Find the Area of a Triangle

In discussing the following problem over on Math Notations, there's been an interesting mix of methodology.

In the coordinate plane, what is the area of ΔPQR given the coordinates P(4.5,4.5), Q(8.5,8.5), and R(6,0)?

The first level of difficulty lies, of course, in getting the image correct. Leaving this up to the kids makes for a tougher problem. Here's the diagram.

So the question fairly begs for a solution and it was interesting to me how many different ideas came up. Before you scroll down and see what the others did, try this problem yourself.

Dee-dee-dee-dee, dee-dee-dee, dee-dee-dee-dee-DUM-da-dee-dee
(Badly transcribed Jeopardy theme tune)

Here we go!

All of them are valid but operate under different frames of reference - your starting point, I think, depends a lot on what you're working on in math or the most recent similar problem you've solved.

Here's probably the simplest version. Big triangle minus the yellow one:

(1/2)(6)(8.5) - (1/2)(6)(4.5) = 12

I did the problem mentally and didn't twig on the two triangles. Instead I found a base and an altitude. 1/2* 4(root2) * 3(root2) like so:

The most interesting one was done by a student:
Translate the points P and Q down the line y=x. The orange and red triangles are still equal areas because they have the same height and equal bases.

Now find the area of the orange triange with a height of 4 and a base of 6.

Ain't that cool?

To repeat my comment on MathNotations:
We all approach problems differently depending on what frame of mind we're in at the time.

A basic math student would probably use the 2triangles method. An algebra student who just got through pythagorean theorem, distance formula and simplifying radicals might gravitate to the other. The algebra student who's been translating stuff might think of that - though the idea that sliding the two points down y=x doesn't change the area is NOT something that many students can do in their heads! Someone else might go to the trouble of flipping out Heron's formula or 1/2ab*sinC if the points were set up differently. Still others use Pick's formula since they can see the points in the diagram.

If your student weren't able to immediately follow your using PQ as a base and finding the altitude to it using perpendicular slopes (the problem, it seemed to me, was set up to push the solver in that direction) then he isn't really comfortable with algebra.

Don't knock your initial instincts until they get in the way of solutions. Students do like to see patterns that flow through many courses and like to see that old ideas are springboards to new solutions. We tell them all the time to "use critical thinking" and "Choose the best method," so we should show them different methods when the opportunity arises.

Hindsight is 20-20 in mathematics, too. For the record, my first instinct was 1/2 PQ * altitude. Had P and Q not been on y=x, I might have changed course but the solution was easy enough.

Here's some variations on this theme which may push the solver in different directions depending on what he's just been working on:

P(4.5,4.5), Q(8.5,8.5), R(5,1)
or
P(4.5,5.5), Q(8.5,9.5), R(6,1)
or
P(1.5,6), Q(4.5,4.5), R(6.5,8.5)

Just a few thoughts on a Saturday morning early Sunday afternoon.

Of Valedictorians and Significant Figures.

Not to change the topic, but why is education a horse-race? How can averages be any real measurement of or comparison between two students, when they are taking different courses from different teachers at (probably) different levels of difficulty?

I have always been at schools with valedictorians chosen after semester 7 and I HATE it. The only thing worse would be if we had chosen them after 8 semesters because then the two or three kids would be driven batty for their entire HS careers rather than for most of it.

We're not measuring them in a vacuum. We're measuring availability of a GOOD algebra in the eighth grade, not the BS that most eighth grade teachers are pawning off on their kids.

We're measuring the amount of constant worry about grades, rather than a more measured balance between doing your best and trying a course that might not be a slam-dunk but might be interesting or challenging.

We're measuring placement, too. My classes tend to be different from the guy across the hall. I have a different homework policy. He grades homework but I don't. His tests were hand-written for many years. I always typed. For some, this makes a difference. We grade differently as well. I use an overhead, he uses a blackboard.

We're measuring placement - will the schedule allow you to take AP English and AP Biology at the same time? Is your French 4 conflicting with your AP Physics? Why are you taking AP courses anyway (for the percentage push or for the knowledge and enjoyment)

We're measuring their lives. The kids who win are rarely the three-sport athlete-scholar, even though that kid at #8 on the list is far superior in almost any measure except GPA. Who are we rewarding? The kid who drops out of nearly everything but the academic grind.

From the first day they show up in the high school, their entire focus is on grubbing points so they can be #1 four years later. They choose courses based on whether it "helps" their average, not whether it is something they're interested in. Every test is an exercise in brown-nosing.

A couple of years ago I had two students who were on the "track". One took pre-calc as a sophomore and got a 97, the other took it as a junior and got a 96. The difference between the two grades could have been as little as 5 points on the final exam and rounding by GradeQuick. They finished within 0.003 points of one another. All other courses being equal (as silly as that statement is) means that the difference in difficulty between two of my finals might have been all that distinguished one from the other. One got the UVM full scholarship, the other didn't. Guess which one wanted to go to UVM. You're right, the other one. The valedictorian wanted to go to RPI and didn't use the money. #2 needed the money badly, but didn't get it. Interestingly, this scenario is repeated in one of the comments on Scheiss Weekly, here. It happens far too often to be tolerated.

In engineering and science, we would deride this as excessive use of accuracy pretending to be significant figures. Only in education can we imagine that we can judge that accurately with numbers that fluctuate constantly.

The center circle in a soccer field may 10 yards radius but we drew the line with a spray can tied to a string that wound around the center post and shortened it and we were walking around hunched over and weaving back and forth a bit. You can't say the area is 314.15926535897 yds² and pretend you know it to one-hundred billionth of a square yard just because there's a π key is on your calculator. GradeQuick does the same kind of thing. The precision is fantasy.

When you look at your teachers' grades, as I have, you find that grade inflation is rampant, but not consistent. There are far more 60s, 61s, 62s than there are 67s, 68s. Why? Because of the "bump". Inconsistency is a bitch if you are trying to be ultra-precise. The same happens at the top end. 95s get pushed to 99s or 100s by the "curve." Wherever your school has cut-offs, you find this shifting occurring. Pass-fail, eligibility minimums, honor-roll minimums, NHS reqs, whatever. The distribution is NOT correct. The dips below the cut-off points and the bumps above them are noticeable, if you look. Of course, guidance would never allow you to look if you ever let them know what was going on.

Try searching for correlations between SAT scores and grades in math - there's a real eye-opener in many cases. Are those As really As? Are those top kids really that good? When the kid gets an A in math all the way up to Calculus, but can only get 480 in 3 tries at the SAT, do you still have the same confidence in your grading system, it's fairness and your valedictorian?

There are other kids also making 100s, not because they are perfect and can solve anything in the course but because they are in a class that they outshine and the teacher can't give 105s. If you put them in with their peers, they might only get a 97. If you put them in the class that would be most beneficial, they might get a B, but would work their butts off and really learn everything trying to keep up. Why should placement be a part of the kid's worth?

Why should an IEP kid who gets 100s all time be the valedictorian if she cannot write a paragraph-long speech, or deliver it? (True story. The school quickly changed its mind. Ruined at least 7 rants for this blog.)

We need a change here, people.

You can pretend that a photo-finish is appropriate but it isn't. It certainly isn't if you consider all the ways in which parents and school can affect the situation out of the control of the kid. Think of all the shifting around that I've mentioned and you can come up with a bunch more.

How can we say that Alphonse is better than Gaston?

I'm in favor of identifying the "bunch" at the top and having them all participate - call it the TRUE Honor Society.

Having just one may be more satisfying to the ONE, but it's not education and it isn't real.

Just sayin'.

Help ! I'm confused. My math mojo is deserting me ...

Okay, here's the question. NEA sends out this blurb about germs in school and you just know there's stupid people involved. I'm reading the thing and seeing that the water fountain has 2 700 000 bacteria per inch.

Huh? Shouldn't that be "per square inch"?

Okay, maybe a misprint. I read on. "But the very worst, most disgusting, just say no for germ's sake, was the public sandbox, where NSF's analysts found a horrifying 7,440 bacteria per inch!"

Huh? Huh?

How is this the very worst? The water fountain is something like 4000x as much. The cafeteria plate that they eat off is twice the number! What the hell? With mathematical analysis like this, it's no wonder that elementary teachers don't find it necessary to do math or find it embarrassing when they screw up teaching fractions.

"I just don't DO math. Tee-hee-hee. Giggle."

Can't you just hear it now? "Let's make a silly crack about how video games are bad for you - 551 sounds like a BIIIIIG number!" Too bad it's one-third the number of bacteria on a typical student's hand.

"I don't DO fractions, either. Giggle. Snark."

Here's the link. Reposted below.

Eeek! There's a germ on my desk!

The bacteria and germs are out there, just not where you might think.

Even as fears of a flu pandemic fade, it's worth noting where all kinds of germs lurk in your classroom. In 2006, the non-profit NSF International took its microscope to two Michigan elementary schools to measure the general bacteria population on different surfaces.

Water Fountain Spigot 2,700,000	Cafeteria Tray 33,800	Cold Water Faucet 32,000
Hot Water Faucet 18,000	Cafeteria Plate 15,800	Computer Keyboard 3,300
Toilet Seat 3,200	Student's Hand 1,500	Animal Cage 1,200

* Units are number of aerobic bacteria per inch

This year, NSF's microbiologists widened their scope and swabbed surfaces in 26 public places. Thanks to your school custodians, schools starred in the comparison. School bus seats practically sparkled with fewer than 10 bacteria per square inch, as did books and basketballs. Still quite clean were desks, gym mats, and computer mice.

But leaving schools behind, they found a little germ factory on the video game controls in movie theaters-551 bacteria per square inch. (Proof that those games aren't so great for mind AND body.) But the very worst, most disgusting, just say no for germ's sake, was the public sandbox, where NSF's analysts found a horrifying 7,440 bacteria per inch!

Fortunately, your classroom doesn't resemble a sandbox. And it's not too hard to keep it from breeding like one. Check out these tips from veteran custodians on fighting germs in the classroom.

Those who can, do. Those who can't, teach teachers or run their union.

Linda's Top Ten Excuses

It's such a shame that students have to work at stuff.

Linda's Top 10 Excuses for Not Passing Physical Science, Frankly, Any Class Really.
Dude.
The excuses parents hear - followed by what it REALLY means

1. My teacher didn't teach this
   1. What it REALLY means - I didn't show up at school until 1/2 way (or more) through the class, and she refused to immediately stop the class, return to the beginning, and teach it all again - but, this time, speeded up.
2. My teacher didn't help me
   1. When I finally got around to working, my teacher didn't serve me as my personal tutor, and ignore all the rest of the class, who HAD been working - so, it's really HER fault
3. My teacher never explains this
   1. I missed the explanation/lab/whatever, due to:
      1. Talking
      2. Drawing
      3. Sleeping, and/or
      4. Texting
4. My teacher's tests are too hard - any or all of the following apply
   1. I didn't read the textbook
   2. I didn't read the end-of-chapter summary
   3. I didn't ask questions
   4. I didn't listen to the answers
   5. I took no notes
   6. I copied everything in class/lab from the smart kid - it wasn't my fault that he/she wouldn't let me copy off his/her test
   7. I had a really important television show to watch
5. My teacher acts like his/her class is the only one
   1. And gets mad when I try to do my math homework in class
   2. And won't excuse my cutting class to complete my English paper
   3. And expects me to show up for tutoring after school
6. My teacher never is around
   1. When I show up after school, without an appointment, during the same time as the Faculty Meeting (which I knew about from the afternoon announcements)
   2. And won't give up her lengthy 22-minute lunch without an appointment
   3. And expects to keep her doctor's appointment, even though I have tears in my eyes and am begging and everything
7. My teacher hates me
   1. Just because I called her a b---h when she wouldn't give me help during the test
   2. Because she wouldn't give me a "breathing" D - and, I really was!  Breathing, I mean - every class
8. Nobody can pass this test
   1. Unless they like, STUDY, or something unreasonable
   2. So, of course, all those OTHER students, who passed, must have been given the answers in advance
9. The amount of work is ridiculous!
   1. A warm-up activity, EVERY DAY!  That is based on what was done the previous class!  (Like I remember!  That was YESTERDAY!)
   2. We only had 2 weeks to complete the study guide!  Which we ONLY got 100 points for!
   3. She even assigned a project!  That wasn't a poster!
   4. She even expects us to read the textbook!  Or, at least the chapter summary!  Now, that's WAY too much!
10. She won't travel to the state to insist that they not count my high-stakes test!
    1. She "claims" that the test is based on what we were doing in class.  As if I did any of THAT stuff!

"It is SOOOO unfair. I like to hear myself whine but I know that Mommy and Daddy will come in and complain. The principal will raise the grade afterwards anyway or demand endless reams of paperwork to 'justify' the failing grade."

9 Kinds of Stupid Right There

Texting while Driving a Bus at 60 mph.

It's more than 50% likely that this guys a fool.

Via The Bad Astronomer and Mark C. Chu-Carroll comes what is probably the most probability-challenged statement of the year.

If it could happen or not happen, then the odds must be 50-50.

WOW.

Watch it all or forward to Walter Wagner at about the 2:15 mark.

The Daily Show With Jon Stewart	M - Th 11p / 10c
Large Hadron Collider
thedailyshow.com
Daily Show Full Episodes Economic Crisis First 100 Days

This is from the Daily Show and is pretty interesting until Wagner makes his mind-blowingly stupid remark.

But then, the guy sued in a Honolulu court to stop the collider, which isn't even in this country, so his lawsuit has a 50-50 change of going his way -- either he'll win or he won't, right?