*Conversation between me and a learning-disabled student. The quotes aren't exact, but I think I've correctly captured the overall pace, tone, and spirit of the exchange.*

ME: OK, so you remember the slope-intercept form for graphing a line?

LD STUDENT: It's "*y* equals *mx* plus *b*."

ME: Perfect. And you've worked with finding a line's equation by using two points, right?

LD: Yup. (*makes a weird, bubble-popping sound when pronouncing the final "p"*)

ME: So let's review that. Here are two points. Point A will be... (*writing while talking*) at (5,5), and Point B will be at... let's see... (-1,-3). Can you find the slope from that?

LD: Uh...

ME: The slope, in general, is what over what?

LD: Umm... 2 over 3?

ME: No, it's rise over...

LD: Oh, yeah-- rise over run.

ME: Right. So in this case, with the two points I gave you, can you calculate the slope?

LD: Uh...

ME: We talked about this before. You're basically making a fraction, but you have to subtract something from something, and something else from something else.

LD: Huh?

ME: The top of the fraction is going to be what minus what?

LD: Uh... this and this? (*points to "5" and "-1," the x-coordinates*)

ME: No, those are the x-coordinates. Is the top of the fraction going to be the difference of *y*'s or the difference of *x*'s?

LD: Difference of *y*'s?

ME: Right. Slope is rise over run, and the y-coordinates are all about the up/down while the x-coordinates are all about side-to-side. So where are the y-coordinates?

LD: Here? (*points to "5" and "-3," the proper y-coordinates*)

ME: Good. So those are the *y*'s. And which ones are the *x*'s?

LD: These? (*points to "5" and "-1," the proper x-coordinates*)

ME: Great! So can you make this fraction?

LD: Is it like this? (*writes "(5 - (-3))/(5 - (-1))"*)

ME: Perfect! So what's next?

LD: (*writes "8/4"*)

ME: Hm. Something's wrong here.

LD: (*writes "-8/4"; he's obviously just guessing*)

ME: OK, you did the subtraction correctly on the top of the fraction: minus a negative is like adding a positive, so 5 minus negative 3 is like 5 plus 3, or 8. That part's fine. So--

LD: Oh! (*redoes the subtraction in the denominator and gets the proper result: 6*)

ME: So it's 8 over 6, right?

LD: Yup. (*weird popping sound*)

ME: Are we done yet? Can this fraction be reduced?

LD: (*wordlessly reduces 8/6 to 4/3*)

ME: You got it! So now we have our slope. What do we do next?

LD: (*stares at me uncomprehendingly*)

ME: What's the whole point of this exercise?

LD: (*still staring, brow furrowed in earnest concentration*)

ME: We're trying to find the equation of this line by using two points, right?

LD: (*nods, not really following*)

ME: So now we've got the slope, right?

LD: (*nods*)

ME: So where do I plug that into the slope-intercept form? Which part is the slope?

LD: *b*?

ME: No, it's *m*. Remember, *m* is the slope. So now we can plug *m* into the original equation, like so-- (*jotting down "y=mx+b," then replacing m with 4/3*). But we're not done, right? We still have to find *b*. So how do we do that?

LD: I don't know.

ME: Sure, you do! We've already been over this three or four times, you've done it at school, and you went over it with my coworker the other day.

LD: Uh...

ME: You've already been given two points, so all you need to do is choose one of those points, then plug the *x* and *y* values into the equation we're making. So choose a point. Point A or Point B?

LD: (*looking distracted*) A.

ME: OK. So Point A is (5,5). How do I plug these values into the equation?

LD: *b* equals 5?

ME: No. Remember that each point is in the form (x,y), so--

LD: Subtract?

ME: No. You--

LD: Multiply?

ME: No; you need to *plug the numbers into the equation.* Do you understand what I mean by "plug the numbers in"?

LD: (*blinks rapidly, uncomprehendingly*)

ME: (*sighing*) OK, so we've got this equation: y = (4/3)x + b. We've chosen (5,5) to plug into the equation, so I write a "5" instead of the *y* and a "5" instead of the *x.* What's the equation?

LD: y = mx + b?

ME: No-- what's our new equation *after I plug in all the numbers?*

LD: Oh. (*starts writing... eventually writes "5 = (4/3)(5) + b"*)

ME: Good! So where do we go from here?

LD: (*stares again*)

ME: We're trying to solve for *b,* right?

LD: (*nods and blinks rapidly*)

ME: So we've got to get *b* all by itself. How do we do that? What's the first step?

LD: (*clearly no clue, and this is pre-algebra stuff*)

ME: Let's multiply 4/3 by 5 first. What's that going to be?

LD: (*calculates on paper*) 20 over 15?

ME: Where'd that extra 5 come from?

LD: I don't know.

ME: Do you remember how to multiply fractions? Top to top, bottom to bottom?

LD: Yeah, so I did 4 times 5 and 3 times 5.

ME: But there's no second 5.

LD: (*sullen silence*)

ME: The whole number 5 can be rewritten as a fraction, right? What's that fraction?

LD: 5 over 5.

ME: No, it's 5 over 1.

LD: Oh, riiiiight...

ME: OK, so if we multiply 4/3 by 5/1, we get...

LD: Oh, wait-- that's 20 over 3.

ME: Right! So if we're trying to get *b* by itself, what happens next?

LD: (*blank stare*)

ME: Don't we need to subtract...?

LD: OK. (*does nothing*)

ME: (*sighing again*) So we subtract 20/3 from both sides (*writing out the subtraction*), which means the left side of the equation now says "5 minus 20/3," and on the right side, all that's left is...?

LD: Zero?

ME: No...

LD: One?

ME: No...

LD: I don't know.

ME: It's *b.* All that's left is *b.*

LD: (*sounding skeptical*) O...kay.

ME: So *b* equals...?

LD: 5 minus 20/3?

ME: But we need to evaluate that expression. We can't leave it that way. So...

LD: It's minus fifteen thirds?

ME: Huh? No, that's not it. You've got to convert your whole number into a fraction-- something over 3.

LD: (*blinking while staring at the paper we've been working on*)

ME: (*writing "5/1 = [ ]/3"*) 5 over 1 equals something over 3.

LD: 3?

ME: No. How did I go from 1 to 3 for the denominator? What times one is 3?

LD: 3?

ME: Right.

LD: (*irritated*) But that's what I *said.*

ME: No, you were saying "three" for the wrong thing before. So if 1 times 3 is 3, then 5 times 3 is...?

LD: 15.

ME: Good! So what's the fraction?

LD: 15 over 3?

ME: Right. So now we can subtract. 15/3 minus 20/3 equals...?

LD: Negative 5?

ME: Negative 5 *over 3.*

LD: OK.

ME: So where do we go from here?

LD: (*that famously blank stare*)

ME: We've just solved for *b,* right? So what do we do?

LD: (*still staring*)

ME: We've got our slope and our y-intercept. Can't we plug those into the equation?

LD: I guess.

ME: So give it a try.

LD: (*poised, looking ready to write something on the page... eventually writes nothing*)

ME: What's our slope?

LD: Negative 5 over 3?

ME: No, that's our y-intercept. (*pointing at the slope written on the page*) What's the *slope?*

LD: Oh, yeah-- four thirds.

ME: And what's the y-intercept?

LD: Negative 5 over 3?

ME: And what's the slope-intercept form?

LD: *y* equals four-thirds *x* plus *b*?

ME: No, no-- the *general* form?

LD: (*stares, unsure what to say next, fearful of being wrong again*)

ME: It's y = mx + b.

LD: Oh. Right.

ME: So now let's plug in our slope and our intercept...

LD: *y* equals four-thirds *x* plus five thirds?

ME: Almost there, but remember that the y-intercept is *negative.*

LD: *y* equals four-thirds *x* minus five thirds?

ME: Good!

LD: (*frustrated*) But I *said* four thirds a second ago and *you* said I was* wrong!*

ME: I was asking you for the general equation before; now we're talking about the equation for *this particular line.* But at long last, we've got our equation, so congratulations. So, one more time-- what's the slope of this line?

LD: (*long pause*) Five thirds?

ME: (*sighing*)

_

## Saturday, February 04, 2012

### the dialogue went like this

#### 2 comments:

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Damn you, Kevin! You promised me you wouldn't post this on your blog! I'll never ask your help on math again!

ReplyDeleteBy the way, the word verification is "aurce"! That's a British word. A quant like you probably doesn't know what it means, but it's what you are for telling on me!

Jeffery Hodges

* * *

Sorry, Jeff, but the world has a right to know about the many crosses I bear.

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