Saturday, July 18, 2015

Median = Geometric Mean? A Common Core Investigation

As tweeted on

J noticed that for an arithmetic sequence like 3,7,11,15,19 the median equals the arithmetic mean. In this case, the median and "mean" are both 11. She found this was well-known and not too difficult to prove.

She wondered if there was an analogous rule for geometric sequences like 2,4,8,16,32. Instead of the arithmetic mean she tried the geometric mean:
(2•4•8•16•32)^(1/5) which equals 8, the median. VERIFY THIS WITHOUT A CALCULATOR!

Unfortunately her conjecture failed for a geometric sequence with an even number of terms like 2,4,8,16 in which the median equals 6 while the GM = 4√2.

(a) Test her conjectures with at least 4 other finite geometric sequences, some with an odd number of terms, some with an even #.

(b) PROVE her conjecture for an odd number of terms.
Hint: If n is odd then a,ar,ar²,...,ar^(n-1) would have an odd # of terms. Why?

(c) How would the definition of median have to be modified for an even # of terms?

How much arithmetic/algebraic background is needed here?

Arithmetic sequences more than enough for middle schoolers to explore? Geometric too ambitious?

PROOF too sophisticated for middle schoolers? How would you adapt it? We are trying to raise the bar, right?


mathwater said...

I think it could be a nice investigation, but I would make sure that there is ample practice beforehand for messing with whole number factorization. (I left a couple of related comments here:

To really get the idea of this proof, I would think students should be familiar with factor pairs. In my experience: one rich, cognitively demanding task is to have students explore the sorts of whole numbers that have an odd number of factors. (Both symbolically and diagrammatically, e.g., drawing rectangular arrays for each pair of factors.)

For example, observe that 16 has an odd number of factors: 1, 2, 4, 8, 16; a total of 5 (an odd number).

1, 4, and 9 also have an odd number of factors...*

Besides the foundational elements involved in factoring (how do these re-arise, e.g., when factoring quadratic expressions?) and an increased understanding of multiplicative reasoning (an important transition that many students struggle with) the factor pairs of (1, 16); (2, 8); and (4, 4) are reminiscent of the pairing up involved in tackling the "verify without a calculator" example in your post (as well as the general proof for an odd number of terms).

You might even connect the two tasks: Suppose the whole number n has an odd number of distinct factors, and write down this list. What is the geometric mean? What is the arithmetic mean? What is the median? Which of these can you write a formula for? Which of these can be the same? When/why? (If never: Why not?)

*Given a whole number's prime factorization, what more can we say about how many distinct factors it has? E.g., if p and q are prime, how many factors does a number of the form p^2 x q^3 have? (Why should this answer be the same for any primes p and q?) Etc.

Dave Marain said...

Deeply appreciate your reasoned and detailed comment. Your suggested Investigations make far more sense for younger students. Quite a bit of multiplicative machinery needs to be developed for geometric sequences but there is a strong analogy to arithmetic sequences. Just as 3,7,11,15,19 can be expressed symmetrically in terms of its arithmetic mean: 11-8,11-4, 11,11+4,11+8 whose sum is 5•11 we can express 2,4,8,26,32 as 8/4,8/2,8,8•2,8•4 whose product is 8^5. This in itself is worth discovering even without the 5th root.

I wasn't suggesting this Investigation could be implemented as is for younger students. But I wanted to present a less well-known application which teachers could adapt to the group. I believe some students would use their calculator to discover that the product is 8•8•8•8•8 where 8 is the median. More background enables them to go further but less machinery does not prevent some level of discovery and the satisfaction that comes with it. Our role is to set the objectives and guide the exploration. When we provide the time and opportunity for children to explore, we can't always anticipate where they will go. For every student who's disinterested and will not invest there will be one who will take the journey. We create the culture of inquiry...