## Saturday, December 26, 2009

### A Quadratic Trinomial/Factoring Investigation for Algebra I/II

In my Christmas post, I raised the issue of how much time should be spent on factoring quadratic trinomials over the integers in light of the new ADP Standards for Algebra I and II. Hopefully, some of you will provide us with the benefit of your knowledge and experience. I may even make this into a poll or survey to be voted on but, in this post, I will appear to contradict myself and propose an investigation of this topic which requires some effort and time on the part of the student. The target audience would be the regular or accelerated Algebra I/II student.

We all need to become more creative in the strategic use of time in our classrooms (I still think of myself as being in the classroom!). What are some alternatives to using class time for this? I'll suggest one approach and I'm hoping others will offer their suggestions:

Assign the following as an extra credit or "long-term" project to be due in a week or two. I would not even take classroom time to discuss it. Just hand it out or post it on your website or the department's website if it is to be given to all the Algebra classes. Students can easily download it or print directly if they wish. After they are collected, graded and returned, you may choose to discuss it briefly for about 10 minutes using an overhead transparency, opaque projector or via your computer and a projector. You can also post some  student solutions on the website.

THE INVESTIGATION/PROJECT

[OPTIONAL HINT OR CUE]
The following may require an application of the ac-method learned in class.

(1) Factor the following over the integers and show all steps used in your method of factoring:

(a) 12x2 + 27x + 15

(b) 12x2 + 28x + 15

(c) 12x2 + 29x + 15

(2)
(i) List all positive integers values of b, including the ones from part (a), for which 12x2 + bx + 15 is factorable over the integers.
(ii) For each value of b, factor the resulting trinomial.
(ii) How many of these trinomials produce a gcf ≠ 1 for 12, b and 15?

(3) If we knew in advance that 180 has 18 positive integer factors, explain how it follows that there are 9 values for b in part (2).

(4)
(i) If the "12" and "15" were interchanged, explain why this would not change the possible values for b in part (2)?
(ii)For each resulting trinomial such as 15x2 + 28x +12, determine its factors and explain how they are related to the factors of the original trinomial (i.e., before interchanging the 12 and 15).

QUESTION FOR OUR DISCUSSION (No, these are not rhetorical! Some are quite knotty)

(1)  What do you see as the benefits of this investigation, if any?

(2) Do the new standards and assessments discourage us from investing time into this type of in-depth problem-solving?

(3) Do you believe this type of assignment should be reserved for the accelerated/honors Algebra I student in 7th or 8th grade or even for the stronger Algebra II student?

(4) With the new ADP Algebra standards, do you believe this type of investigation is reasonable, particularly since it is unlikely that any variation of this would appear on an End of Course Test?

(5) If you were to give this problem, how would you edit the investigation? Parts you would delete or change? Parts you would add?

(6) My goal for this blog has always been to provide you with useful and engaging examples of in-depth problems for your students that require going beyond the mechanical aspects of the course. These problems are developed for this blog -- they do not come from my notes from 30 years ago! Would you be interested in a supplementary resource of such problems for each course you teach? Do you already have one from the publisher or from another source which you really enjoy? Share it!

## Thursday, December 24, 2009

### How Much Factoring In 1st Year Algebra?

SEASON'S GREETINGS
Math Notations 3rd Birthday- Thank You!

The American Diploma Project is and will be impacting on what is being taught in both Algebra I and II in the 15 states who have joined the ADP Consortium. The classic flow from Standards to Assessments to Course Content is leading to the type of content standardization in our schools which I envisioned decades ago. A natural part of this process is deciding what topics in our traditional courses need to be deemphasized or eliminated to allow more time for the study of linear and non-linear function models, one of the central themes of the new Algebra standards.This leads to curriculum questions like...

How much time should be spent on factoring quadratic trinomials in Algebra I?

My assumption is that factoring ax2+bx+c where a ≠ 1 is still taught in Algebra I. Please challenge that assumption if wrong! If we also assume there is sufficient justification for teaching this, then we move on to the issue of how much time should be devoted to instruction. Two days? More? Time for assessment?

Here are some arguments pro and con...

PRO

(1) It is required by the ADP Standards (see below).

(2) Learning only simple trinomial factoring of the form x2+bx+c is not sufficient for solving more complex application problems.

(3) The various algorithms, such as the "ac-method", which have been developed for factoring quadratic trinomials, are of value in their own right; further, the "ac-method" introduces or reinforces the important idea of factoring by grouping.

(4) Students gain technical proficiency by tackling more complicated trinomials.

(5) Students should be given the option of more than one method, not just the quadratic formula.

CON

(1) The AP Calculus exam generally avoids messy quadratics in their problems. If such occur, students normally go directly to the Quadratic Formula.

(2) The SATs generally avoid asking students to factor such quadratics directly, particularly since it is easy to "beat the question" by working backwards from the choices. Instead, they ask the student to demonstrate an understanding of the process.

Here's a typical question they might ask:

If 6x2 + bx + 6 = (3x + m)(nx + 3) for all values of x, what is the value of b?

(3)The ADP standards for Algebra I do include this topic but it does not appear to be stressed. The following are taken from the ADP Algebra I standards and practice test:

(3) Do other nations teach our traditional methods of factoring or are students told to go directly to the quadratic formula?

(4) Current Alg I texts seem to have deemphasized factoring in general and some have moved this topic to later in the book.

So I am opening the floor for your input here!

(a) How much time is spent on factoring quadratic trinomials in Algebra I in your school?
(b) Do you teach the "ac-method"? If yes, do you motivate it or teach it mechanically?
(c) Do you believe factoring quadratic trinomials is essential or should it be deemphasized?

By the way, here is an example of the ac-method:

Factor completely over the integers:   6x2 + 13x + 6

Step 1: Find a pair of factors of ac = (6)(6) = 36 which sum to b = 13.
Hopefully, students think of 9 and 4 without a calculator!

Step 2:  Rewrite the middle term 13x as 9x + 4x (works in either order)
Then 6x2 + 13x + 6  =   6x2 + 9x + 4x + 6

Step 3: Group in pairs and factor out greatest common monomial factor from each pair:
3x(2x + 3) + 2(2x + 3)

Step 4: Factor out the common binomial factor 2x + 3:
(2x + 3) (3x + 2)

Step 5: Check carefully by distributing.

Here is a "proof" of this method (some details omitted like the meaning of h and k):

## Wednesday, December 16, 2009

### Divisibility, Counting, Strategies, Reasoning -- Just Another Warmup

Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school

List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.

(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"

(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number?

(3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?

(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!

(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in?

## Monday, December 7, 2009

### Demo For Building An Investigation In Geometry For All Levels

Note: Diagram has been modified from original.

For Figures 1 and 2, the following is given:

AD + AC = BD + BC
Perimeter of triangle = 36
AC = 15

Show that the length of AD = 3.
In other words, demonstrate that the length of AD is independent of sides AB and BC.

Instead of imposing or suggesting my way of using this question to build an investigation, how would you do it?

If you're new to this blog, I have published dozens of examples of investigations which are intended to develop process, conceptual understanding, generalization  and a different view of what mathematics is for our students. An investigation allows students to explore particular cases before attempting to generalize and abstract. Some might call this scaffolding. I see it as creating an experimental environment in the classroom, encouraging our students to become mathematical researchers! I know every argument against this approach but, remember, I'm suggesting that this type of activity only be used perhaps once a month...

The question above can be given as is to some groups of students but may not be appropriate in its present form for many others. The question can be reworded or changed completely.

What would you do?

## Tuesday, December 1, 2009

### Using WarmUps in Middle School/HS to Develop Thinking and Review/Apply Skills

My 500 or so subscribers may not have seen the following anagrams which have been in the right sidebar of my home page  for the past month or so. No one has yet taken the time to solve them. They're not that hard! Pls email me at dmarain at gmail dot com with your answers.

VORTEC SCAPE

(1) Hidden Steps OR

(2) General Arrows

The following problems are similar to ones I posted recently...

Mental Math and No Calculator!

1)  The following sum has a trillion terms:

0.01 + 0.01 + 0.01 + ... + 0.01 = 1000...0
How many zeros will there be in the sum?

2)  The following product has a trillion factors:

(0.01) (0.01) (0.01) ... (0.01) = 0.000...1
How many zeros after the decimal point will there be in the product?

(a) You may want to adjust the "trillion" for your own groups but I'm intentionally using this number for a few reasons, not the least of which is to review large powers of 10 (Will most think: "A million has 6 zeros, a billion has 9 zeros, so a trillion has..."?).

(b) The second one is more challenging and intended for Prealgebra students and above but, using the "Make it simpler" and "Look for a pattern" strategies, make it possible for younger students.

(c)  How many of you are reacting something like: "Is Dave out of his mind? My students don't know their basic facts up to 10 and he wants mental math with a trillion!" I have found that large numbers engage students since they know there is a way of doing these without a lot of work if you know the "secrets"! Besides, we either push our students or we don't. You decide...

(d) These questions review several important concepts and skills. You may want to use these to introduce or review the importance of exponents and their properties.

## Wednesday, November 25, 2009

### INSTRUCTIONAL STRATEGIES SERIES: Teaching for Meaning - More Than Just A Geometry/Algebra Problem

HAPPY THANKSGIVING!

Alright, you're teaching about the rule for slopes of perpendicular lines in Algebra or Geometry.

Here are some of the instructional strategies or approaches you may have used...

(1) State the theorem without explanation followed by 3-4 demo examples of how it's used
(2) Motivate the theorem using the lines y = (3/4)x and y = (-4/3)x, choosing the points (4,3) and (-3,4) to demonstrate why these lines are perpendicular
(3) A more abstract approach using the following diagram

NOTE: Q(-b,a) is the point on line M in quadrant II. The label is too far from the dot!

FROM THE GIVEN INFORMATION IN THE DIAGRAM PROVE THAT ∠QOP IS A RIGHT ANGLE, THAT IS, LINES L AND M ARE PERPENDICULAR.

(a) If your group was advanced, would you omit the perpendiculars QR and PS?
(b) Would you draw the diagram to scale to prevent confusion for most students?
(c) Would you even consider Option (3) with a regular or weaker group of students? Would Option 2 be more than enough to get at the main idea?
(d) To more strongly suggest the use of slopes and/or similar triangles, would it be better to use the points (4,3) and (-6,8) on the lines? I personally would prefer this (and not give the equations of the lines). What do you imagine most students would do with this problem a few weeks (or even days!) later? Would they make the connection to slopes immediately if they had moved on to another unit or if this appeared on an assessment?
(e) Would some students need more than one example to suggest a generalization? Exactly what questions would you ask to promote a generalization?
(f) What have you done with this topic and/or how would you modify the above ideas??? The floor is open..
By the way, do you believe it is likely or unlikely that some version of this problem might appear on a standardized test like ADP's Algebra 2 End of Course Test or the SATs?

## Sunday, November 15, 2009

### The Return of the WarmUp Challenges!

Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students.

Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.).  If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.

1.   NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!

2.   If 10-1000 - 10-997 is written as a decimal, answer the following:

(a) How many decimal places are there, i.e., how many digits to the right of the decimal point?
(b) One can show that the decimal digits end in a string of 9's. How many 9's?
(c) How many zeros are to the right of the decimal point and to the left of the string of 9's?

Notes:
(1) If we write the negative exponent expressions as rational numbers, this is perfectly appropriate for middle schoolers and, in fact, I think they need more of these experiences!
(2) The "Make It Simpler - Look for a Pattern" Strategy should be second nature to our youngsters, but when they see questions like these on the SATs, how many of our students really think of it!
(3) The fact that some calculators return a value of zero for the expression in the problem is a teachable moment - seize it!!
(4) See below for an algebraic approach.

--------------------------------------------------------------------------------------------

1. 9√3

2. (a) 1000   (b) 3   (c) 997

An Algebraic Approach to #2:
First, students need to be familiar with the basic pattern:
10-1 = 1/10 = .1 Note that there is one decimal digit.

10-2 = 1/102 = 1/100 = .01  Note that there are two decimal places, etc.

10-1000 - 10-997 = 1/101000 - 1/10997
Using 101000 as the common denominator, we obtain
1/101000 - 103/101000 =
-999/101000 from which the results follow (with some additional reasoning)...

Note: I could have worked directly with the exponent form by factoring out 10-1000 but I chose rational form for the younger student.

## Wednesday, November 4, 2009

### 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.

(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...

## Tuesday, November 3, 2009

### RESULTS OF THIRD MATHNOTATIONS CONTEST and OTHER NEWS...

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.

THIS WAS A CHALLENGING CONTEST, PARTICULARLY FOR YOUNGER STUDENTS, ALTHOUGH, AS YOU CAN SEE BELOW, THEY HELD THEIR OWN!! CONGRATULATIONS TO ALL PARTICIPANTS FOR A JOB WELL DONE!

FIRST PLACE - 12 OUT OF 12 POINTS!

CHILES HIGH SCHOOL
TALLAHASSEE, FL

Marshall Jiang - 11th
William Dunn - 12th

Wayne Zhao - 9th

Andrew Young - 11th

Jack Findley - 12th

Danielo Hoekman - 11th

SECOND PLACE - 11 OUT OF 12 PTS

HARVEST PARK MIDDLE SCHOOL

PLEASANTON, CA

Eugene Chen - 8th
Jerry Li - 8th

Brian Shimanuki - 8th

Christine Xu - 8th

Jeffrey Zhang - 8th

Ian Zhou - 8th

Advisor, Randall S. Lomas

THIRD PLACE - 9 OUT OF 12 PTS

KOBE, JAPAN

Kevin Chen - 11th
Sean Qiao - 11th

Alice Fujita - 11th

Cathy Xu - 11th

Steven Jang - 11th
Sooyeon Chung - 10th

Advisor, Ms. Elizabeth Durkin

FOURTH PLACE - 7 OUT OF 12 PTS

KOBE, JAPAN

Hyun Song - 11th
Max Mottin - 11th

Ron Lee - 10th

Kyoko Yumura - 10th

Selim Lee - 10th

Evangel Jung - 10th

Advisor, Ms. Elizabeth Durkin

FIFTH PLACE - 4 OUT OF 12 POINTS

MEMORIAL MIDDLE SCHOOL - TEAM DAVID

FAIR LAWN, NJ

David Bates - 8th
Isaiah Chen - 8th

Kajan Jani - 8th
Thomas Koike - 8th
Priya Mehta - 8th

Joseph Nooger - 8th

Advisor, Ms. Karen Kasyan

SIXTH PLACE TIE

WALLINGTON JR/SR HS - SENIOR TEAM

WALLINGTON , NJ

Nicole Bacza - 12th
Tomasz Hajduk - 12th

Martyna Jezewska - 12th
Thomas Minieri - 12th
Urszula Nieznelska - 12th
Damian Niedzielski - 12th

FAIR LAWN HS - TEAMS A & B
FAIR LAWN, NJ

Team A
Egor Buharin - 12th

Kelly Cunningham - 12th

Elizabeth Manzi - 12th
Gurteg Singh - 12th
Daniel Auld - 12th

Richard Gaugler - 12th

Team B

David Rosenfeld - 12th

Gil Rozensher - 12th

Roger Blumin - 9th

Mike Park - 9th

Jason Bandutia - 9th

Alexander Lankianov - 9th

SEVENTH PLACE TIE

WALLINGTON JR/SN HS

WALLINGTON, NJ

Junior Team
Konrad Plewa - 11th

Matthew Kmetz - 11th

Patrick Sudol - 10th

Marek Kwasnica - 10th

Anna Jezewska - 10th

MEMORIAL MIDDLE SCHOOL - TEAM SIMRAN
FAIR LAWN, NJ

Simran Arjani - 8th
Aramis Bermudez - 8th

Allan Chen - 8th

Kateryna Kaplun - 8th

Harsh Patel - 8th

Advisor, Ms. Karen Kasyan

## Monday, October 12, 2009

### 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.

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?

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?

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?

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%

## Monday, October 5, 2009

### 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?

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

## Sunday, October 4, 2009

### 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.

## Wednesday, September 30, 2009

### Two Trains and a Tunnel! Is There Room For This In The Tunnel And In Your Curriculum?

At the same instant of time, trains A and B enter the opposite ends of a tunnel which is 1/5 mile long. Don't worry -- they are on parallel tracks and no collision occurs!

Train A is traveling at 75 mi/hr and is 1/3 mile long.
Train B is traveling at 100 mi/hr and is 1/4 mile long.

When the rear of train B just emerges from the tunnel, in exactly how many more seconds will it take the rear of train A to emerge?

Click on More to see answer (Feed subscribers should see answer immediately).

1. Appropriate for middle schoolers even before algebra? Exactly when are middle schoolers in your district introduced to the fundamental Rate_Time_Distance relationship?
2. What benefits do you think result from tackling this kind of exercise? If it's not going to be tested on your standardized tests, is it worth all the time and effort?
3. How much "trackwork" needs to be laid before students are ready for this level of problem-solving?
4. As an instructional strategy, would you have the problem acted out with models in the room or use actual students to represent the trains and the tunnel? OR just have them draw a diagram and go from there? Do a simulation on the TI-Inspire or TI-84 using graphics and parametric equations for the older students?
5. If you believe there is still a place for this type of problem-solving, should it be given only to the advanced classes and depicted as a math contest challenge?
6. I'm dating myself but I remember seeing problems like this in my old yellow Algebra 2 textbook? Uh, I believe this was B.C. -- before calculators! Can you imagine! Do you recall these kinds of problems? Do you recall the author or publisher?
7. Of course, the proverbial "two trains and tunnel" problems are frequently parodied and used as emblematic of the "old math"! They've been replaced by "real-world" applications. "Progress makes perfect!"

Answer: 9.4 seconds (challenge this if you think I erred!)

## Thursday, September 24, 2009

### More Challenges/SAT Practice, Core Curriculum Standards, Reminders, Comments...

Challenge 1:

HOW MANY DIGITS OF 10001000 - 1 WILL BE EQUAL TO 9 WHEN THIS EXPRESSION IS EXPANDED?

Challenge 2:

HOW MANY 5-DIGIT POSITIVE INTEGERS HAVE A SUM OF DIGITS EQUAL TO 43?

Challenge 3:

Jorge can run a 6-minute mile while Alex can run a 5-minute mile. If they start at the same time, how much less distance, in miles, will Jorge run in 10 minutes?

(Yes, you can respond with answers and solutions to these in the comments!)
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Tired of hearing about THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST!? IF I RECEIVE 10 MORE REGISTRATIONS, I MAY JUST STOP!
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The Common Core State Standards Initiative
First look here for a quick overview and here for an index to the latest draft of the standards. Of course, this blog only discusses the mathematics part of the document.

Overview

The Common Core State Standards Initiative is a joint effort by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) in partnership with Achieve, ACT and the College Board. Governors and state commissioners of education from across the country committed to joining a state-led process to develop a common core of state standards in English-language arts and mathematics for grades K-12.

These standards will be research and evidence-based, internationally benchmarked, aligned with college and work expectations and include rigorous content and skills. The NGA Center and CCSSO are coordinating the process to develop these standards and have created an expert validation committee to provide an independent review of the common core state standards, as well as the grade-by-grade standards.

HIGHLIGHTS

• Each standard is broken into Core Concepts and Skills, provides research-based evidence and many illustrative examples to clarify the language
• Alignment of these standards to those of 5 representative states: California, Florida, Georgia, Massachusetts and Minnesota
• Standards reduce the number of Core Concepts and Skills in accordance with many recommendations to pare down the number of required topics to allow for greater depth
Example of a Standard (Standard 5)

Equations | see evidence
An equation is a statement that two expressions are equal. Solutions to an equation are the values of the variables in it that make it true. If the equation is true for all values of the variables, then we call it an identity; identities are often discovered by manipulating one expression into another.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs, which can be graphed in the plane. Equations can be combined into systems to be solved simultaneously.

An equation can be solved by successively transforming it into one or more simpler equations. The process is governed by deductions based on the properties of equality. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, stimulating the formation of expanded number systems (integers, rational numbers, real numbers and complex numbers).

A formula is a type of equation. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1 + b2)/2) h, can be solved for h using the same deductive process.

Inequalities can be solved in much the same way as equations. Many, but not all, of the properties of equality extend to the solution of inequalities.

Connections to Functions, Coordinates, and Modeling. Equations in two variables may define functions. Asking when two functions have the same value leads to an equation; graphing the two functions allows for the approximate solution of the equation. Equations of lines involve coordinates, and converting verbal descriptions to equations is an essential skill in modeling.

Core Concepts
Students understand that:
1. An equation is a statement that two expressions are equal.
see examples

2. The solutions of an equation are the values of the variables that make the resulting numerical statement true.
see examples

3. The steps in solving an equation are guided by understanding and justified by logical reasoning.
see examples

4. Equations not solvable in one number system may have solutions in a larger number system.
see examples

Core Skills
Students can and do:
1. Understand a problem and formulate an equation to solve it.
see examples

2. Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
see examples

3. Rearrange formulas to isolate a quantity of interest.
see examples

4. Solve systems of equations.
see examples

5. Solve linear inequalities in one variable and graph the solution set on a number line.
see examples

6. Graph the solution set of a linear inequality in two variables on the coordinate plane.
see examples

FUNDAMENTAL ASSUMPTIONS AND CONSIDERATIONS

Very Important!
(Click on image to see a clearer view)

INITIAL MATHNOTATIONS REACTIONS

1. Exceptionally clear and definitive document
2. Influenced by NCTM (Curriculum Focal Points), Achieve, College Board, ACT
3. Illustrative examples are of high quality
4. Will serve as a basis for states' revisions of current standards hopefully creating more consistency than currently exists
5. Leaving curriculum to local districts and states was a politically necessary decision, however, in my opinion, developing a reasonably consistent curriculum by grade level and/or course across districts and states from these standards may prove to be difficult and may again lead to considerable disparity. Hopefully, this will be self-correcting when standardized assessments are created as is currently being done with the End of Course Tests from Achieve

## Sunday, September 20, 2009

### A Practice PSAT/SAT Quiz with Strategies!!

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 - 1002n-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

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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!!

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1. B

2. B

3. C

4. 1

5. C

6. 125

7. 6

## Thursday, September 17, 2009

### Demystifying Per Cent Problems Part II - Using Multiple Representations and an SAT Problem

Have you forgotten to register for MathNotation's Third FREE Online Math Contest coming in mid-October? We already have several schools (from around the world!) registered. For details, link here or check the first item in the right sidebar!!

Before tackling a more challenging problem in the classroom, I would typically begin with one or more simpler examples. My objective was to review essential concepts and skills and demonstrate key ideas in the harder problem. This incremental approach (sometimes referred to as scaffolding) enabled some students to solve the problem or at least get started. Usually within each group of 3-4 students, there was at least one who could help the others. Some groups or classes might still not be ready after one example, so more would be needed. I never felt that this expense of time was too costly since my goal was to develop both skill and understanding.

SIMPLER EXAMPLE
Consider the following two statements about positive numbers A and B:

(1) A is 80% of B.
(2) A is 20%
less than B .

Are these equivalent, that is, if values of A and B satisfy (1), will they also hold true for (2) and conversely?

How would you get this idea across to your students?

Again, depending on the students, I would often allow them to discuss it first in small groups for two minutes, then open up the discussion.

Note: If the group lacks the skills, confidence or background (note that I left ability out, intentionally!), I might first start with concrete values before giving them the 2 statements above: E.g., What is 80% of 100?

How would I summarize the methods of solution to this question. Here's what I attempted to do in each lesson. I didn't reach everyone but I found from further questioning and subsequent assessment that this multi-pronged approach was more successful than previous methods I had used. Most of these methods came from the students themselves!

INSTRUCTIONAL STRATEGIES

I. Choose a particular value for one of the numbers, say B = 100. Ask WHY it makes sense to start with B first and why does it make sense to use 100. Calculate the value of A and discuss.

II. Draw a pie chart (circle graph) showing the relationship between A and B. Stress that B would represent the whole or 100%.

III. Write out the sentence:
80% of B is the same as 100% of B - 20% of B
In other words:
80% of B is the same as 20% less than B.

IV. Express algebraically (as appropriate):
0.8B = 1B - 0.2B

Numerical (concrete values)
Visual (Pie chart)
Verbal (using natural language)

Symbolic (algebra)

Yes, it's Multiple Representations! The Rule of Four!

To me, it's all about accessing different modes of how students process. Call it learning styles, brain-based learning, etc., it still comes down to:
RARELY DOES ONE METHOD OF EXPLANATION, NO MATTER HOW CLEAR OR STRUCTURED, REACH A MAJORITY OF STUDENTS. YOUR FAVORITE EXPLANATION WILL MAKE THE MOST SENSE TO THE STUDENTS WHO THINK LIKE YOU!!

Now for today's challenge.
(Assume all variables represent positive numbers)

M is x% less than P and N is x% less than Q. If MN is 36% less than PQ, what is the value of x?

Can you think of several methods?
I will suggest one of the favorite of many successful students on standardized assessments:

Choose P = 10, Q = 10. Then...

Click on More (subscribers do not need to do this) to see the answer without details.

Answer: x = 20

## Sunday, September 13, 2009

### Demystifying Harder Per Cent Word Problems for Middle Schoolers and SATs - Part I

Example I
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students left, how many were in the class to start?

Solution without explanation or discussion:

0.4x = 240 ⇒ x = 600

Example II
40% of the the Freshman Calculus class at Turing University withdrew. If 240 students were left, how many were in the class to start
?

Solution without explanation or discussion:

0.6x = 240 ⇒ x = 400

Thinking that the issues in the problems above are more language-dependent than based on learning key mathematics principles or effective methods? I would expect that many would say that using the word "left" in both problems was unnecessarily devious and that clearer language should be used to demonstrate the mathematics here. Perhaps, but when I taught these types of problems I would frequently juxtapose these types of questions and intentionally use such ambiguous language to generate discussion - creating disequilibrium so to speak. If nothing else, the students may become more critical readers! Further, the idea of using similar but contrasting questions is an important heuristic IMO.

Even though I've been a strong advocate for a standardized math curriculum across the grades, I fully understand that the methods used to present this curriculum are even more crucial. Instructional methods and strategies are often unpopular topics because they seem to infringe on individual teacher's style and creativity. BUT we also know that some methods are simply more effective than others in reaching the maximum number of students (who are actually listening and participating!). I firmly believe there are some basic pedagogical principles of teaching math, most of which are already known to and being used by experienced teachers.

Percent word problems are easy for a few and confusing to many because of the wide variety of different types.

Here are brief descriptions of some methods I've developed and used in nearly four decades in the classroom.

I. (See diagram at top of page)
The Pie Chart builds a strong visual model to represent the relationships between the parts and the whole and the "whole equals 100%" concept. How many of you use this or a similar model ? Please share! There's more to teaching this than drawing a picture but some students have told me that the image stays longer in their brain. I learn differently myself but I came to learn the importance of Multiple Representations to reach the maximum number of students.

II. "IS OVER OF" vs. "OF MEANS TIMES"
The latter is generally more powerful once the student is in Prealgebra but, of course, the word "OF" does not appear in every percent so many different variations must be given to students and practiced practiced practiced practiced over time. The first method can be modified as a shortcut in my opinion to find a missing percent and that may be its greatest value. However many middle schoolers use proportions for solving ALL percent problems. I personally do NOT recommend this!

Well, I could expound on each of these methods ad nauseam and bore most of you, but I think I will stop here and open the dialg for anyone who has strong emotions about teaching/learning per cents...

## Monday, September 7, 2009

### 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!

(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.
Note: Sending me the email is not a commitment! It simply means you are interested and will receive a registration form.

## Saturday, August 29, 2009

### Batteries Required! A Combinatorial Problem MS /HS Students Can Use...

Have you ever inserted batteries in a device only to find that it didn't work? You reverse the batteries and try again, but no luck. You can't find the polarity diagram to guide you and you're dealing with 3 or 4 batteries and all the possible combinations! Well, that just happened to me as I was inserting 3 'C' batteries into a new emergency lantern I just purchased. There was no guide that I could see. I knew there were 8 possibilities but it was late and my patience quickly ran out. I tried it again the following morning, shone my small LED light on it and saw the barely visible diagram.

After seeing the lantern finally operate, I realized I should have used a methodical approach -- practice what I preach!! Then I thought that this might be a natural application of the Multiplication Principle one could use in the classroom. Of course, it would work nicely if you happened to have the identical lantern but you might have some of these in the building or at home which take 2 or more batteries. IMO, there's something very real and exciting about solving a math problem and seeing the solution confirmed by having "the light go on!" I'll avoid commenting on the obvious symbolism of that quoted phrase...

Instructional/Pedagogical Considerations

(1) I would start with a small flashlight requiring only one battery to set up the problem. For this simplest case, students should be encouraged to describe the correct placement in their own words and on paper.

(2) Would you have several flashlights/lanterns available, one for each group of 2-4 students or would you demonstrate the problem with one device and call on students to suggest a placement of the batteries? Needless to say, if you allow students to work with their own flashlights, they will look for the polarity diagram so you will need to cover those somehow. That is problematic!

(3) Do you believe most middle school students (if the polarity diagram is not visible) will randomly dump in the batteries to get the light to go on and be the first to do so? Is it a good idea to let them do it their way before developing a methodical approach? Again, if a student or group solves the problem, it is important to have them write their solution before describing it to the class. If there is more than one battery compartment, students should realize realize the need to label the compartments such as A, B, C , ... Once they reach 3 or more batteries, they should recognize that a more structured methodical approach is needed so that one doesn't repeat the same battery placement or miss one. One would hope!

(4) Is it a drawback that the experiment will probably end (i.e., the light goes on) before exhausting all possible combinations? How would we motivate students to make an organized list or devise a methodical approach if the light goes on after the first or second placement of the batteries?

(5) I usually model these kinds of problems using the so-called "slot" method. Label the compartments A, B, ... for example and make a "slot" for each. For two compartments we have

A B
_ _

Under each slot, I list the possibilities, e.g., (+) end UP or DOWN (depending on the device, other words may be more appropriate). Here I would only concern myself with labeling the (+) end, the one with the small round protruding nub. For this problem I would write the number (2) on each slot since there are only TWO ways for each battery to be placed. Note the use of (..). In general, above each slot I would write the number of possibilities. For two compartments (or two batteries), the students would therefore write (2) (2). They know the answer is 4 but some will think we are adding rather than multiplying. Ask the class which operation they believe will always work. How would you express your questions or explanation to move students toward the multiplication model? The precise language we use is of critical importance and we usually only learn this by experimentation. If one way of expressing it doesn't seem to click with some students, we try another until we refine it or see the need for several ways of phrasing it. This is the true challenge of teaching IMO. We can plan all of this carefully ahead of time, but we don't know what the effect is until we go "live" (or have experienced it many times!).

Perhaps you've already used a similar application in the classroom - please share with us how you implemented it. Circuit diagrams in electronics also lend themselves nicely to this approach. Typically, I've used 2, 3 or more different coins to demonstrate the principle but the batteries seem to be a more natural example, although I see advantages and disadvantages to both. At least with the batteries, students should not question the issue of whether "order counts!"

I could say much more about developing the Multiplication Principle in the classroom, but I would rather hear from my readers.
If you've used other models to demo this key principle, let us know...

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!