THE OPEN-ENDED CONTEST PROBLEM AND SOLUTIONS

As promised, here is the open-ended, rubric-based, holistically scored, performance-assessed, student-constructed first problem from MathNotation's Third Contest:

1. A primitive Pythagorean triple is defined as an ordered triple of positive integers (a,b,c) in which a² + b² = c² and the greatest common factor (divisor) of a, b and c is 1. If (a,b,c) form such a triple, explain why c cannot be an even integer.

Comments
(a) The content here is number theory. Is some of this covered in your district's middle school curriculum or beyond? More importantly, at what point do students begin to formulate and write valid mathematical arguments?

(b) The immediate reaction of most students was that this seemed like a fairly simple problem. However, only a couple of teams scored any points. Perhaps the challenge here was the construction of a deductive argument, although as you will see below, there is one challenging part.

(c) There were two successful approaches used by the teams. Both involved indirect reasoning. Do your students begin to do these in middle school or are "proofs" first introduced in geometry?

(d) I allowed students to assume without proof the following:

(i) The general rules of parity of the sum of two integers
(ii) The square of a positive integer has the same parity as the integer

(e) Interestingly, none of the teams considered an algebraic approach to the one challenging case, i.e., demonstrating that the sum of the squares of two odd integers is not divisible by 4.

If a and b are odd, they can be represented as
a = 2m+1 and b = 2n+1, where m and n are integers.
Then a² + b² = (2m+1)² + (2n+1)² =
(4m² + 4m + 1) + (4n² + 4n + 1) =
4(m² + n²) + 4(m + n) + 2, which leaves a remainder of 2 when divided by 4.
BUT, if c is even, say c = 2k, then c² = 4k², which is divisible by 4.

(f) The two best solutions came from our first and second place teams, Chiles HS in FL and Hanover Park Middle School in CA. Both used the ideas of congruence modulo 4.

Here is the indirect method used by Chiles:

Let's assume that c can be an even integer. We'll prove by contradiction. An even integer can be summed in two ways:
1. with two even integers or
2. two odd integers
If it is the latter case, then looking at the residuals of modulo 4, the two odd integers summed will be equal to 2, but this is not the case as 2 is not a modulo of 4 residue. If it is the former case, then it does not satisfy the problem as then a, b, and c have common factor of 2. Therefore c must be an odd integer. Q.E.D.


Here is the indirect method used by Hanover Park:

Suppose, for the sake of contradiction, that there is a PPT (primitive Pythagorean Triple) s.t. c is even. Then c² ≡ 0 (mod 4).

We break this into cases based on the parity of a,b.

Case I: Both a and b are even; gcd(a,b,c) ≥ 2 because a,b,c are even, a contradiction.

Case 2: One of a and b is even. Then, a² + b² ≡ 0 + 1 ≡ 1
not ≡ 0 (mod 4), a contradiction.

Case 3: Both of a, b are odd. Then a² + b² ≡ 1 + 1 ≡ 2
not ≡ 0 (mod 4), a contradiction.
We have covered all cases for a, b with no valid cases. Thus, in a PPT, c cannot be even.

Both of these arguments represent a more sophisticated understanding of mathematics and the methods of proof. Clearly, these students are quite advanced and exceptional, however, I feel many middle school teachers begin early on to encourage their students to explain their thought processes both orally and in writing. Am I right? I would like to hear your thoughts on this...

MathNotations Third Online Free Math Contest Update and Sample "Proof"

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

• Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
  
• Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
  
• The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
  
• Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
  
• Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
• Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
  
• Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.
  

The following is a sample of the open-ended "proof-type" questions on the test:

Explain why each of the following statements is true. Justify your reasoning carefully using algebra as needed.

The square of an odd integer leaves a remainder of 1 when divided by
(a) 2
(b) 4
(c) 8

I may post a sample solution to this or you can include this in your comments to this post.

Using Number Theory To Promote Logic and Writing in Middle Schoool and Beyond

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.

In-Depth Investigation of Patterns: Lattice Points, Algebra,...

Update!

Answers to General Formulas for |x| + |y| = N:
ON: 4N
INSIDE or ON: N² + (N+1)² = 2N(N+1) + 1 = 1+4+8+...+4N

OUTSIDE the 'diamond' but INSIDE or ON surrounding square:
2N(N+1) = 4+8+...+4N

TOTAL: (2N+1)²
[Pls indicate any discrepancies you find!]


The following starts with a fairly simple pattern but watch out -- students from Middle School through Algebra 2 and beyond can take this as far as their imagination and skill can carry them. The questions below review absolute values, inequalities, graphs, geometry, etc., but that is the tip of the iceberg. Prealgebra students should start with simpler values to get the idea. Have them make a table of their findings as suggested at the bottom.

One could begin with a variation on an SAT-I, SAT-II type or math contest question:

How many ordered pairs (x,y), where x and y are both integers,
satisfy |x| +|y| ≤ 6?

Note: If this were to appear on the 'new' SAT, the inequality would be replaced by an equality. or the '6' would be replaced by a smaller number.

Here's a restatement using the terminology of lattice points:

In the coordinate plane, a point P(x,y) is said to be lattice point if both coordinates are integers.
How many lattice points are inside or on the graph of |x| + |y| = 6?

Answer: 85
Sorry for giving it away, but the objective is to discover ways to derive this result, not the result itself.

When I see questions like this, my inclination (as a mathematician) is to generalize, that is, try to understand a general relationship if the '6' were replaced by N.
Here is the more advanced generalization (one could go further!):

We are investigating the number of lattice points inside or on the graph, G, of
|x| + |y| = N, where N is a positive integer. We will also consider the smallest square, S, whose sides are parallel to the coordinate axes in which the graph of
|x| + |y| = N is inscribed. [This really needs to be drawn but I'll leave it to the reader's imagination. The top side of this square is part of the line y = N.]

(a) Derive a formula for the number of lattice points inside or on G.
(b) Derive a formula for the number of lattice points inside or on S
(c) Derive a formula for the number of lattice points outside G but inside or on S.


Notes, Comments:
For the younger students, or for any group that isn't quite ready for the above, I strongly recommend building the pattern one step at a time and to record their results in a table:
|x| + |y| = 0 (o is not a positive integer but it makes sense to start here) to |x| + |y| = 1 to
|x| + |y| = 2,...
Also, there are alternate formulations (or a variety of patterns) for the general number of lattice points. Don't be surprised if your students find them! Of course, they should be encouraged to show they are all equivalent. Questions about the geometry of the figures could also be raised (if time permits). For example, some students do not fully appreciate that the diamond-shaped graph of |x| +|y| = N is in fact a square, Now, what would its area and perimeter be...
This investigation can be started in class then assigned to be finished outside of class, preferably with a partner. I plan on starting this with my 9th graders on Mon or Tue but I will need to review absolute values, graphs, etc., and I will develop it incrementally. I will be more than happy if, in one 40 minute period, they can complete a table showing the correct number of lattice points for N = 0,1,2,3!

[Additional but critical point: I am not suggesting that these kinds of extensive explorations should ever replace the need to deliver content and develop skills. These are intended to be enrichment activities, no more, no less!]