one of those days...

so there's this one 10th grader that i meet with twice a week after school (assuming he shows up) for geometry help. the funny thing is that his mom (verrrrry asian) always asks other people to ask me to help her son. about once a week the school registar asks me if i can meet with the kid. earlier today, i happened to run into one of the indonesian ladies who works in the payroll office, and right there in the bathroom, SHE asked me if i'd tutor the kid. does his mother not understand that I ALREADY TUTOR HIM?!? siiiigh. i can't tutor him more than twice a week, partly because my life is insanely busy and also partly because if i tutored him more than twice a week i'd go crazy. i kept track of our conversation during our tutoring session today. it goes as follows.

(preface - this student comes to these tutoring sessions with the expectation that i will bend over backwards for him and give him the answers to everything because his parents pay a lot of money for him to go to this school. so i basically do my best to NOT do that. not that i won't help him, but i'm not going to spoon feed him the answers to his homework. if he asks a question, i help him work through it. i don't just give him the answer. this tends to aggravate him. so with that intro to my mindset, read on...)

*long silence as he stares at the book*
me: what problem are you working on?
student: i'm kind of stuck.
me: which number?
student: 16.
me: ok. what do the instructions say?
student: i don't have enough information.
me: what does the problem say?
student: the altitude is 9.
me: the altitude of what?
student: the triangle.
me: and what else?
student: that's all i'm given.
me: well, what are you asked to do?
student: uhmmmm…… oh, find the area of the triangle.
me: so what other information are you given?
student: none!
me: bring me your book.
*student brings book - we are sitting in different places in my classroom*
problem says: the altitude of an equilateral triangle is 9. find the area of the triangle.
me: ok, so you know it's an equilateral triangle.
student: i guess.
me: what's special about an equilateral triangle?
student: i don't know.
me: better look it up then.
*student rolls eyes, gives a deep sigh, and takes his book back to his desk. another long pause*
me: so did you find it?
student: opposite angles are congruent.
me: it's a triangle… it doesn't have any opposite angles.
student: fine.
me: what do you know about its sides?
student: it has three.
me: right… and what's special about the sides of an EQUAL-lateral triangle?
student: they're equal?
me: right! and every equilateral triangle is an isosceles triangle too, right?
student: i guess.
me: ok, and what is special about the altitude of an isosceles triangle. (i know he knows this because i've beaten it into his head.)
student: it bisects the base perpendicularly.
me: right! so if every equilateral triangle is an isosceles triangle, and the altitude of an isosceles triangle bisects the base, what do you know about the altitude of THIS triangle?
student: it's 9.
me: ok, but what else do you know about it?
student: nothing. i don't know.
me: what is every equilateral triangle?
student: i forget.
me: isosceles.
student: oh.
me: and what is true about the altitude of an isosceles triangle?
student: it bisects the base.
me: so what do you know about this triangle and its altitude?
student: i don't know.
me: ok. keep trying. (starting to get exasperated, and turning back to my work.)
student: (thinks for a second) it bisects the base.
me: mmhmm. ok, so let the length of each side be x since you don't know how long they are, but you DO know that they're all the same.
student: ok. (draws a picture to represent it)
me: and since you have that altitude drawn right down the middle, now you have 2 smaller right triangles. so let's just think of one of those triangles first. the hypotenuse is one of the sides of the equilateral triangle, so it's how long?
student: x.
me: and how long is the base of that right triangle?
student: x.
me: no, remember we're only looking at half of the big triangle.
student: 2x?
me: no… what is half of x?
student: i don't know. what's x?
me: you don't know x. that's why you're using a variable. how would you write half of x?
student: 1/2x.
me: ok. so now use the pythagorean theorem to find x. then you'll have the length of all the sides!
student: ok.
*goes back to desk. 10 minutes later comes back*
student: i don’t know what to do next.
me: which number are you on?
student: 16 still.
me: ok… where are you stuck at?
student: the pythagorean theorem.
me: well, what do you have?
student: 81 plus half of x squared equals x squared.
me: right.
student: what do i do next?
me: solve for x.
student: oh. right.
*goes back to desk. several minutes later….*
student: how do i solve for x?
me: just like any equation, get x on one side and everything else on the other side.
student: how do i get x on one side?
me: what do you have?
student: same as before.
me: well, square the half x first.
student: so it's just x.
me: uhm… no… what is one-half x squared? remember the one-half x is in parentheses.
student: yeah, i know. it's x.
me: no… i think you're taking one-half x times two. remember that squared means you multiply it by itself. so one-half x times one-half x.
student: so it's x?
me: no. still no. what is one-half times one-half?
*student doodles on paper for a minute or two*
student: one-fourth.
me: ok. and x times x?
student: 2x.
me: nope.
student: x squared.
me: yes. so one-half x squared is…
student: x squared.
me: don't forget about the one-fourth.
student: one-fourth x squared.
me: ok. now subtract that from both sides.
student: how?
me: how do you subtract?
student: i don't know.
me: write out your equation.
student: (writes) ok. now what?
me: subtract one-fourth x squared from both sides. write it.
student: ok. so i have 81 equals x squared minus one-fourth x squared. now what?
me: simplify. what is x squared minus one-fourth x squared?
student: i don't know.
me: what is one x squared minus one-fourth x squared?
student: hold on.
*student works for awhile*
me: ok, so what do you have now? what did you get for x?
student: root 108.
me: great! so now you can use that to find the area of the triangle.
student: how?
me: well, that tells you each side of the equilateral triangle since you let one side be x before and you just solved for x.
student: oh.
me: so what is the length of the base?
student: half of root 108.
me: noooo… what is x?
student: root 108.
me: so what is the length of the base?
student: half of root 108.
me: still no. look at your drawing. the base is x. you just solved for x. what is x?
student: root 108.
me: right. so how long is the base?
student: root 108.
me: good. and what is the height?
student: the altitude. 9.
me: good. and how do you find the area of a triangle?
student: half base times height.
me: right. so do it.
*student grabs calculator*
me: don't forget that your answer needs to be in square root form…
student: i know.
*continues punching values into his calculator, thus getting a bunch of messy decimals*
me: you can't use your calculator to get square root form…
student: i said i know. i heard you the first time.
*continues with the calculator. i am speechless. why am i doing this again?*

ps - it took us 45 minutes to get through one problem in this fashion. maybe that's why his mom keeps asking other people to ask me to help him.