Sunday, March 15, 2009

Those "Function" Questions on the SATs - Practice, Tips


The following is not a classic function question even though it uses function notation. This is an original problem I wrote but it is the kind of question that might appear. The level of difficulty would be medium. The math content is middle school level but the wording and notation are the challenge for most students. Beyond preparing students for a test like the SATs, my strong belief is that such questions should be included in textbooks from middle school on (even with that function notation!). This question reviews basic math concepts (primes, factors, gcf) and can also be used as a springboard for discussion of the concept of "relatively prime", Euler's phi function, π(x) and other number-theoretic topics.
Note: The "For example" hint may or may not be included in the question. It certainly makes the notational issue less formidable.

If n is a positive integer greater than 1, then the sets F(n) and P(n) consist only of positive integers and are defined as follows:

A positive integer, k, belongs to the set F(n) if k ≤ n and the greatest common factor of k and n equals 1.

A positive integer, k, belongs to the set P(n) if k ≤ n and k is prime.
For example, F(6) contains the numbers 1 and 5 and therefore has two elements. P(6) contains the numbers 2, 3 and 5 and therefore has three elements.

What is the ratio of the number of elements in F(20) to the number of elements in P(20)?

Click Read more below to see answer (suggested solution will be posted later).

Answer: 1
Explanation: Not yet...


Anonymous said...

I appreciate the attention to the difficulty of the mathematics and the difficulty of the wording.

If I were to share this problem, I predict that I would end up explaining how to determine what F and P were for 20, and I would leave the kiddos with just the middle school math.

So, let me think about this: Maybe I should be more regularly offering challenging problems, not these, but on the order of H(n) = {x| 2n≤x≤3n, gcf(x,n) ≠ 1} and ask for, for instance a translation into words, and then H(20).

Practice reading what's hard to read, and then when they run into a whammy (and that's how I'd describe this problem) they might concentrate on the math.

The mean thing, though, is I'm proposing, in the meantime, stripping away anything easy... which means I risk frustration... might need to slowly build towards H.

I start already with ℚ = {x|x = p/q, p,q ∈ ℤ, q ≠ 0}- but I am not consistent about practicing the reading...

Hey, you made me think! Thanks.


(I avoided less than and greater than because they were getting read as HTML containers... just wanted to explain that bit of weirdness in advance)

Anonymous said...

mathmom via kindle: Hi, I really like this! I have been playing wity looking for other n where F(n) and P(n) have the same cardinality - no obvious pattern but a worthwhile exercise nonetheless. I hope to try some version of this with my middle schoolers. Thanks!

Dave Marain said...

mathmom and jonathan--
I'm glad you like this.

Jonathan, I avoided "set-builder" notation issues, but defining a sequence of sets using a function whose domain is positive integers greater than 1 and whose range consists of sets of integers was intended to help students adjust to different "mathnotations!"

When I was in the classroom I attempted to demystify function notation by make functions into very concrete objects: "When you see f(x) immediately make a T-table for x and y, substitute a few simple whole numbers for x and evaluate."

In this problem, I would have students replace n by some value like 6 and have them orally state the definitions of F(6) and P(6) as I replaced n by 6 on the board. Even with examples I gave in the problem, many would still struggle with the symbolism and 'math speak.'

BTW, I used the term "elements" of a set. On the SATs, it is usually expressed as "members" of a set.

Your thinking is so like mine it's as if we read each other's minds! I could not begin to describe all those values of n for which the sets have the same number of members but I thought it was interesting to consider a few special values and make students wonder about this 'curiosity'!

F(x) in thhis problem is really related to Euler's phi function. It is "multiplicative" and therefore can be evaluated readily from the prime factors of n. Thus phi(6) = phi(2) times phi(3) = (1)(2) = 2.

P(x) in this problem was, of course, inspired by π(x), the prime number function, and is still at the forefront of research. Every few years we read about another way of estimating or expressing the values of this function using the Riemann zeta function or logs or some new discovery.

Anonymous said...

This is a great problem. I can see how the math content is middle school but a bit challenging. Although, I see it as a great review problem for high school students.

Dave Marain said...

Thanks, Anon!

Now all I need to do is come up with a "book" of these or at least a few practice worksheets. Anyone care to volunteer to help me!!

The truth is our students need these kinds of 'different' experiences. I enjoy writing them but it isn't easy for me and I've been doing this for decades! I'll bet if we ask some of our students to create a few of these kinds of items they will devise some wicked hard ones!

Functions are a hot topic on standardized tests these days. Combining this topic with fundamental mathematics is challenging but worthwhile.

Hilary said...

I agree with one of the earlier posts. This would be quite challengeing for students in my middle school. But, I love the idea of a contest. Can anyone elaborate on how they might be able to create a grade wide math contest using similar problems (or modified content)?