Important Update: Day 10 - received a reply today from Jennifer Graban of the National Math Panel - will be posting her entire statement and my reply by Monday.
Update to Activity Below: This lesson was implemented today with my group of 9th graders. Math is generally a struggle for them. Do you think they completed it in 40 minutes? If not, then how much? Do you think this activity engaged them or they lost interest after a few minutes? Do you think anyone identified what kinds of numbers are 'unsummable'? Before I tell you, I'd be interested in your best guesses!
The following investigation enables the student to explore concepts in factoring, primes, composites, odd vs. even, consecutive integers, averages, median, pattern recognition, arithmetic series, generalization and proof just to name a few ideas!
It may appear to be written for middle schoolers but it can modified for grades 9-12 using algebraic methods (particularly sequences and series) and more sophisticated reasoning. Students may discover new ideas I never imagined when I wrote it. The basic idea of this question is very well known. What might make it different is the journey you and your students will be taking. Ok, if you're not an educator, enjoy the ride (even if it may be simplistic!).
Students should work in pairs. Calculator for checking sums is optional. Allow one period for this, however, additional time may be allocated for further investigation outside of class. This problem is about much more than making an organized list! How would you modify it to make it better? Richer? More suitable for younger children? Older children? What questions might you ask to guide them through it when they appear to be 'stuck'? Is it better not to say anything and let them struggle with it? Is there a place for this kind of discovery? Is it worth all the effort and time 'lost'?
You and your partner are trying to unlock the secret of the 'UNSUMMABLES'!
The number 5 can be expressed as a sum of two consecutive positive integers: 5 = 2+3
Similarly, 6 can be written as a sum of three consecutive positive integers: 6 = 1+2+3
22 can be written as a sum of four consecutive positive integers: 22 = 4+5+6+7
9 = 4+5 but it can also be written as 2+3+4
Ah, but no matter how hard you try, the number 8 cannot be written as the sum of 2 or more consecutive positive integers (try it!!). The number 8 is one of the mysterious unsummable numbers!
(a) In a table format, express each of the integers from 5 through 35 as a sum of 2 or more consecutive positive integers if possible. If it is not possible for some integer, call it unsummable! If you are able to find more than one way to sum a number, that's even better.
(b) Write at least 5 observations and conjectures, i.e., what did you notice and what do you think will always be true. We'll start you off:
We noticed that every odd number can be ________________________.
Note: Think about primes, composite numbers, factors, ...
(c) How many unsummable numbers did you find? What did you notice about these numbers? Can you unlock their secret? A special prize if you can explain WHY they are unsummable!
Thursday, February 8, 2007
A Math Exploration 2-9-07
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11 comments:
Wow, this looks like a great problem/exercise. I plan to have my middle school students work on this, though probably not until after February vacation.
After you and your students analyze this, you might want to talk about Gauss' Theorem on triangular numbers, the Four Squares Theorem, and Waring's Problem.
Theorem [Gauss]: Every positive integer is the sum of three or fewer triangular numbers.
Gauss wrote in his diary on 7/10/1796 this charming statement:
ΕΥΡΗΚΑ num = Δ + Δ + Δ.
Theorem [Lagrange]: Every positive integer is the sum of four or fewer squares.
Problem [Waring]: For any positive integer k, show that there is a number g(k) such that every positive integer is the sum of at most g(k) k-th powers. Show that there is an integer G(k) such that every sufficiently large positive integer is the sum of at most G(k) k-th powers. Determine these for various values of k.
We don't know what these numbers are for more than a few values of k. The difference between G(k) and g(k) is an important idea though. For example, g(3)=9, but we only know G(3)≤7.
Some of this I took from MathWorld, of course. But interested students should consult Hardy and Wright…
Hi Dave,
On (a), you still have to say positive integers, otherwise no number is unsummable.
This is actually related to a problem on Jonathan's web site. See
http://jd2718.wordpress.com/2007/01/30/puzzle-consecutive-integers-1-2/
My guesses (with no basis in reality, having never observed a ninth grade class):
(1) A few of them spent the whole time and did it for all the numbers, taking a bit less than 40 minutes, and identifying the unsummables
(2) A few of them lost interest after the first few numbers
(3) A few of them worked all the time but did not complete the exercise.
(4) All the above fews added up to the whole class
tc--
right on! thanks for the correction, although the error I made leads to a 'teachable moment!' Asking them to recognize that (-7)+(-6)+...+6+7+8 = 8 is not insignificant, however, it has a different purpose.
As far as your guesses, you definitely showed your experience in working with students! Actually, I started by reviewing primes and factors for 5 minutes. As you might have predicted, they needed help in getting started because of the term 'consecutive' - it's usually a key word that prevents them from getting into the problem. Experience helps in anticipating these difficulties but you have to try these to learn from experience! Once they got started, they ALL worked for the full 40 minutes, investing more into this than any activity I tried all year! Three did finish up to 35 and one young lady, K.C., recognized which numbers were unsummable and formulated a general description as "even and when divided by 2, they're still even!" Ok, not sophisticated, but darn good, huh!
I can't use that problem today, but I've printed it out and I'm saving it for the future.
How do you guys organize your problems for future reference?
Do you organize them by topic or grade in a binder or are they all in books you own and you simply remember where to look?
myrtle--
excellent question! when i come across great problems from people on the web (like jonathan at jd2718), I bookmark the site and move these bookmarks into a folder called Math Problems to make it easier to return to that problem and that site for more problems. I also take a screenshot of the problem and store it as .png file or .pdf and keep these in a file on my desktop. I use these electronic methods becaause I'm pathologically disorganized when it comes to pieces of paper and file folders/binders! I have SAT and other math questions i have written for 35 years scattered thorughout my house in various folders. They could probably amout to 5000 problems at least 20% of which are publishable in my opinion and I'm not sure where they all are - how sad! Whenever I tell my wife I'm going to compile these one day, she laughs at me because she knows me so well!! Oh, well, this blog is certainly forcing me to be more organized.
As far as finding great problems in books, there are many outstanding problem books for MathCounts, SATs, other Math Contests, NCTM calendar problems for the past 50 years, etc. You can Google to your heart's content and find an endless supply of varying quality. Here's the issue however. To be useful for a particular age or grade level or aiblity group or course, these questions often need to be modified or expanded or enriched. That's why I'm also including these 'activities' or explorations -- to show what is possible when we take the time to construct meaningful experiences that go beyond an exercise in a text. Is this some extreme constructivist point of view or have I been doing this for 35 years? I'll let you decide...
Dave wrote:
Asking them to recognize that (-7)+(-6)+...+6+7+8 = 8 is not insignificant, however, it has a different purpose.
There is a cute related contest-type problem that comes up from time to time:
The number 8 can be expressed as the sum of 16 consecutive integers. What is their product?
As to your needing to review the meaning of "consecutive", that is one I have to review frequently with my upper elementary and middle school kids. It is amazing how much of "contest" math comes down to knowing the vocabulary. And paying careful attention -- when it says "product", don't try to use "sum", etc....
Dave, you might also want to introduce your kids to Cross Sums / Kakuro. Playing with this problem myself reminded me a lot of the thinking behind doing the cross sums because you have to make a target sum out of a certain number of distinct digits. There are lots of free puzzles available on the web.
Looking forward to seeing the details of your exchange with the Math panel.
Hey, Dave, if I use one of your investigations more or less verbatim, how would you like to be credited? I'd rather not list your URL, since I don't really want them to go looking it up. :) How about "Investigation courtesy of Dave Marian"? Or is there something else you prefer?
Mathmom--
Pls note the 'Proper Attribution' statement that now appears in the sidebar. Thank you for your consideration and ethics.
Has that always been there, or did you just add it? In any case, thank you for the clarification.
Hi Dave,
I recently posted a follow-up detailing many of the conjectures my middle schoolers came up with on this. I hope you get a chance to pop over and let me know what you think!
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