## Thursday, March 12, 2009

### Pi Day, SAT Day - Last Minute Reminders and a Problem to Try...

It's fitting that many juniors will be taking their SAT on 3-14! The following are some thoughts for students to take with them on Saturday morning.

Each question on the Math section is worth about 11 points. Six to seven careless errors (which is common) can cost you up to 90 points! How many students would like to increase their score 90 points without that high-powered SAT Course? It is within you! So what can you do to minimize the damage?

• HIGHLIGHT (circle, underline) KEY WORDS AS YOU READ THE QUESTION.
• LOOK FOR WORDS/PHRASES LIKE EVEN, POSITIVE INTEGERS VS. INTEGERS, NOT, LEAST POSSIBLE, ...
• SLOW DOWN! WRITE OUT DETAILS - DO NOT SKIP STEPS; DON'T WORRY ABOUT FINISHING EVERY QUESTION. IT'S THE 6-7 CARELESS ERRORS THAT HURT MOST!
• IF YOU HAVE ANY DOUBT ABOUT DOING THE PROBLEM ALGEBRAICALLY, PLUG IN SIMPLE WHOLE NUMBERS LIKE 1, 2 OR 3. See Example below.
• REMEMBER:
INTEGERS: ...,-3,-2,-1,0,1,2,3,...
EVEN INTEGERS: ...,-4,-2,0,2,4,..
PRIMES: 2,3,5,7,11,... NOTE THAT ONE IS NOT PRIME!!
• ZERO IS THE MOST IMPORTANT NUMBER ON THE SATS AND MATH IN GENERAL! LEARN THE TRUTH ABOUT ZERO:
WHOLE, EVEN, INTEGER, NOT POSITIVE, NOT NEGATIVE, 0/5 = 0, 5/0 IS UNDEFINED!
Example 1: If m > 0, the average (arithmetic mean) of 2⋅3m and 2⋅3m+2 can be written as c⋅3m+1. What is the value of c?

Possible solution: Unless you're a math team whiz, you should immediately substitute a simple whole number for m and not worry about c. Also, ignore the phrase "arithmetic mean" which is math terminology for the common average.

"Plug in" m = 1: We want the average of 2⋅31 and 2⋅31+2. This translates to the average of 6 and 54 which equals 30. If you're prone to any careless arithmetic errors (order of operations, exponent issues), do this on the calculator even though your math teacher would cringe!
We want to express 30 as c⋅31+1 or 9c. Thus 9c = 30 or c = 30/9 = 10/3 = 3.33 if you're gridding in.

• The above example would normally be among the last 5 questions of a section, therefore, considered to be more difficult. Students should not give up too quickly on these. They often can be solved by straightforward methods like the one described above.
• For the mathematical purists out there who are offended by "plug-in" methods (btw, I'm one of those purists!), this post was about providing test-taking strategies or 'survival' techniques. For the classroom development of algebraic skills, I would certainly have demonstrated an algebraic method:
2⋅3m+2 = 2⋅32⋅3m = 18(3m). The average of 2(3m) and 18(3m) = 10(3m) = 10(3-1)(3m+1) =
(10/3)(3m+1) ,etc. Wonderful review of exponents and operations but not necessarily for everyone taking this test...

Eric Jablow said...

But plugin techniques are often quite useful, time-saving, and elegant. For example, suppose you are given a set of points in the plane, and suppose you are told there all fall on the graph of a quadratic. Then, write the equation y=ax^2 + bx + c, and plug in three of the points. You can then solve for a, b, and c, and that is all you must do. Partial fraction problems are most efficiently solved that way:

x/[(x-2)(x-4) (x-7)] = A/(x-2) + B/(x-4) + C/(x-7)

is most easily solved by multiplying by (x-2)(x-4)(x-7) and then substituting x=2, x=4, and x=7. There are times when the easy thing is the right thing.

Dave Marain said...

Excellent points Eric. There are many other sound examples of substitution techniques (e.g., method of undetermined coefficients, looking for a particular solution to a differential equation, etc.).

The 'cringing' is a reaction I've seen and heard from some (not all) math teachers who see 'plugging in' as simply a way of 'beating the test,' i.e., it's not 'real math.' That's why I believe this approach is sometimes demeaned.

The assumption is that if students really 'knew their stuff' they wouldn't have to resort to this method. However, I've observed some very capable students who are taking advanced math honors classes who prefer to use this technique although they often will say that it depends on the problem. That is, when they recognize that the algebraic procedure is the most efficient they will choose that first. One student in particular has refined the 'plug in' method into an art form. He is highly successful at it and it would be absurd for me to discourage him or be critical.

On the other hand, we as educators can use this technique to drive home a very important point about unique solutions and well-defined problems. If a problem states that some relationship is true for all real values of x, then of course it must be true for a particular value! Also, if there is only one answer to a problem, then the result is independent of the particular values or the particular rectangle or triangle so why not choose "special values' or 'special geometric cases.' There is serious mathematics underlying all of this so I guess we can stop cringing now!

In the end, Eric, I was trying to give some last-minute advice to high school juniors to help them relax and get through this rite of passage that has blighted the adolescence of millions.

Eric Jablow said...

There's an old story about von Neumann; he was given the following old chestnut at a party:

There are two towns, A and B, 120 miles apart. At noon, a train leaves town A heading for B at a speed of 30 mph, and at the same time a train leaves town B for A at the same speed of 30 mph. Also at noon, a pigeon on the engine of the first train heads for the second train at a speed of 60 mph; when it reaches the second train, it immediately turns around and heads for the first, etc.

When the two trains pass each other, how far through the air has the pigeon flown?

One way to do this is to ask how long it will take for the pigeon to reach the second train the first time. Relative velocities tells you that the answer is 80 minutes (in which it traveled 80 miles), at which time the trains have travelled 40 miles, leaving a 40 mile gap between them. The trip back takes 80/3 minutes and miles, since the problem now is 1/3 of the original problem's size, and then the trip back again takes 80/9 minutes and miles, leading to a geometric series adding to 120 miles.

Of course, there's an obvious trick. The trains meet in 2 hours, and the bird is traveling at 60 mph all that time, so it must have traveled 120 miles.

As the story went, von Neumann answered the question immediately. The questioner replied, "You must know the trick." "What trick?"

Now, what von Neumann did in the story is hardly justifiable; students should be taught to think before doing unnecessary work. A teacher who teaches his students to work inefficiently with no pedagogical point does his students a disservice, and pedagogy certainly does not apply on an exam.
I may avoid using l'HÃ´pital's rule as much as possible in class, but I would never require my students to forsake it on an exam.

However, this assumes that the student will choose a valid shortcut. We have all mourned the students who panic and replace the square root of a sum with the sum of the square roots. Teachers are often rigid about techniques because they fear panic-induced errors. The only cure is to teach judgment along with theory. Too many teachers are incapable of that, I fear.