A Rant, An Update and Model Problems for You

And the seasons they go round and round
And the painted ponies go up and down
We're captive on the carousel of time
We can't return we can only look behind
From where we came
And go round and round and round
In the circle game...

Oh, how I love Joni Mitchell's lyrics made famous by the inimitable Buffy Sainte-marie. Oh, how The Circle Game lyrics above describe my feelings about the state of U.S. math education. I feel I've been on this carousel forever. But I do believe that all is not hopeless. I do see promise out there despite all the forces resisting the changes needed to improve our system of education.

Our math teachers already get it! They get that more emphasis should be placed on making math meaningful via applications to the real-world, stressing understanding of concepts and the logic behind procedures, reaching diverse learning styles using multiple representations and technology, preparing their students for the next high-stakes assessment, trying to ensure that no child is ... They've been hearing this in one form or another forever. BUT WHAT THEY NEED IS A CRYSTAL CLEAR DELINEATION OF ACTUAL CONTENT THAT MUST BE COVERED IN THAT GRADE OR THAT COURSE.

The vague, jargon-filled, overly general standards which have been foisted on our professional staff for the past 20 years is frustrating our teachers to the point of demoralization. THIS IS NOT ABOUT THE MATH WARS. THIS IS NOT AN IDEOLOGICAL DEBATE. JUST TELL OUR MATH TEACHERS WHAT MUST BE COVERED AND LET THEM DO THEIR JOB!

BY "WHAT MUST BE COVERED" I AM INCLUDING THE SKILLS, PROCEDURES AND ESSENTIAL CONCEPTS OF MATHEMATICS. NONE OF THIS CONSTRAINS TEACHER STYLE OR CREATIVITY. BUT WITHOUT THIS STRUCTURE THERE IS ONLY THE CHAOS THAT CURRENTLY EXISTS. AND IF YOU DON'T THINK THERE IS CHAOS OUT THERE, TALK TO THE PROFESSIONALS WHO HAVE TO DO THIS JOB EVERY DAY.


UPDATES...

Results of MathNotation's Third Online Math Contest

The Common Core State Standards Initiative

NCTM's latest response to the Core Standards Movement - the forthcoming Focus in High School Mathematics

Validation Committee selected for draft of Core Standards

The results of the latest round of ADP's Algebra 2 and Algebra 1 end of course exams

It will take several posts to cover all of this...


RESOURCES FOR YOU

MODEL PROBLEMS TO DEVELOP HIGHER-ORDER THINKING AND CONCEPTUAL UNDERSTANDING

Consider using the following as Warm-Ups to sharpen minds before the lesson and to provide frequent exposure to standardized test questions (SAT, ACT, State Assessments, etc.). I hope these problems serve as models for you to develop your own. I strongly urge you to include similar questions on tests/quizzes so that students will take these 5-minute classroom openers seriously.

I've provided answers and solutions/strategies for some of the questions below. The rest should emerge from the comments.

MODEL QUESTION #1:

For how many even integers, N, is N² less than than 100?

Answer: 9

Solution/Strategies:
Always circle keywords or phrases. Here the keywords/phrases include
"even integers"
N²
"less than".

This question is certainly tied to the topic of solving the quadratic inequality, N² "<" 100 either by taking square roots with absolute values or by factoring. Of course, we know from experience, when confronted with this type of question on a standardized test, even our top students will test values like N = 2, 4, 6, ... However, the test maker is determining if the student remembers that integers can be negative as well and, of course, ZERO is both even and an integer! Thus, the values of N are -8,-6,-4,-2,0,2,4,6, and 8.


MODEL QUESTION #2

If 99 is the mean of 100 consecutive even integers, what is the greatest of these 100 numbers?

ANSWER: 198
Solution/Strategies:
There are several key ideas and reasoning needed here:

(1) A sequence of consecutive even integers (or odd for that matter) is a special case of an arithmetic sequence.

(2) BIG IDEA: For an arithmetic sequence, the mean equals the median! Thus, the terms of the sequence will include 98 and 100. (Demonstrate this reasoning with a simpler list like 2,4,6,8 whose median is 5).

(3) The list of 100 even consecutive integers can be broken into two sequences each containing 50 terms. The larger of these starts with 100. Thus we are looking for the 50th consecutive even integer in a sequence whose first term is 100.

(4) The student who has learned the formula (and remembers it!) for the nth term of an arithmetic sequence may choose to use it: a(n) = a(1) + (n-1)d. Here, n = 50 (we're looking for the 50th term!), a(1) = 100, d = 2 and a(100) is the term we are looking for.
Thus, a(50) = 100 + (50-1)(2) = 198.

However, stronger students intuitively find the greatest term, in effect inventing the formula above for themselves via their number sense. Thus, if 100 is the first term, then there are 49 more terms, so add 49x2 to 100.



MODEL QUESTION #3: A SAMPLE OPEN-ENDED QUESTION FOR ALGEBRA II

If n is a positive integer, let A denote the difference between the square of the nth positive even integer and the square of the (n-1)st positive even integer. Similarly, let B denote the difference between the square of the nth positive odd integer and the square of the (n-1)st positive odd integer. Show that A-B is independent of n, i.e., show that A-B is a constant.


MODEL QUESTION #4: GEOMETRY

If two of the sides of a triangle have lengths 2 and 1000, how many integer values are possible for the length of the third side?


MODEL QUESTION #5: GEOMETRY

There are eight distinct points on a circle. Let M denote the number of distinct chords which can be drawn using these points as endpoints. Let N denote the number of distinct hexagons which can be drawn using these points as vertices. What is the ratio of M to N?

Answer: 1
Solution/Strategies: The student with a knowledge of combinations doesn't need to be creative here but a useful conceptual method is the following:
Each hexagon is determined by choosing 6 of the 8 points (and connecting them in a clockwise fashion for example). For each such selection of 6 points, there is a uniquely determined chord formed by the 2 remaining points. Similarly, for each chord formed by choosing 2 points, there is a uniquely determined hexagon. Thus the number of hexagons is in 1:1 ratio with the number of chords.

MODEL QUESTION #6: GEOMETRY AND THE ARITHMETIC OF PERCENTS

If we do not change the angle measures but increase the length of each side of a parallelogram by 60%, by what per cent is the area increased?

(A) 36% (B) 60% (C) 120% (D) 156% (E) 256%

Another Sample Contest Problem - Counting...

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!!_

...Read more

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.