51+52+53+...+100 is how much more than 1+2+3+...+50? Why, 50^2 of course! Now Explain and Generalize...

Quick Updates....
Mystery Mathematician Contest ending soon...
Pi fact for today? Try explaining why the imaginary number i raised to the power of i is REAL without mentioning π somewhere! Of course, you could just ask Google to do it for you!


You'd think that the deafening silence from the 5-7-8 triangle post would discourage me - NOT! Here is an investigation for middle schoolers and up.

Typical Content Standard: Patterns, Relations, Algebra

Objectives:
(1) Developing strategies for comparing sums
(2) Developing algebraic generalizations
(3) A few dozen more!

Where might the first question in the title of this post be asked?
(a) SATs?
(b) End of Course Test for Algebra 2?
(c) Other standardized tests?
(d) Math contests? If so, what grade level? 7th? 8th? Higher?

If you value a question such as this, would you introduce it to middle schoolers in 6th? 7th? Prealgebra? Would you use a very different instructional approach with students in higher math courses who have reasonable algebra background? Even if you don't like this question, try it with one of your groups tomorrow and let me know what happens!

Since I have personally posed this type of question to both middle schoolers and older students, I can tell you that even strong math students often have not seen the 'compare differences of corresponding terms method'. I made up that designation but, hopefully, you can make sense of it. Do you think many high school students would attempt to find two separate sums by some method/formula (or using their calculator if allowed) they've seen?

Well, I won't give any more away, but I believe the issues of pedagogy here may transcend the problem and the math strategies:

How does one introduce this? Do you simply have this question on the white board as students enter the room and allow them to work on it individually or in small groups for 5-10 minutes? We all hear about our re-defined role as 'guides on the side' but what exactly does this look like for this activity? How do we facilitate? When do we ask leading questions? What questions would be highly effective here? I haven't even mentioned the calculator issue yet!

So many questions. So few answers...

Actually I was going to do a short video presentation of this question to demonstrate one instructional model, but, unfortunately, my dog ate my main computer which has all my files and applications. Wait - let me apologize to my pooch. He really didn't eat it or even bless it with his bodily functions. But my iBook is very sick and will need intensive care from Apple. In the meantime, I'm on a backup machine, with limited memory and lacking many of my files and applications. Excuses, excuses, excuses! Please bear with me!

Taking the 'Unsummable' Numbers to a higher level: An Algebraic Proof

[You may also want to look at the preview of the interview with Alec Klein, author of A Class Apart, to be hosted on MathNotations. Alec has agreed to answer my questions about Stuyvesant HS in NYC, other specialized schools and gifted education.]


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Anyone recall a detailed investigation I posted in Feb '07 about numbers which can or cannot be written as a sum of 2 or more consecutive positive integers? You will probably want to quickly review that for this discussion. That investigation was implemented in a 9th grade prealgebra class with students who had struggled with math for a long time. They worked for the entire period (and into the next class as well) and expressed satisfaction and a sense of accomplishment. One student, KC, even found a way to express those numbers which were unsummable!

Today, we will take this question of which numbers are unsummable to a higher level. The challenge for my readers and for students is to use methods from Algebra 2 and basic number theory (primes, factors) to prove a conjecture made by one of my former students.

Quick background:
Students were asked to investigate those positive integers which can be written as a sum of 2 or more consecutive positive integers. I started them off with examples like
3 = 1+2; 5 = 2+3; 6 = 1+2+3, etc.
They worked in pairs and completed a table up to 36 over a couple of days. Most quickly realized that every odd positive integer starting with 3 could be represented but not every even positive integer. Working in pairs helped students to catch common arithmetic/logic errors and the results were reviewed after every 10 numbers or so in order to insure that all students had accurate results to work with.

Here's your challenge for today:
Rather than demonstrate which numbers can be represented as such a sum, your mission is to prove the following:
Powers of 2 are unsummable, i.e., a power of 2 can never be represented as a sum of 2 or more consecutive positive integers.
Notes:
(1) The whole notion of algebraic proof may be new for some students, so this may require some demonstration first.
(2) Students will need to know the formula for the sum of an arithmetic series, so this challenge would be appropriate after learning or reviewing that. The instructor however could develop that formula earlier on or simply provide the formula.
(3) Some understanding of the Fundamental Theorem of Arithmetic is needed here, i.e., every positive integer is either prime or can be written as a product of primes in a unique fashion. This theorem often goes unmentioned or taken for granted in middle school - time to bring it back?
(4) Some readers may find a way to prove this without using the algebraic formula mentioned above. Share that as well!

Enjoy...