The following examples also provide practice for open-ended questions and a view of the Explain or Show type questions on our next Online Math Contest to be held in 5 weeks (see info below). Since formal proof is not the goal here, students are encouraged to write a logical chain of reasoning in which they can use/assume basic knowledge about odd and even integers. Further, these questions strongly suggest the strategies consider a simpler case first and patterning.
Another benefit of these types of questions is to review important terminology and to help students improve reading comprehension, a major obstacle for many youngsters in math class (and everywhere else!). Some middle schoolers and high schoolers will have difficulty making sense out of what the question is asking because of both the wording and the information load in the problem. We need to help them group key phrases together and, yes, I guess that means we are also reading teachers!
Example 1
Is the sum of the squares of the first 2009 positive integer multiples of three odd or even? Explain your reasoning.
Example 2
Is the sum of the squares of the first 2010 positive integer multiples of three odd or even? Explain your reasoning.
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REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!
Updates (Pls Read!!)
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.
Monday, September 7, 2009
Using Number Theory To Promote Logic and Writing in Middle Schoool and Beyond
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6 comments:
Our co-op classes start back up this week, and monthly math clubs next week. This looks like a great problem to get my students' rusty brains moving and to review some basic vocabulary. Thanks!
Oooh, I love these problems! I have to figure out a way to organize myself this year so that I remember to use them after I get this new group warmed up a bit. (It's a very young group this year, so I don't think they're ready for it yet.)
The problem is a trifle contrived, but that's okay. So is each annual Putnam problem of the year. (The William Lowell Putnam Competition always has one problem where the current year is involved.) There are many amusing ways to simplify the problem:
1. The 'multiple of three' is a red herring; all that means is that each summand is divisible by 3² = 9, and an odd factor won't change anything.
2. The 'squares' is also a red herring; an integer is odd if and only if its square is odd. That's how we show √2 is irrational, right? So, that reduces things to the sum of the integers from 1 to 2009.
3. We could calculate this sum as 2009 ⋅ 2010/2, and we don't even need to multiply that out, or
3′. We could realize that exactly 1005 of the terms are odd
4. The number in the second part differs from the number in the first part by an even number. There's no need to repeat the analysis.
Eric,
Thank you as always for the clarity of that explanation. I agree the problem is contrived. In this era of seeking real-world contexts for each problem we give our students, we sometimes forget that mathematics is essentially "a way of thinking." My online contests always include this type of reasoning and we know middle schoolers need training early on with simple one-, two- and three-step chains of reasoning. Since this part of their training is often underemphasized, we see high schoolers and undergraduates struggle also. Geometry helps to build some of this foundation but the difficulty some have with proof suggests they are missing a foundation in logical reasoning from early on. Middle school teachers can start with one-step logic problems and show youngsters how to write a mathematical argument like "This follows from ____ because we know that all ______'s are _____." Nothing wrong with with teaching the form of a syllogism! We need to provide many written models for deductive arguments before some students can write one of their own. This also leads to a discussion of formal 2-column proofs vs. paragraph proofs but I'll leave that for another day!
You also know how many students can't even get started because of their difficulty in parsing out the language of this problem or not being confident of the meanings of key terms. That was also one of my goals.
Finally, the ability to "see" all the "red herrings" is based on knowledge and understanding of fundamental properties of integers. Maybe, I should end all my problems with the direction: "GO FISH!"
Denise and Mathmom--
Thanks for the positive reaction and let me know how your students react if you try it. I might start with a slightly simpler version of this however!
I tried it with my new middle school group, but too much of the vocabulary was new for them. We spent way too long defining terms, which meant we didn't get to the "fun" stuff of eliminating red herrings and solving the problem. In the future, I will need to look for easier puzzles for this group.
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