MathNotations

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...

Posted by Dave Marain at 6:04 AM 3 comments

Labels: indirect proof, math contest problems, MathNotations Contest, number theory, open-ended, proof

FINALLY -- THE RESULTS ARE IN!!

I apologize for the delay in getting these results out. The participating schools have all been notified.
NOTE: If any participating school did not receive an email from me, the advisor should email me. Also, if I misspelled anyone's name pls let me know and I'll correct it immediately!


INITIAL COMMENTS ON CONTEST, ETC...

• MEAN SCORE: 5.6 PTS OUT OF 12
• TOPICS INCLUDED Number Theory, Geometric Sequences, Function Notation, Geometry, Discrete Math, Quadratic Functions, and Absolute Value Inequalities (advanced level)
• Twenty schools registered from around the world, but only about half were able to actually give the contest.
  
• I will post the open-ended number theory problem later on but I didn't want to take away from recognizing the efforts of these outstanding students and their dedicated advisors.
• The next contest will be announced in a few weeks. Sign up early!
• After the 5th contest, you will be able to purchase all contests and solutions via download.
  

Posted by Dave Marain at 9:00 AM 0 comments

Labels: math contest, MathNotations Contest, online math contest

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!).

BASIC RULES
* 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.


Ok, here's another sample contest problem, this time a "counting" question that is equally appropriate for middle schoolers and high schoolers:

How many 4-digit positive integers have distinct digits and the property that the product of their thousands' and hundreds' digits equals the product of their tens' and units' digits?

Comments
The math background here may be middle school but the reading comprehension level and specific knowledge of math terminology is quite high. This more than counting strategies is often an impediment. If this were an SAT-type question, an example would be given of such a number to give access to students who cannot decipher the problem, thereby testing the math more than the verbal side. On most contests, however, anything is fair game!

Beyond understanding what the question is asking, I believe there are some worthwhile counting strategies and combinatorial thinking involved here. Enjoy it!

Click More to see the result I came up with (although you may find an error and want to correct it!)




My Unofficial Answer: 40
(Please feel free to challenge that in your comments!!_

Posted by Dave Marain at 6:29 AM 3 comments

Labels: combinatorial math, math contest problems, MathNotations Contest, middle school, more

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.
  

Posted by Dave Marain at 8:44 AM 0 comments

Labels: math contest, MathNotations Contest, open-ended, proof

UPDATE #2: Answers to the quiz are now provided at the bottom. If you disagree with any answers or would like clarification, don't hesitate to post a comment or send an email to dmarain "at gmail dot com".

UPDATE: No comments from my faithful readers yet -- I suspect they are giving students a chance to try these! I will post answers on Friday 9-25. However, students or any readers who would like to check their answers against mine need only email me at dmarain "at" gmail "dot" com and I will let them know how they did!


With the SAT/PSAT coming in a few weeks, I thought it would be helpful to your students to try a challenging "quiz". Most of these questions represent the high end level of difficulty and some are intentionally above the level of these tests. Then again, difficulty is very subjective. A student taking Honors Precalculus would have a very different perspective from the student starting Algebra 2!

Also, these questions can also be used to prepare for some math contests such as the THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST! Yes, another shameless plug, but time is running out for your registration...

A Few Reminders For Students

(1) Do not worry about the time these take although I would suggest about 30 minutes. The idea is to try these, then correct mistakes and/or learn methods/strategies. It's what you do after this quiz that will be of most benefit!

(2) I added strategies and comments after the quiz. I suggest trying as many as you can without looking at these. Then go back, read the comments and re-try some. I will not provide answers yet!

(3) Don't forget these problems are copyrighted and cannot be reproduced for commercial use. See the Creative Commons License in the sidebar. Thank you...


PRACTICE PSAT/SAT QUIZ
1. If n is an even positive integer, how many digits of 100²ⁿ - 100^2n-2 will be equal to 9 when the expression is expanded?

(A) 2 (B) 4 (C) 8 (E) 2n (E) 2n - 4


2. The sides of a triangle have lengths a, b and c. Let S represent (a+b+c)/2. Which of the following could be true?

I. S is less than c
II. S> c
III. S = c

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


3. The mean, median and mode of 3 numbers are x, x+1 and x+1 respectively. Which of the following represents the least of the 3 numbers?

(A) x (B) x - 1 (C) x - 2 (D) x-3 (E) 2x - 2


4. (10/√5)⁵⁰⁰ (1/(2√5))⁵⁰⁰ = _________


5. A point P(x,y) lies on the graph of the equation x²y² = 64. If x and y are both integers, how many such points are there?

(A) 4 (B) 8 (C) 16 (D) 32 (E 64


6. Each side of a parallelogram is increased by 50% while the shape is preserved. By what percent is the area of the parallelogram increased? __________


7.


AB is parallel to CD , AB = 3, CD = 5, AD = BC = 4. If segments AD and BC are extended to form a triangle ABE (not shown), what would be the length of AE?
Ans_________

Figure not drawn to scale



-----------------------------------------------------------------------------------------------
STRATEGIES/COMMENTS

1. Most students learn to substitute numbers for n here although it can be done algebraically by factoring. However, the real issue here is figuring out what the question is asking. Reading interpretation - ugh!!

2. When you are not given any information about what type of triangle it is, just choose a few special cases and draw a conclusion. O course, if one recalls a key inequality theorem from geometry, this problem can be done in short order.

3. If you don't feel comfortable setting this up algebraically (preferred method), PLUG IN A VALUE FOR x...

4. Your calculator may not be able to handle the exponent so skills are needed. The large exponent also suggests a Make it Simpler strategy. This is a "Grid-In" question so if one is guessing remember that most answers are simple whole numbers! Finally, if one knows their basic exponent rules and basic radical simplification, none of the above strategies are needed!

5. Possibilities should be listed carefully. It is possible to count these efficiently by recognizing the effect of reversals and signs. Easy to get this one wrong if not careful.

6. For those who do not remember or want to apply a key geometry concept about ratios in similar figures, there are a couple of essential test-taking strategies which all students should be aware of of:
(a) Consider a special case of a parallelogram
(b) choose particular values for the sides.
In the end, even strong students often make a different error, however. That darn ol' percent increase idea!

7. Should you skip this if you have no idea how to start? Absolutely not! Draw a complete diagram and even if you don't recognize the similar triangles, make an educated guess! It's a grid-in and there's no penalty for guessing. Further, answers tend to be positivc integers!!

-----------------------------------------------------------------------------------------------------
ANSWERS

1. B

2. B

3. C

4. 1

5. C

6. 125

7. 6

Posted by Dave Marain at 6:18 AM 1 comments

Labels: math contest problems, MathNotations Contest, more, PSAT, SAT strategies, SAT-type problems

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:
(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 will receive a registration form.

An aside...
I've been asking my kids questions every day to sharpen their minds for school which starts next week. I asked my son how he would spell, arachnophobia, the fear of spiders. He was confident he knew the first four letters: iraq....

Posted by Dave Marain at 12:38 PM 0 comments

Labels: MathNotations Contest, update

It took a couple of weeks but the results are in -- finally!

FIRST PLACE

PINK PANDA TEAM
Canadian Academy
Kobe, Japan
---------------------------------------------------------
SECOND PLACE (TIE)

WALLINGTON HS (SENIOR TEAM)
Wallington, NJ

THE BLACK SWAN TEAM
Canadian Academy
Kobe, Japan
---------------------------------------------------
THIRD PLACE (TIE)

WALLINGTON HS (JUNIOR TEAM)
Wallington, NJ

DECATUR AREA HOMESCHOOLERS
Decatur, Il
----------------------------------------------------------
FOURTH PLACE (TIE)

LAKE STEVENS HS TEAM I
LAKE STEVENS HS TEAM II
Lake Stevens, WA
----------------------------------------------------------
Comments

• Winning score was 9 pts out of a possible 14
• Median score was 4
• Participation was down for this contest but opening it up to more than one team from a school proved very successful. Some schools which had planned on participating found the timing at the end of May to be very problematic and had to drop out after registering. I'll remember that for next year!
  
• Hardest problems involved trigonometric derivations and a probability question requiring an infinite series.
  
• This contest definitely proved harder than the first and several questions were designed for the upper level secondary student.
• Future contests may be split into a 9th-10th grade version and an 11th-12th grade version similar to AMC-10 and -12.
• Students indicated the contest was challenging but expressed interest in participating again.
  
• The open-ended questions required considerable effort on students' parts and mine in grading them!
• The winning team sent a highly detailed and original solution to one of the trig questions. They wrote it by hand and scanned it. This technology works very well for this kind of contest.
• I will probably publish 1 or 2 of the questions with answers and partial solutions on this blog in the near future.
• Any schools interested in participating in the fall should send me an email now ("dmarain at gmail dot com") to get on my mailing list. I've already received several emails. At this time I plan on keeping the contests "free to a good home"!
• I am very excited about the international flavor of these contests -- this seems highly appropriate given the culture being established by our current administration. The world really is becoming "one out of many!"
  
• As mentioned previously, one of my goals is to publish 10-12 of these contests with detailed solutions as a book or pdf document which can be downloaded for a nominal fee.
  

Posted by Dave Marain at 6:34 PM 2 comments

Labels: contest, MathNotations Contest

First, some reminders and updates re the upcoming MathNotations (FREE) Online Math Contest:

• I am still accepting registration up to this Fri 5-15-09 and that may be extended. Just email me at "dmarain at gmail dot com" and I will email you the registration and information forms in short order!
  
• Because a few schools have expressed concern that some students are still taking AP's next week (makeups?) or their brains will be fried after this weeks AP's, I am willing to allow sponsors to administer the contest either the week of the 18th or the week of the 25th (after Memorial Day of course here in the US).
• Participating students should review their trig identities, infinite geometric series and probability. However, there are other questions or parts that do not involve these more advanced topics.
  
• Several questions are multi-part with later parts of increasing difficulty.
  
• At least one question requires a detailed explanation, i.e., showing one's method clearly.

• Have you been keeping up with Burt's insightful comments, clear explanations and advocacy for balancing concept and procedure in our classrooms, K-12? Read Burt's comments to this post...
• I've been remiss in keeping up with all the carnivals. I will get caught up in a few days.
• Been thinking about the AP issues I raised in a recent post (from the NYT article)? I will have more to say about this, particularly based on the reader comments to that article. Link to the Times article from my post and skim through the 60 or so reader comments. Fascinating stuff...
  

Posted by Dave Marain at 8:15 AM 9 comments

Labels: MathNotations Contest, update

Posted by Dave Marain at 3:38 PM 0 comments

Labels: MathNotations Contest, more

Updates...

• There are some 'new' math blogs (in some cases, new to me!) that I wish to bring to your attention. One of these was just started by one of my former students who is still in high school. She made excellent contributions to MathNotations last year. I'll post a link shortly. Look for creative thinking, challenging problems and an engaging writing style!
• Look for the Math Problem of the Day in the right sidebar. This is a 'gadget' made available by Blogger. Nice problems which are accessible to advanced middle schoolers and secondary students. They change every day so try them and check the solution link which is provided.
• There's a wonderful fraction calculator out there on the Web to which I will post a link and publish an article in a few days. If you haven't seen it, it will blow your mind! Imagine seeing the decimal expansion of any rational number to any desired number of places (within limits) instantly and that's not all it does!
  
• Yes, Pi Day is coming so I will post links to previous articles I've published and other excellent resources out there.
• Some of you know there's an excellent free resource of Singapore Math assessments for primary grades and more. You can download these and use them for your students. they make wonderful Problems of the Day and discussion points for your next department meeting! I'll post a link in a few days. Wait till you see the level of thinking and the content in the Grade 3 assessment, for example.
  

Posted by Dave Marain at 7:54 AM 5 comments

Labels: math contest problems, MathNotations Contest, update

"It was such a pleasure for me to see students so excited about math. Great job, Dave!"
Maureen Capuzzi, Montville HS

"I thought you might be interested to know that the moms appreciated your contest enough to ask for another one---for next week!"
Denise, Decatur Area Homeschoolers

"we all had a few 'a ha!' moments, which I think is the best part of math..."
ecv at fhs

And the envelopes please...

FIRST PLACE (11 out of 11 pts)
ARCADIA HS, Arcadia, CA
Advisor: Kerry King


SECOND PLACE
MONTVILLE Twp HS, Montville, NJ
Advisor: Maureen Capuzzi


THIRD PLACE (Tie)
FAYETTEVILLE MANLIUS HS, Manlius, NY
Advisor: Kate Nowak

FLORIDA MU ALPHA THETA TEAM, Tallahassee, Vero Beach, Fl
Advisors: Steve Friedlander, Brandi Williams

WYEDEAN HS, Chepstow, England
Advisor: Rhys Jeremiah


FOURTH PLACE (Tie)

DECATUR AREA HOMESCHOOLERS, Decatur, Il
Advisor: Denise Gaskins

WALLINGTON JR/SR HS, Wallington, NJ
Advisor: Stephanie Regetz


FIFTH PLACE (Tie)

FORT VANCOUVER HS, Vancouver, WA
Advisor: Nathan Shields

LAKE STEVENS HS, Lake Stevens, WA
Advisor: Kaleb Allinson

TAHOMA HS, Maple Valley, WA
Advisor: Dave Wright

Congratulations to our other participating teams, some of whom consisted of middle schoolers. This contest proved very ambitious for students below grade 11 but I hope it was a valuable learning experience for all. In the future, I plan to have separate contests for middle and secondary schools. THE NEXT FREE CONTEST IS SCHEDULED FOR APRIL OR MAY - STAY TUNED!

I also plan on posting some of the contest problems over the next few days. This has been an interesting experience for me and I learned much from the comments of the advisors and students. Next time I will limit the time to 30-45 minutes for ease of scheduling! If you think you might be interested in our next contest, you can always email me early and let me know. (dmarain at gmail dot com).

Posted by Dave Marain at 11:54 AM 2 comments

Labels: MathNotations Contest

Important Updates:

• Several schools have requested registration up to this point so the contest will probably run on Tue Feb 3rd as planned.
• All you need to do to sign up initially is to email me! I will email you the Reg. Form and Rules/Procedures within 24 hours. Complete the form (about 5 minutes) and email it back and you're officially registered! (dmarain "at geemaill dot com")
  
• A team of students should be able to complete most of the problems in 45 minutes or less. It is not necessary to keep students for the full 90 minutes! The extra time was provided for students to enter their answers/solutions electronically.
• Scanned student solutions will be accepted if format is followed.
• Return registration form ASAP even if you have not yet identified the 6 participants. The team can have fewer than 6 members (but at least 2).
• The contest questions are copyrighted, therefore I will probably not publish all of them on this blog although I will provide some samples of questions and student responses for discussion purposes on this blog.
  
• After the contest is over, participating schools will receive results, answers, suggested solutions and certificates via email. At that time, if anyone else is interested in receiving a copy of the questions, email me.
  
• If you like the idea of this kind of contest and would be interested in signing up for the next one (probably in March), let me know via email or comments.
• I will send a template for Certificates of Participation for your school and individual participants. Top-scoring schools will receive a Certificate of Merit.
  

• WINNING TEAMS WILL BE RECOGNIZED ON MATHNOTATIONS AND WILL RECEIVE SCHOOL AND INDIVIDUAL CERTIFICATES!

• INTERESTED SPONSORS SHOULD EMAIL IMMEDIATELY (see address below) TO RECEIVE REGISTRATION FORM AND RULES/PROCEDURES!

• DEADLINE FOR REGISTRATION: TUE JAN 27th

• CONTEST WILL BE EMAILED TO SPONSORS BY JAN 30TH

• SUITABLE GRADE LEVELS: 7-12 (Some questions can be handled by Middle School students)

• 90 MIN TIME LIMIT - FLEXIBLE RANGE OF TIMES FOR ADMINISTRATION!
  TEAMS WILL BE ABLE TO PERFORM WELL EVEN IF ONLY 45 MIN ARE AVAILABLE!
  

• CONTENT: Up to and including precalculus; emphasis on Algebra II
  

• CALCULATORS ALLOWED

• FEE: NONE!

• Team event - Up to 6 participants may work together!
  
• All answers/solutions must be submitted electronically
• Some multistep and open-ended questions
• FREE! (At least this first one is!)
• Separate acknowledgments on MathNotations given to Middle and High School teams
• All students will receive a Certificate of Participation and top-scoring schools and students will receive a Certificate of Merit via email.
  

Posted by Dave Marain at 11:00 AM 4 comments

Labels: contest, MathNotations Contest