Algebra in The RealWorld

Dan Meyer poses Three Questions about a problem from his professional development:

A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?

"One, is the problem realistic? Would a real person need to solve this problem?
"Two, is the solution realistic? Would a real person solve the problem using a system of two equations?
"Three, in what ways does this problem help our students become better problem solvers?"

I think he's right, in many ways. The problem he's discussing is contrived and convoluted at best. It does have that "algebra-book word problem from hell" feel about it, but word problems in math textbooks are a funny thing. They have to straddle the line between being realistic and being useful in a classroom for teaching. When constructing a word problem or using one, you need to keep this line in mind. If the word problem doesn't fit your topic, you should change it.

I take a slight issue with the questions, though, in that I don't want to always be asking for a real-world solution that a real-world person would find for a real-world problem.

First, define a "real person". Do I pick me or a mathphobe? If you purposefully want a problem that takes creativity to solve, then separate it from the track you're in. Call it the Puzzle of the Week.

This particular problem doesn't have enough information to come to a single solution anyway - how about adding in the cost per mile of van and car, as well as an additional payment for driver responsibility. Are we trying to minimize cost or make sure all chaperones drive? Do we have a reason to not use the vans unless necessary?

I'm never gonna use this.

Well, Eighties Music Forever, I never said that you would have to. I notice that you didn't "use" poetry, chemistry, biology, or science of any kind, nor history, psychology, art of any kind, nor Literature; pretty sad that all that schooling has gone to "waste" because no one gave you an artificially simplified problem that you recognized as "Algebra" instead of as a real-world problem that you couldn't have dealt with unless you'd understood algebra.

Of course, you wouldn't be much of a person without it all, though, so it's a good thing you learned it in school.

We could use "I'm bored" as an excuse or "When am I ever gonna have to use this?" as a reason to kick you out the door into the Real-World and let you get a Real-Job and pay Real-Bills, but I prefer to put it more simply:

You're (Black/Latino/Female ... you can fill in the blank) so you aren't allowed to take Algebra - it will be too difficult for you and the community doesn't feel that they should be paying for your education when you'll only ever be a (minimum-wage/slave-labor/custodian/mechanic ... fill in the blank) worker.

If you recoiled when I forbid you from learning something because I didn't consider you worth the effort, why should I allow a student to do this to themselves?

Every student hates algebra because learning it is hard ... learning anything is hard if you've never done anything like it before.  

Should Math Really Be A Required Subject?

I recently got an email:

And as a homeschooling father to a nine-year-old who puts two hours-plus of serious math instruction into each day I find myself in complete agreement with your approach to teaching the subject.  BUT...

I'm curious, do you find anything to agree with in Nicholson Baker's famous critical essay on Algebra II?  I do, and I make up for time that would be spent on the subject with an excess of number theory.  Any opinion greatly appreciated!  Best,

Oooh, boy.

Good evening,

I haven't read it, and it's behind a paywall, but what little I can see of or about it leads me to think "Just another whiny English Major and writer railing against algebra." "I don't DO math. Hee, hee."

I agree with the idea that students who have absolutely no interest in a mathematical major in college should probably be able to take a good statistics course instead of what alg2 has become under CCSS, but I hesitate to let all of them off the hook so quickly. I've found that most students can get SOMETHING out of my algebra class, and I hate to see them quitting on themselves after Geometry. I also believe that many kids vastly overestimate the difficulties they will have.

I think putting an elective as the third year and allowing kids to take alg2 as seniors is often a good compromise.

I guess for me the biggest complaint is this idea that every kid must take it and do well.  Why is A the only acceptable result? Why can't a kid struggle with it, get a D and move on? (And by D, I mean the kinda-gift, 'thanks for trying, but you really don't know this well enough for me to say Alg2 on a transcript') At least then he's been exposed to some things, has a partial idea of what logs are and how they apply to pH, and some pretty functions and so on.

The next time, if there is a next time, he'll do better. If there isn't a next time, well ... there's that small fraction he did understand.

If we look at a common complaint, "I'll never get this in a million years!" and tone down the exaggeration to "I'll never get this in one year!", it should become apparent that some kids may only get parts of alg2 this year. But they can always revisit it later in college or as the parent of student.

It does develop abstract thinking and the concepts are critical in most of the modern world. Yes, I understand that everyone thinks they "didn't use algebra in the RealWorld today" but they are wrong.  Algebra in its pure form never appears ... but the concepts, ideas, and behaviors show up everywhere.

Finally, I ask this one question .... If I said "Your son is too stupid to take algebra." "Your daughter is incapable of such abstract thought; have her take something more girly." "Your black child can't understand that." "Your daughter shouldn't fill her head with difficult math."  What would you do?

Well, I think you'd be calling for my head.

You know your kid best and even you don't think that you can predict his future; neither of you have any idea what he'll need or desire to do. You aren't really sure whether his difficulties are due to bad teaching or a lack of ability. You've read that girls learn their math-phobia from their mothers/ kindergarten teachers/ peers/ MyLittlePony and you do not want to limit her in any way before she's had a chance.

So why do we all have this mindset to "Let them admit defeat so early"? Me, I'd rather have them try again.

I tell kids "You have no idea what you'll do tomorrow. How can you predict what you'll be good at in ten years?  Algebra is difficult, but lots of things are difficult -- besides, you're 16 ... what else do you have in mind to take? What else can you do while the education is FREE? Do you want to put this off until college and pay $1200 per credit for it?

I was good at math and physics in HS, but I really hated history (BOOOOOORING). I got my degree in engineering and I spend hours during my vacations solving math puzzles .... but my passion is medieval reenactment. I can hold my own with the history department on anything older than 1600, pre-Renaissance. I have complete sets of armor that I wear in full-combat martial arts ... and the people I fight against are teaching me techniques they've researched from centuries old fechtbuch (in the original German). Last summer, we were camped around the fire, period canvas pavilions in a big circle, and we spent spent hours discussing the accuracy of the story of Bayeaux Tapestry as compared to and checked by contemporary sources.  Who would have predicted that?

Should every child take history? Yes. Math? Yes, including all the algebra they can stand, and then some. Art? Science? Languages? English? Yes.

No, I don't when you'll use it.

But I do know you'll do very little without it.

I've gotta get back to work. Thanks for reading.

Factor Diagrams

Factor diagrams have been making the rounds and I was struck by this one from Brent:

Is it just me or should the diagrams for 15 and 45 be a blue pentagon with surrounds?

Still, it's cool.

Algebra 2 for All

When all adults in California can understand and complete an algebra II course, then it makes sense that all high school students should be able to.
Otherwise ...

It's not that these California math teachers had the stones to sign, it's that more of us don't feel we can.
http://mrmeyer.com/blog/wp-content/uploads/palo_alto_high_anti_ag.pdf

Maybe this is why tenure is so necessary?

That's WYCDWT - Speeding Tickets

"For each ticket, Mr. Foreman digitally superimposed the two photos - taken 0.363 seconds apart from a stationary point, according to an Optotraffic time stamp - creating a single photo with two images of the vehicle. Using the vehicle’s length as a frame of reference, Mr. Foreman then measured its distance traveled in the elapsed time, allowing him to calculate the vehicle’s speed. In every case, he said, the vehicle was not traveling fast enough to get a ticket."

Judges agreed.  Booyah.

Let's make Algebra II mandatory - that'll learn them.

From Mother Jones and The Washington Post:  ed reform has gone to this:

Of all of the classes offered in high school, Algebra II is the leading predictor of college and work success, according to research that has launched a growing national movement to require it of graduates.
In recent years, 20 states and the District have moved to raise graduation requirements to include Algebra II, and its complexities are being demanded of more and more students.
....One of the key studies supporting the Algebra II focus was conducted by Anthony Carnevale and Alice Desrochers, then both at the Educational Testing Service. They used a data set that followed a group of students from 1988 to 2000, from eighth grade to a time when most were working. The study showed that of those who held top-tier jobs, 84 percent had taken Algebra II or a higher class as their last high school math course. Only 50 percent of employees in the bottom tier had taken Algebra II.
....But not everyone is convinced that Algebra II is the answer. Among the skeptics is Carnevale, one of the researchers who reported the link between Algebra II and good jobs. He warns against thinking of Algebra II as a cause of students getting good jobs merely because it is correlated with success. “The causal relationship is very, very weak,” he said. “Most people don’t use Algebra II in college, let alone in real life. The state governments need to be careful with this.”

Holy crap!  It's so damned simple!

Calculus students do really well in jobs that require mathematics.  Let's require every student to take calculus and we'll have a nation of math geniuses.

Algebra II! As a minimum requirement to hold a high school diploma! We're literally saying that if you can't factor polynomials, manipulate complex numbers, do matrix arithmetic, and understand basic trig, then you can't get a high school diploma? Really?

The push comes from Achieve, a group of idiots would really, really need to take statistics again.

Let's think here ... smart kids take algebra II ... smart kids are motivated ... smart kids are likely to maintain their motivation and work their way up the corporate ladder ... smart kids are likely to take art ... smart kids are likely to do well in literature ...  smart kids are likely to finish college ... smart kids do well in science ... smart kids work hard ... smart kids focus ... smart kids are likely to be smart ... smart kids eat healthy foods ... smart kids play a lot of video games ... smart kids read a lot ... smart kids play soccer ... smart kids volunteer their time ... smart kids eat crappy, greasy foods ... smart kids are likely to be athletic ... we could go on.

Why is anyone making the link only between algebra and success?

Why are we so stupid?

PsuedoContext

Dan Meyer gives an example from "The Real World":

Santa Cruz Sentinel, today:
The City Council will consider a proposal today to establish a citywide pay-by-cell phone system that would allow motorists to start, finish and extend time for meters or fee-based parking spots. [..] Consumers would pay a fee of 35 cents per transaction, or 25 cents for frequent users if they are willing also to pay a monthly access fee of $1.75.

"Is pseudocontext a failure of imagination or is it a symptom of laziness? Because this sort of thing just isn't hard to find."

I think it's pretty lame of him to toss out this false dichotomy like that. (My browser settings and his comment system don't seem to be on speaking terms, so I'll mention this here.)  And I notice that he's been teaching for how long and only now noticed this gem? (I know that's unfair, but so is the original question.  I withdraw my snarky comment.)

No, the real problem here is one of timing, of not having this pop up in your newspaper on the day you need it, at the time you're doing the lesson planning. It's got little to do with a lack of imagination.  In fact, a lack of imagination is probably the best trait for someone doing lesson plans.  Teaching takes imagination -- why waste this limited resource on something as foolish as lesson plans?  What God in whichever Heaven you stare at when you can't see a ceiling decreed that you must know what the kids will be doing during the 13th minute of your class period? What if you needed to repeat something or -- swoon -- got off on a really good tangent?

Besides, not everyone spends every waking moment solely focused on teaching and mathematics. Sometimes, I just read.

It does take a certain frame of mind to watch for these things ... an old teacher friend of mine used to clip articles ... this is similar.  I never could get the hang of it with newspaper, but I do find it easier on the Internet. Now that I have been using the thumbdrive method of file transport, I have amassed a large library of these types of things, neatly sorted into the classes and folders so I can find them later when I get around to it. (Yeah, right)

Having said that, this "Real-World" problem is no more or less engaging to students than the cellphone plan that charges by the month or by the 100 minute-block or the psuedocontextual dreck in the book.

[So I had a five-minute bit of Photoshop fun at Dan's expense. Don't think nothing of it.]

On a scale of 1 to 10, these are funny.

So, there's this magic dust called "Math Teacher Magic Scale Factor" ... it makes things really big. Sometimes too big.

Skating Puzzle

It is recorded that in a mile race between two graceful skaters the rivals started from opposite points to skate to the other's place of beginning. With the advantage of a strong wind Jennie performed the feat two and a half times as quick as Maude, and beat her by six minutes. The problem, which has created no end of discussion, is to tell the time of each in skating the mile.

How Old is Mary?

"You see," remarked GrandPa, "the combined ages of Mary and Ann are 44 years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when when Ann is three times as old as Mary was when Mary was three times as old as Ann.
How old is Mary?

~ Sam Loyd, Cyclopedia of Puzzles, 1914

Clock Puzzle

The picture of a clock dial illustrated the important point of evidence in a detective story where a stray bullet from the assassin's pistol struck the face of the clock.

It struck the exact center, driving the post through the works and stopping the clock. The two hands became united, as it were, in one line, pointing in opposite directions, although not in the direction shown, for it is evident that the hands could not point at three and nine at the same time.

Can you tell what time of day it must have been, thereby proving an alibi for the hero who wishes to show he was eating a plate of pig's knuckles in Hoboken at the time the pistol was fired in Sir Reginald's flat in Harlem?

- Sam Loyd, Cyclopedia of Puzzles, 1914


answer in this backdated post.

Letter to the Editor - WCYDWT?

Found this. What Can You do with It?

It costs more to use less
An Albany (NY) Times Union Letter to the Editor

It's bad enough that the town of Colonie, in tandem with the Latham Water District, has raised the cost per 1,000 gallons of water usage a number of times over the past few years. Now, the minimum charge has been raised so that anyone trying to conserve water will pay more than the base rate.

In the last six months, my water usage decreased by 5,000 gallons, but instead of paying for the water at the rate of $2.65 per 1,000 gallons, I've been assessed a minimum charge of $3.25 per 1,000 gallons because I failed to use as much water as the town thought I should use. Unless your usage is more than 20,000 gallons in a six-month period, you will be zapped with a higher rate.

Why would I want to be careful about water usage when it will cost me more if I use less?

This administration apparently is taking a page out of the book of the state Legislature by trying to make up its deficit on the backs of the people it will hurt the most, the seniors. I believe the seniors are most likely the group who will use the least amount of water, yet pay more for it. Water conservation helps everyone and it should not be used as a tool for the town of Colonie to make money.

Robert Leffingwell
Colonie

What can you do with this? Fish Fungus

Last year, one of the fish developed a fungal infection:

We don't care much about this particular one as it was a 12 cent feeder fish but the same pond has other fish.  The koi, for example, are quite valuable. If we did nothing, this one might infect the rest.  So we needed PIMAFIX, but how much?
More below.

Game-Changing Graphics

Those of you who know me know that I love data visualization done well. I am always looking for, and hopefully finding, graphics that clarify or that bring a new perspective to data. Tables of numbers are rarely helpful. An appropriate graph or visual, on the other hand, leads everyone to the classic indicator of epiphany ... "Hummm, now that's interesting ..."
 
These are some that I consider the game changers ...

Below the fold so the graphics don't kill one-time visitors:

Math in the Crosshairs again, this time Maryland

"... many graduates do not have a grasp of the basics."
"... schools have deemphasized drilling students."
"... taught too early to rely on calculators."

A calculator is a tool. It should be used as a tool. As soon as it replaces thought, it should itself be replaced.

I have decided to make a new slogan, signifying my reluctance to rely on the thrilling new technology of calculators because of the very real effects on the kids' development.

"Thrill and Kill."

"... ninety-eight percent had to pay for remedial classes." Okay, it's a community college and you expect that many of the attendees would be looking to improve their math skills. But 98% ??

"Across the nation, slightly more than one-third enroll in remedial classes." That's bad, people.

The report gets specific but, in my view, misses the mark. " ... particularly critical of the Algebra I standards, saying that they are watered down because educators must teach material for the High School Assessments, which includes data analysis. It is not what any mathematician would consider an algebra course."

No, I think the algebra I course is watered down because, (A) it is taught to eighth graders and they had to water it down so they could pass more easily and (B) mainstreaming and the refusal to place students in an appropriate class means that every room has kids who slow down the group. This insistence on placing kids in a course based on emotion, faulty pedagogy, self-delusion and parental desire instead of mathematical ability will ruin your classrooms every time.

Anyway, the article that started this train of thought appears below the fold.

A silly little ratio problem for you

I was wandering around the SmartBoard website and I came across this little bit of information:

Model  Active screen area
690  94" (238.8 cm), 16:9 aspect ratio
685  87" (221.0 cm), 16:10 aspect ratio

If that 94" is the diagonal of the active area, what are the dimensions of the whole board? Let's assume it's 3" from the active area to the actual edge of the equipment on the left, top and right and 6" at the bottom (to include the "chalk tray").

How about for the 87"?

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.