 1 or 2. 

Pam  11:46  
Can you say why? 

Kim  11:47  
Yeah, because... Well, if I know what it looks like.

Pam  11:51  
Do you

Pam  11:52  
want to type it in? Well, go ahead... Sorry, I'm making you do four things at the same time.

Kim  11:58  
x^3.

Pam  11:59  
Is that

Pam  12:01  
what you thought it did? 

Kim  12:03  
Um, hang on. 

Pam  12:05  
While she's typing, listeners, I'll just sort of describe. So, a parabola looks kind of like a U, but it's not... The sides aren't parallel, right? They keep getting bigger and bigger as you go to the left to the right. A cubic does not look like a U. How do you even describe the shape of a cubic? It's like a funny.

Kim  12:24  
Yeah, it's a weird shape. So, is there... 

Pam  12:26  
Ski slope. 

Kim  12:28  
I guess what I was picturing when I said 2 is that there is, at some point, where it is flat. But I guess it makes sense that there's not.

Pam  12:38  
At some point, the slope is... 

Kim  12:41  
Is it?

Pam  12:42  
It's 0. It's not (unclear).

Kim  12:45  
But at a single point?

Pam  12:48  
Say that again.

Kim  12:50  
Like at a single point.

Pam  12:51  
So, at the origin, y equals x^3 is 0. 

Kim  12:55  
Okay, 

Kim  12:55  
but it makes sense that it would only have one then.

Pam  12:59  
Okay. But then we can tweak a cubic by adding a quadratic to it or subtracting it. So, just to the one that you have...

Kim  13:06  
Yeah.

Pam  13:07  
...do minus 2x^2. So, you should have x^3 minus 2x^2. And now we get a different look of a cubic. So, it's still a cubic in the long run, but we (unclear).

Kim  13:20  
That's what I drew my paper? Yeah, okay.

Pam  13:21  
Yeah, that's why you had. You had a form of a cubic. 

Pam and Kim  13:25  
Yeah. 

Pam  13:25  
And so I think that's why you said there could be 2. So, listeners, if you graph x^3 minus 2x^2, we've got a zero at 0. It's still going through the origin, but we also have a zero over there at 2. Yes?

Kim  13:39  
Yeah. 

Pam  13:39  
Cool. So, when we did a quadratic, we said there could be a possibility that there were no x-intercepts. Is that possible with a cubic? 

Kim  13:49  
No.

Pam  13:50  
You're shaking your head no.

Pam  13:51  
Why? 

Kim  13:51  
Because it always is going to cross the x-axis at one of the two points.

Pam  13:57  
But that's part of the end behavior of a cubic. 

Kim  14:00  
Yeah.

Pam  14:00  
The definition of the cubic is as we go to the left, it's getting more and more negative. As we go to the right, it's getting more and more positive. Somewhere it's going to cross the x-axis, at least once. But it might only be once. Okay. Then you said twice. Hey, and we're looking at an example of twice. So, it's, it's sort of bumping the x-axis at the origin. The x^3 minus 2x^2 is sort

Pam  14:26  
of bouncing there.

Kim  14:27  
Yeah.

Pam  14:28  
And that's a that's a double root. And then over there at 2, we have a single root. So, 2. Okay, cool. Is that it? So, it will for sure have one x-intercept. It could have two x-intercepts. Can it have more than two?

Kim  14:45  
I don't think I know enough about the transformations to know if something causes it to go back down.

Pam  14:50  
To dip

Pam  14:51  
again? 

Kim  14:51  
Yeah.

Pam  14:52  
Yeah, so, teachers, look how we just set Kim loose. Now, she could start to play. How could she mess with a cubic? So, what are all the ways you can mess with a cubic? Well, that's interesting. We just subtracted a quadratic. You can add a line. You can add a constant function. Subtract or add a line or a constant function. And we would then let kids play with that. And then say to them, what is the long run behavior of a cubic? And then what are the different kinds of short run behaviors that we could do? So, Kim, I'm just going to invite you to take that one that we had and just add 1. 

Pam  15:27  
Yeah, I don't think... Go ahead. You said add 1. 

Pam  15:30  
Add 1. So, you should have x^3 minus 2x^2 plus 1. And now how many x-intercepts?

Kim  15:36  
It's got 3.

Pam  15:36  
Now, we have 3. So, we know that it's going to at least have 1. It could have 2. We're seeing 3. I think we let the listeners go play. Could we have 4 with a cubic? How could you? And, ya'll, you could just start typing stuff in. Now, what's the definition of a cubic? Well, definition of the cubic is it's a polynomial where  the term of highest degree is 3. I didn't just say that well. The highest degree. The term. The term with the highest degree is no higher than 3. It has to have a 3. If the highest degree term is 3, then it's a cubic. It can't have other. You know, it has to be polynomial.

Kim  16:14  
Right.

Pam  16:14  
So, then let kids play with that. And then I'm just going to expand that. If we get kids good at the parent functions, what is the behavior of an exponential function? And like Kim just very nicely said, what are all the possible transformations?

Kim  16:22  
Right.

Pam  16:22  
How about a logarithmic function? How about a sinusoidal function? Like, if we could get kids to actually know that behavior, then they have a sense of the possible numbers of 0s. Which means they are solutions to equations, which means they are intersecting the graph. Though they're x-intercepts. So, solutions are connected to 0s of a function, are connected to x-intercepts of a graph. All of those can be informed by the sense and feel that we have for parent functions. Alright, there was our quick little fun that we had today.

Kim  17:00  
Yeah, so I feel like different than other Problem Strings. This one is about helping kids name what is happening to the graph as you add the next problem. Like, what shift is happening, what move is happening in order to give them more experiences. So, you would probably come back then with another string, and they might have a little bit more of experience with. Now students who didn't know what a cubic looked like, they're like, "Oh, yeah. I remember the shape of that.  And now, I may or may not remember that it can have 1, 2, or 3 intersections with the x-axis," but you're layering on more of that feel for what it can and can't do.

Pam  17:40  
Nicely said. And I'll add one other just quick thing. And we're using technology as a tool to understand what's happening, not as a crutch just to get answers.

Kim  17:51  
Yeah.

Pam  17:51  
Oh, if we could do more of that. Alright, ya'll. 

Kim  17:53  
Yeah.

Pam  17:53  
Thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!