Thursday, August 20, 2009

Challenge Their Minds Day 1 - A 'Means to an End'

With the school year starting for some and soon for others, here are a couple of ideas to set the tone in our math classes early on. Do not assume these are intended only for your advanced youngsters!


Middle School


1) (No calculator!) What is the average of ninety-nine 1's and one 2?

2) (No calculator!) Find 5 different sets of 5 numbers each of which has a mean of 5.

Note: The wording will be problematic here since students often associate the adjective different with the numbers themselves. Basic grammar, cough, cough...


High School (or advanced middle schoolers)


(No calculator!)
Set S consists of 100 different numbers each of which is between 0 and 1.
Which of the following could be the mean of these 100 numbers?

I. 0.01
II. 0.5
III. 0.98


(A) I only (B) II only (C) I and II (D) I and III (E) I, II, and III

[Yes, there will always be some discussion of "between!"]

A few comments...
(1) These problems are intended to be a springboard for your own creativity. You can do better!!

(2) Each of you probably has your own favorite resources of problems so that you don't have to reinvent the wheel. However, finding high-quality Problems of the Day which are matched to your curriculum is not always easy despite the abundant ancillaries supplied by the publisher and resources on the web.

(3) From the previous comment you can guess that I feel strongly about giving more challenging warm-ups to our students - all of our students (adjusted for backgrounds, abilities, skills). Don't worry that discussion of these will destroy your lesson. Students can work together for 5 minutes while you're taking attendance, checking homework, etc. I usually invited students who solved some or all of these to go to the board and explain their methods. To encourage students to look these over, tell them you will include a variation of one of these questions on the next quiz or test. Start by having it as an Extra Credit problem, then worth a couple of points, gradually increasing their value.

(4) Imagine if our students were exposed to these higher-order types of questions about 180 times a year from middle school on. By the time they take their college-entrance exams or other state assessments (or tests like the ADP End of Course Exams), they will have a much higher degree of comfort and should perform better, although we know that there are so many other factors that go into performance on high-stakes tests.

(5) Yes, the above high school problem is in SAT format. Why do you think I included these kinds on my daily warm-ups? By the way, I'm not promoting ETS but middle and high school teachers may well want to invest in (or ask their supervisor to order) the College Board's book of
10 Real SATs. There is no better source for these kinds of problems and many questions are appropriate for middle schoolers.


6 comments:

watchmath said...

This is a good problem. I taught about this when I was in middle school (I think). So basically start with a guess average and inspect for each data whether it is off or exceed the average, total this and the average will be the guess average + (total off/number of data).

So for
1,1,1,...,1,2
Guess average is 1 only 2 off from this number by +1. So the average is 1+(1/100).
All the other problems can be solved in this manner.

Totally_clueless said...

Hi Dave,

For the high school problem, did you intend that the numbers be between 0 & 1 (i.e. fractions), or either 0 or 1 (two-valued). The answer doesn't change either way, but how students may approach the problem would change.

Or maybe, it begs a two-parter with the distinction above.

TC

Dave Marain said...

tc--
I intended it to mean all real values between 0 and 1.

watchmath--
I really like the guesstimate/differences method. It's valuable both in statistics and for deepening understanding of averages for younger children. As you noted it can be used to compute the actual mean, to construct a data set with a desired mean, or to verify that a given number is the mean (differences sum to zero). The differences must be defined carefully of course as value - mean.

I do think the high school problem leads to higher conceptualization about the density of rational or real numbers. I observed students solving this and the variety of approaches was so gratifying to me. The ensuing brief discussion was rich. I'll let you conjecture about what they came up with.

watchmath said...

Doesn't allowing the data to be reals in (0,1) make the problem trivial? Because now any x in (0,1) can always be an average (just take one hundred of x's).

Dave Marain said...

Watchmath,
The problem indicated 100 different numbers!
This slowed many down but the more conceptual
students intuited that it could be done.

rmssanderson said...

this is a good thing to do at the beginning of the year to see what kind of students you are dealing with.