Saturday, February 04, 2012

the dialogue went like this

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.


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)



Horace Jeffery Hodges said...

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

By 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

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Kevin Kim said...

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