A Review and a Critique of the National Math Panel Report - PART I

On March 13, 2008, the National Mathematics Advisory Panel presented its Final Report, Foundations for Success, to the President of the United States and the Secretary of Education.

As with any education report there is a both a technical and a political aspect. Each of us will view this type of document through the lens of our personal bias. As much as I'd like to believe I am fair and objective, I know that I bring my own perspective to this as one who passionately believes that one can expect both mastery of skill and conceptual understanding, the what, the how and the why. Instruction that combines skill practice with rich problem-solving and exploration. I know I am not the only one who holds these beliefs and, to me, they are self-evident.

Since the inception of this blog 15 months ago, I have not wavered from this view. You can see this in the many letters I wrote to the National Math Panel which have been published on this blog to my interview of Prof. Schmidt, to my interview with Prof. Steen, to my post in Jan 2007 describing the genesis and reason for this blog. Repeatedly I have called for more coherence and consistency in K-8 math curriculum and beyond. Repeatedly, I have asked those in a position to make a difference to listen to our dedicated professional educators who have always known the truth of what is needed for our children's math education. Perhaps some of this has not fallen on deaf ears...


You may want to view the short video of an interview with Dr. Larry Larry Faulkner,
the chair of the National Mathematics Advisory Panel. In it, he discusses the key findings of the panel, published in their recently released report.

In this post I will summarize and comment on this video interview. In later posts, I will comment on other aspects of the report.

Highlights of Dr. Faulkner's Comments
(much of this was taken verbatim from the video):

(1) We need to streamline mathematics education in the years leading up to algebra
(2) Streamlining is needed because it is well-known both inside the US and outside that we have too many topics in early grades that are covered too shallowly
(3) A principal recommendation is that we cover fewer topics and cover the most important ones more thoroughly.
(4) The panel focused on the preparation of American children for success in algebra. Algebra has a central role in mathematics curriculum, the first course leading into secondary mathematics.
(5) Success in algebra not only plays a central role because of its correlation to success in high school math courses, but also because of its correlation to college-attending and graduation rates and eligibility for the national technical workforce.
(6) Cognitive science informs us that children who believe that working harder can make one smarter, actually achieve more.
(7) To encourage their children, parents should make note of how common mathematics is in the world and how many jobs parents hold which make use of mathematical concepts.

MathNotations Reactions to Dr. Faulkner's statements:
(1) "...It is well-known outside the US and inside"! When did it become so well-known? Could it be when Prof Schmidt uttered WELL OVER A DECADE AGO his now-famous description of our math curriculum as "one inch deep and one mile wide"? You mean now his word will be heeded? Now that NCTM has recognized the need for a narrower focus as recently published in its Curriculum Focal Points? Now there will be new textbooks written that reflect this narrower focus? Is that what you mean Prof. Faulkner? Exactly who will be apologizing to the generations of students who have been exposed to such superficiality? Who will be apologizing to the dedicated educators who have been compelled to sacrifice mastery and deeper understanding for superficial coverage of a myriad of topics? Who exactly will be apologizing for not listening to Prof. Schmidt until now?

(2) We need to be informed by the research of cognitive scientists that a strong work ethic will lead to success, particularly in mathematics? That persistence and effort can lead to success in math!

This is the revelation I've been waiting two years to hear? Yes, I agree this is a wonderful message for all students to hear in all aspects of their learning. Yes, Prof. Faulkner, I believe we can also accumulate 400 million pieces of anecdotal evidence from our educators to support the truism that there is no substitute for hard work, particularly in mathematics...

Irving 'Kap' Kaplansky - He Strikes A Familiar Chord! The Mystery is Solved...

We had three winners this week who correctly identified Professor Kaplansky (1917-2006). As usual, I had many reasons for selecting this outstanding mathematician. I'm always partial to algebraists or number theorists and 'Kap' worked in both areas. I've also been told I bear some physical resemblance to Dr. Kaplansky! I was also influenced by the fact that music played an important role in his life.
A press release from the U. of Chicago provided the following excerpt:

Kaplansky loved working with young people, and he served as Ph.D. advisor to 55 graduate students, the most of any mathematics professor ever to have taught at the University of Chicago, said J. Peter May, Professor of Mathematics at the University.

He published close to 150 papers, the earliest appearing in 1939 and the last in 2003, an astonishing span of activity for a mathematician,” May said. “Kaplansky had a great sense of humor, or perhaps more accurately fun. He enjoyed life and lit up any room he was in. He liked quirky mathematical problems with a real life twist. For example, a 1943 paper gave an elegant solution of the problem of finding the number of ways that a given number of married couples may be seated at a round table, men alternating with women, so that no wife sits next to her own husband."

In 1984 he became Director of the Mathematical Sciences Research Institute in Berkeley, California, a post he held until 1992.

David Eisenbud, current Director of MSRI and a former student said, “I remember well his highly entertaining and beautifully polished lectures from my student days in Chicago. Whatever he taught, I signed up for the course, it was such a pleasure to listen to him.

“After stepping down as MSRI director, at 80, Kaplansky went back to full-time research mathematics, and returned to number theory, one of his first loves,” Eisenbud said.

As an avid musician and pianist, Kaplansky played in or directed many University musical and theater productions, including its annual productions of the works of Gilbert and Sullivan.

Music was a very important part of Prof. Kaplansky's life. Here is an excerpt from Ivar's Peterson's Math Trek:

A distinguished mathematician who has made major contributions to algebra and other fields, Kaplansky was born in Toronto, Ontario, several years after his parents had emigrated from Poland. In the beginning, his parents thought that he was going to become a concert pianist. By the time he was 5 years old, he was taking piano lessons. That lasted for about 11 years, until he finally realized that he was never going to be a pianist of distinction.

Nonetheless, Kaplansky loved playing the piano, and music has remained one of his hobbies. "I sometimes say that God intended me to be the perfect accompanist--the perfect rehearsal pianist might be a better way of saying it," he says. "I play loud, I play in time, but I don't play very well."

While in high school, Kaplansky started to play in dance bands. During his graduate studies at Harvard, he was a member of a small combo that performed in local night clubs. For a while, he hosted a regular radio program, where he played imitations of popular artists of the day and commented on their music. A little later, when Kaplansky became a math instructor at Harvard, one of his students was Tom Lehrer, later to become famous for his witty ditties about science and math.

He wrote A Song About Pi. Additional lyrics were added by one of his inspired students. If you ever get a chance to hear a performance by singer-songwriter Lucy Kaplansky (Irving Kaplansky's daughter), you might very well get a rendition of "The Song About Pi" as part of the program. A club headliner, recording artist, and former psychologist, Lucy Kaplansky has her own distinctive style but doesn't mind occasionally showcasing her father's old-fashioned tunemanship.

Now for our winners:

Eric Jablow

Irving Kaplansky...

Barton Yeary

Howdy -- the mystery mathematician is Irving Kaplansky. The reference to rings made me immediately think of him and a bit of googling located his obit from AMS Notices (http://www.ams.org/notices/200711/tx071101477p.pdf). The pictures matched up.

I don't have an anecdote about him. But one fact is this: he wrote a nice book for undergraduates, Set Theory and Metric Spaces.

Vlorbik

Irving Kaplanski.

i met him in '87 at herstein's festschrift:

http://www.math.niu.edu/~behr/Math/Yitz/

.

his daughter's semi-famous:

http://en.wikipedia.org/wiki/Lucy_Kaplansky

keep 'em coming!

I'll leave Prof. Kaplansky's picture up there for another week or so. Stay tuned for the next contest...

Classic AMC Contest Square Dissection Problem and more...

Recognize this diagram from a famous math contest problem which I first saw many years ago on an old AHSME contest (now known as AMC)? We'll start with this one and then modify it, creating variations on the basic theme. Finally, we will ask our readers/students to generalize the result algebraically.


In the diagram at the left, nothing is labeled, so we will describe it verbally and hope it will make sense.
We start with a square and dissect it by drawing 4 segments, each connecting a vertex to a midpoint of a side.

THE CLASSIC
Explain why the area of the shaded region is one-fifth of the area of the original square.

Notes/Comments:
(1) This is a wonderful exercise to develop spatial reasoning and to demonstrate a visual approach to a geometry problem when dimensions are not given. Of course, one could use an algebraic or numerical approach if one chooses.
(2) Students who 'see' the jigsaw puzzle approach of rearranging the pieces rarely consider what assumptions are being made. To make the problem even more meaningful, the instructor could ask why the shaded region is, in fact, a square.
(3) Simpler versions of this often appear on the SATs.

VARIATION #1

This time, both diagonals are drawn. The additional two segments join the midpoint of the bottom side to the midpoints of two other sides.

(a) The red shaded region (does it have to be a square?) is not one-fifth of the original square. What fractional part is it?
(b) The total shaded area is what part of the original square?

VARIATION #2

This time the smaller segments divide the sides into a 1:2 ratio. The figure is not drawn to scale. The 3 segments on the base are supposed to be equal!

(a) The blue shaded region (is it a square?) is now what fractional part of the original square?
(b) The total shaded area is now what part of the original square?



THE GENERALIZATION OF VARIATION 2
Use the diagram from Variation 2. Assume the original square has a side length of 1 unit. If the smaller segments divide the sides of the square into an x:(1-x) ratio, do parts (a) and (b) again, expressing your results in terms of x. What restrictions on x make sense here? Make sure your expressions agree with the results above.

A 'Simple' Traversal through a Number Grid -- Patterns, Functions, Algebra Investigation Part I

Here is an activity for Prealgebra and Algebra students. This introductory activity is not meant to be a conundrum for our crack problem-solvers out there, but the extensions below may prove more challenging.



Target Audience: Grades 6-9 (Prealgebra through Algebra 1)

Major Standards/Objectives:
(1) Representing numerical relationships and patterns algebraically
(2) Recognizing, interpreting and developing function notation
(3) Applying remainder concepts

A 2-column number matrix (grid) is shown above and assumed to continue indefinitely. We will be visiting (traversing) the numbers in the grid starting in the upper left corner with 1. Following the arrows we see that the tour proceeds right, then down, followed by left, then down and repeats.

First, some examples of the function notation we will be using to describe this traversal:
T(1) = 1 denotes that the 1st cell visited contains the number 1.
T(4) = 3 denotes that the 4th cell visited contains the number 3.
Similarly, T(6) = 6.

STUDENT/READER ACTIVITY/INVESTIGATION

(a) Determine T(1), T(5), T(9), T(13), T(17).
(b) 1, 5, 9, 13, 17, ... all leave a remainder of ___ when divided by 4. (Fill in the blank)
Therefore, these numbers can be represented algebraically as 4n + 1, n = 0,1,2,3,...
(c) Based on (a) and (b), it appears that T(4n+1) = _______, where n = 0,1,2,3...
(d) Determine T(2), T(6), T(10), T(14)
(e) 2,6,10,14,... all leave a remainder of ___ when divided by 4. Therefore, these numbers can be represented algebraically as ______, n = _________ (Fill in blanks)
(f) Based on (d) and (e), it appears that T( _____ ) = _____, n = __________.

Note: The instructor may choose to start n from zero or one throughout this activity. I will vary it depending on our needs. It is important for students to see how restrictions (domain of a variable) is critical for an accurate description and that more than one set of restrictions is possible (provided they are equivalent).

Since T(3) = 4 and T(4) = 3, we cannot say that T(n) = n for all n. The numbers 3 and 4 leave remainders of 3 and 0 respectively when divided by 4. We will need a different rule for these kinds of numbers. Let's collect some more data:

(g) By extending the table, determine T(7) and T(8); T(11) and T(12); T(15) and T(16)
(h) Without extending the table, make a conjecture about the values of T(35) and T(36).
(i) Numbers such as 4,8,12,16,... can be represented algebraically as ____, n= 1,2,3,...
(j) Numbers such as 3,7,11,15,... can be represented algebraically as ____, n = 1,2,3,...

Note: Again, the instructor may not like varying the restrictions here. Adjust as needed.

(h) Ok, so you're an expert now. Well, prove it:
T(100) = ______; T(153) = _____; T(999) = ______
Show or explain your method.

EXTENSIONS

Surely, a 3-column number grid or even a 5-column number grid can't be that much more difficult to solve using the same kind of traversal (move to the right until you come to the end, go down, move left until you come to the end, move down, lather, rinse, repeat...). ENJOY!

Ok, for our experts: Try an n x n grid!

DISCLAIMER: As with all of the investigations I publish, these are essentially original creations and therefore have not been proofread or edited by others. You are the 'others!'. You may not only find errors but alternate and perhaps superior ways to present these ideas.
Also, please adhere to the Guidelines for Attribution in the sidebar.

Odds and Evens - Week of 3-24-08

We've already had a couple of submissions for our Mystery Mathematician. I selected him for several reasons - see if you can intuit some of these!

• Take a look at the 29th edition of the Carnival of Mathematics over at Quomodocumque, an interesting 'non-blog' written by a professional mathematician who has a unique perspective on the role of blogging in math research. This edition of the Carnival is weighted in favor of research mathematics, not that there's anything wrong with that! Enjoy the quality and variety of excellent posts. I particularly like the name of the host's web site. Now whenever one of my adolescents utters the classic 'whatever', I can reply "quomodocumque!"
  

• One of our Mystery Math solvers shared a fascinating link to Dr. Hung-Hsi Wu's web site. Dr. Wu from the U. of California, Berkeley, has been a major voice in the counter-reform movement in math education. I particularly encourage readers to scroll down the page until you reach his 2006 paper on Professional Development: The Hard Work of Learning Mathematics. I may have more to say about this.
• Now that the National Math Panel has released its final report, entitled Foundations for Success, it behooves me to comment, considering all of my past correspondences with the Panel. There have already been reactions on several leading blogs including Edspresso and Joanne Jacobs.
  
• Still working on developing some rudimentary math casts using Mimio technology and SnapZ Pro X image capture software. I just don't have enough hours in the day!
• I'm also planning a review of Explore Learning, an excellent web-based set of interactive simulations for math and science. I'm in the process of using some of their Gizmos and I've received helpful background information directly from the company. I'm sure many of you are familiar with this product and will add your thoughts.
  
• I'm working on an in-depth investigation relevant to the Patterns, Functions, Algebra strand in most standards. It starts out as a simple winding traversal through a 3-column number matrix, then onto 5 columns and perhaps beyond. If you've ever counted on the fingers of one hand, then you know how easy this is: 1-2-3-4-5-6-7-8-9. Uh oh, I only have 9 fingers...
  

Mystery Math Icon Week of 3-24-08

Several compelling reasons for our choice of this luminary. In my opinion, he could run rings around anybody! Email me at dmarain at gee-mail dot com with your answer and an interesting anecdote. If this is your first submission, please include some background about yourself as well.

HOW MANY FASCINATING FACTS can you find about 97? about 153?

Note: There has been a revision in one of the properties of 17 below - I'm sure you already caught the error! Joshua caught another error involving the 4th powers - both have now been corrected.

For middle or high school students:

Number sleuths -- get into your detective groups. No calculators for the first 5 minutes.
You will have a total of 15 minutes to uncover as many interesting or fascinating facts as you can about the number 97.
One member of the group must record these and report back. Make two columns.
The first labeled: Discoveries found without the calculator.
The other column: Discoveries found with the calculator.

Ah but you're wondering what makes some fact interesting or fascinating. That's pretty subjective, right? Let's model one:

17

• Prime number
• Both of its digits are odd
• The product of its digits is prime
  
• Can be written as a sum of 2 consecutive integers: 17 = 9+8
• If its digits are reversed, the resulting number, 71, is also prime and the difference of 71 and 17 is 64 or 8²; oh, and 8 just happens to be 1+7!!
  Note: This is clearly incorrect - 3-point penalty!!
  
• Can be written as a sum of squares: 17 = 4² + 1²
• Can be written as a difference of squares: 17 = 9² - 8²
• Can be written as a sum of a perfect square and a perfect cube: 17 = 3² + 2³
• Can be written as a sum of 2 consecutive fourth powers: 17 = 1⁴ + 2⁴; in fact, 17 is the least integer which can be written as the sum of two distinct nonzero 4th powers
  
• You can get your driver's license in some states on your 17th birthday (this is clearly the only fact that's interesting to an adolescent - I had asked one of my students what's special about March 14 and he replied, "It's my birthday and it's also the Junior Prom.")
  
• 17 is the hypotenuse of a Pythagorean triple; 8-15-17 [high school level?]
  
• 17 is part of another Pythagorean triple: 17-144-145 [high school level?]
  
• Is there any end to this list?

Well, you get the idea.
You will have 15 minutes to complete your investigation of the number 97.

To rate each of your lists, we can assign a point value to each fact. For example, the first four facts listed above for the number 17 could each receive one point. Some of the other facts could receive 2, 3, 4 or even 5 points (max) for being more difficult to find or just more amazing. If any fact is incorrect, 3 points are deducted, so you'd better do independent fact-checking on your team! We will then determine the top 3 lists from this rating system and those teams will receive worldwide recognition by having their results appear on You Tube (just their lists, not names or faces for reasons of confidentiality of course). This will surely go viral in 17 seconds or less. Oh, so you'd rather receive 97 bonus points on your next test?

Oh, I forgot to mention. As a super extra credit bonus project/assignment, do the same this evening for the number 153. You will need to email me with your individual lists by 10 PM (yes, yes, we all know this is unfair to students who do not have access to email or whose parents may not give permission for this or for some other reason, so this is just an option!).

A, B are two points on a circle... Extending Student Thinking in Geometry

One of my stellar SAT students (who has been mentioned before) led me to develop the following extension of a challenging SAT-type question. Problems similar to this have appeared on previous SATs and math contests. All geometry students can benefit from both the spatial reasoning inherent in this problem as well as the algebraic analysis needed to demonstrate the results numerically.

Part of the issue of problems like this is that test constructors often attempt to develop questions similar to previous questions that had proved effective. An earlier version of the problem below read something like this:

A, B are 2 distinct points on a line. How many points on the line are twice as far from A as from B?

Occasionally, the ratio changes: three times as far from A as from B...

I've even seen other locus versions of this basic premise:

If l and m are parallel lines in a plane, how many lines are twice as far from l as from m?

In both cases, there are two solutions, one involving internal division and the other involving external division.

One can make this a bit more challenging by asking the more general version:

How many points are twice as far from one point as from the the other. (Similarly for the parallel lines problem).

By symmetry, there would now be 4 solutions. Nice questions that promote spatial sense and the ideas of internal and external ratios. We could stop there, but changing the line to a circle adds another dimension to the question:


The Challenge:
A and B are distinct points on a circle of diameter 2.
The length of arc AB is 1.
Note: For this question, all distances are arc lengths, not chord lengths.

(a) How many points on the circle are three times as far from A as from B?

(b) Now specify the locations of such points. Thus, if P is a point such that the length of arc PA is three times the length of arc PB, determine the position of point P in each case and the actual lengths of these arcs.


Comments:
Do you think most students or even the author of this question (I've altered the original problem slightly and extended it with part(b)), considered what the student with greater insight might have considered, namely, the issue of major vs. minor arcs? Do some students see the more profound subtleties of these questions -- the ambiguities that the author or others do not consider? Worth discussing the topological similarities and differences between a circle and a line?

Carl Gustav Jacob Jacobi 1804-1851 - Our Mystery Mathematician Revealed

A ∂ listing of his accomplishments...

• Jacobi's Elliptic Functions (applying elliptic functions to number theory used to prove such results as the Fermat's 2-square and 4-square theorems)
• One of the early founders of the theory of determinants; in fact, he invented the n x n functional determinant formed by the differential coefficients of n functions of n variables (think of the Jacobian in those multiple integrals when transforming variables)
• Jacobian
• Jacobi Symbol
• Jacobi Identity
• Jacobi Integral used for the sidereal coordinate system in astronomy
• Jacobi Method for eigenvalues
  

Of course I always lean toward any mathematician of whom it is said:
"...widely considered to be the most inspiring teacher of his time" and, oh, by the way, "one of the greatest mathematicians of all time..."
Further, there is no documented evidence of any steroid use during his most prolific period!

And our winners are...

TC

Hi Dave,

That is Carl Jacobi of the Jacobian fame. Interestingly, when my
daughter was first introduced to the Jacobian when dealing with
cylindrical coordinates, she was just told that she needed to include
a Jacobian when she changed variables in the integral, and for this
case the Jacobian was 'r'. It was only a few classes later that the
actual computation was introduced.

I guess one reason you picked him was by tracing back from Ito's
stochastic differential equations; you logically get to a pioneer in
partial differential equations.
--------------------------------------------------------------------------------------------
Kevin

Dave:

KARL GUSTAV JACOB JACOBI

I include just this insightful quote:

From a paper by François Ollivier:

It is said that Jacobi once told to a student who wanted to read all the mathematical
literature before starting his research: "Where would you be if your father
before marrying your mother had wanted to see all the girls of the world?"

Kevin

At r% interest compounded annually, $400 earns $63.05 interest in 3 years. r = ?? Developing Greater "Interest" in Algebra...

SILLY RIDDLE OF THE WEEK
Why were the Romans so good at algebra?
You have to think outside the box and be in the mood for this groaner! Of course you've probably seen this elsewhere on the web...

It's been awhile since we've worked on financial math applications. Anyone recall those 3 mortgage investigations from last year? [Note: To see other mortgage/finance posts, click on the mortgage label/tag in the sidebar].
Considering the current economic situation, perhaps we should devote more attention in our math classes to the subtle trap of running up credit card debt. I'm working on that. There are strong mathematical similarities between loans, mortgages and investments and in this investigation students will focus on the investment problem in the title of this post.

The Problem in the Title of this Post:
At r% compounded annually, $400 earns $63.05 interest over 3 years. What is the value of r?
Let's agree, that r% has already been converted to a decimal so that we do not have to work with r/100 in the formulas below. That is, if r = 10% for example, we will work with r = 0.1.


OVERVIEW OF ACTIVITY
We will first consider a quick estimate of the interest rate by using simple interest to approximate compound interest. This develops sense about the formulas and could be helpful if a question like this appears as a multiple choice question on the SATs or other standardized tests. We will then apply standard compound interest formulas to validate our estimate. Students will be asked to use more than one method for this. Finally, there will be an extension for your students to try.
----------------------------------------------------------------------------------------------------
KEY for this activity (not necessarily standard notation)
[Assume one interest period per year; no additional money deposited or withdrawn]

P = original amount invested (principal)
r = annual rate of interest (decimal form)
n = number of years
A_n = Amount original money is worth after n years
I_n = Interest earned during the nth year
T_n = Total Interest earned over n years
-----------------------------------------------------------------------------------------------------
Background for Simple vs. Compound Interest

Simple: Interest each year is constantly Pr so total interest for n years is T_n = Prn.
Example: If $400 is invested at 10% annually simple interest, then over 3 years one would earn (400)(0.1)(3) = $120 in interest.

Compound Interest
Example: Suppose $400 is compounded annually at 10%.
1st year: Interest earned = I₁ = (400)(0.1) = $40; money is now worth A₁ = $440.
2nd year: Interest earned = I₂ = (440)(0.1) = $44; A₂ = $484
In general:
A₁ = P + Pr = P(1+r)
A₂ = P(1+r) + rP(1+r) = P(1+r) (1+r) = P(1+r)²
(*) A_n = P(1+r)ⁿ

Beginning of Activity
I. Approximating the Rate using Simple Interest:
If the total interest over 3 years is about $63, show that r = 0.05 is a reasonable estimate for our problem using the simple interest formula above.

II. Using Compound Interest Formula
There are several approaches to solving the title problem:

Method I: Use the above compound interest formula (*) directly to solve for r.
Remember: The formula expresses A_n but it's the total interest, T_n that's given.

Method II: Derivation of Related Formulas

(a) Show that or explain why the total interest earned after n years can be expressed as
T_n = P[(1+r)ⁿ - 1].
(b) Use the formula in (a) to solve for r in our problem. Here you will be substituting the values for n, P and I_n first, then solve for r.
(c) Alternate Approach: Use the formula in (a) to derive a general formula for r in terms of n, P and I_n. Then use this formula to find the value for r in our problem. When do you think it makes more sense to use (b)? (c)?

Extension
In the above problem, we knew what the total interest was after 3 years and we needed to manipulate a formula to determine the rate. In other applications, we might want to determine the interest earned each year. This is usually done for us by our bank -- we certainly need this amount for federal and state income taxes. We will now derive the formula for I_n by two different methods:

(a) Derive a formula for I_n using the fact that I_n = A_n - A_n-1, for n = 1,2,3,...

(b) Derive a formula for I_n using the following pattern:
I₁ = rA₀ = rP = rP(1+r)⁰
I₂ = rA₁ = rP(1+r)¹
....
In general: I_n = ____________.
Note: This formula makes sense. Why? Can you show that the results in (a) and (b) are equivalent?

(c) For the original problem in the title of this post, complete the following table:

n................A_n....................I_n
0...............$400...............

1...............$400...............$40

2

3
.
.
.
10


Comments:

• The instructor may choose to use this activity to develop recursive functions. For example, A_n = (1+r)⋅A_n-1
  
• The chart above can be generated using the graphing calculator of course. More importantly, ask students to discover relationships among the columns.
• Much of the above is standard 'stuff' and not very challenging. However, the goal here is to help our students develop a feel for these formulas, rather than mechanically 'plugging in.' Considering that this topic is related to exponential functions, recursive thinking, and geometric sequences, there is unlimited potential for bringing more financial math into the algebra or precalculus classroom. And, yes, it's all standards-based...
  

The Best Pi Quiz on the Web?

I strongly encourage you or your students to try Eve Andersson's Pi Trivia Game.

I tried the quiz on Tue and got 21/25 right. I'll bet some of you get a Pi-Fect score the first time! Some of the questions are easy but many require technical or historical knowledge of pi. Many of the questions involve fascinating facts about π, so it is both educational and fun. The multiple guess format makes it less intimidating but the distractors are tough to crack! As soon as you submit your answers, you get immediate feedback.

What is really nice is that the quiz changes when you refresh the screen, since the questions are randomly generated from her large database. There will be some repetitions of course, but there are enough questions to keep it interesting and any duplication will test your recall!

Also visit her main site, Pi Land, to get more background and some excellent book references for the mathematics and history of Pi.

This is probably not intended for middle schoolers, although I'm guessing one could do some web searches to find most of the answers. Might be a fun activity for your students. If you have a bank of computers in the classroom, tell your group they can try the challenge in the last 10-15 minutes of class after they've completed their work. Highest 3 scores get a Pize - ugh...

51+52+53+...+100 is how much more than 1+2+3+...+50? Why, 50^2 of course! Now Explain and Generalize...

Quick Updates....
Mystery Mathematician Contest ending soon...
Pi fact for today? Try explaining why the imaginary number i raised to the power of i is REAL without mentioning π somewhere! Of course, you could just ask Google to do it for you!


You'd think that the deafening silence from the 5-7-8 triangle post would discourage me - NOT! Here is an investigation for middle schoolers and up.

Typical Content Standard: Patterns, Relations, Algebra

Objectives:
(1) Developing strategies for comparing sums
(2) Developing algebraic generalizations
(3) A few dozen more!

Where might the first question in the title of this post be asked?
(a) SATs?
(b) End of Course Test for Algebra 2?
(c) Other standardized tests?
(d) Math contests? If so, what grade level? 7th? 8th? Higher?

If you value a question such as this, would you introduce it to middle schoolers in 6th? 7th? Prealgebra? Would you use a very different instructional approach with students in higher math courses who have reasonable algebra background? Even if you don't like this question, try it with one of your groups tomorrow and let me know what happens!

Since I have personally posed this type of question to both middle schoolers and older students, I can tell you that even strong math students often have not seen the 'compare differences of corresponding terms method'. I made up that designation but, hopefully, you can make sense of it. Do you think many high school students would attempt to find two separate sums by some method/formula (or using their calculator if allowed) they've seen?

Well, I won't give any more away, but I believe the issues of pedagogy here may transcend the problem and the math strategies:

How does one introduce this? Do you simply have this question on the white board as students enter the room and allow them to work on it individually or in small groups for 5-10 minutes? We all hear about our re-defined role as 'guides on the side' but what exactly does this look like for this activity? How do we facilitate? When do we ask leading questions? What questions would be highly effective here? I haven't even mentioned the calculator issue yet!

So many questions. So few answers...

Actually I was going to do a short video presentation of this question to demonstrate one instructional model, but, unfortunately, my dog ate my main computer which has all my files and applications. Wait - let me apologize to my pooch. He really didn't eat it or even bless it with his bodily functions. But my iBook is very sick and will need intensive care from Apple. In the meantime, I'm on a backup machine, with limited memory and lacking many of my files and applications. Excuses, excuses, excuses! Please bear with me!

A 5-7-8 Triangle for Starters -- An In-Depth Exploration in Algebra, Geometry and Trigonometry

As promised, here is an investigation/activity/challenge/.../ (Anyone remember a certain star college and pro football player, with initials K.S., who was nicknamed 'Slash' because he could play several positions?).

The following extensive activity is designed as a long-term assignment, however, modify it as you wish. Don't forget to forgive proper attribution as indicated in the sidebar.


A Possible Instructional Scenario...
"Girls and boys, what do you think of when you're told that a triangle has a 60° angle? What if you're given that all the sides have integer lengths? Why can't it be 30-60-90 in that case? Of course, it could still be equilateral, but for this exploration, we are looking for scalene triangles with integer sides and a 60° angle. By the way, if it's not equilateral, how do we know it must be scalene? Why couldn't it be isosceles?"
------------------------------------------------------------------------------------------------------------
The Exploration

Let's agree on the following labeling for our triangle:
ΔABC with ∠C = 60° with opposite side c. Other angles are labeled A and B with opposite sides a and b, respectively. For this investigation, b > a.

(a) Show that the angle opposite the '7' side in the 5-7-8 triangle in the title of this post is 60°. Using the labeling above: If a=5, b=8, c=7, show that ∠C = 60°.

Note: This can be done with or without the Law of Cosines. As Joshua pointed out in a previous post, the student can work with the altitude on the '5' side and use the Pythagorean Theorem and algebra to show that a 30-60-90 triangle is formed. This method is instructive (and constructive too!).

For parts (b) - (e), we will no longer be focusing only on the 5-7-8 triangle. Consider any integer-sided ΔABC with ∠C = 60° and with opposite side of length 7.
(b) Show that a² + b² - ab = 49. [*]
See note after part (a) for two possible methods.

(c) We are looking for integer solutions to [*]. There are several methods including a Pell equation approach (we will not go in that direction). Students can certainly try a guess-test strategy, however we can initially restrict the possible values of b, can't we?
As an initial boundary, show that 13 >b ≥ 8 using basic geometry.

(d) Show that a better restriction for b is 14/√3 ≥ b ≥ 8 by:

(i) [Trig] Using the Law of Sines
(ii) [Advanced Algebra] Use the quadratic formula in [*] to solve for a in terms of b. Using the discriminant, show that b ≤ 14/√3.

From this result, explain why it follows that b must equal 8.

(e) Solve for a.
Suggestions: From [*] OR use your result from the quadratic formula in (d)(ii) to show that there are exactly 2 scalene triangles satisfying the above conditions.
The answers for this are:
a=3, b=8, c=7;
a=5, b=8, c=7
BTW, is it a coincidence that the two values of a happen to add up to b?

(f) REPEAT PARTS (b) - (e) for an integer-sided scalene triangle with a 60° angle and an opposite side of length 13. There will be some slight modifications needed such as formula [*]. State and use a similar inequality from (d) to show that there are two possible values for b, namely 14 and 15. Go further and show b=14 is not possible (e.g., using the discriminant).
Then show that b = 15 leads to two solutions (triangles) - sorry, I'm not giving these away yet!

(g) REPEAT PARTS (b) - (e) for an integer-sided scalene triangle with a 60° angle and an opposite side of length 19. Again, you should find that the inequality from (d) leads to two possible values for b, only one of which works. Give the two solutions (triangles).

(h) Some of you will no doubt wonder why we did 3 separate analyses, when a slightly more general approach could have been used, specifically a more general inequality in part (d). Ok, so do that (keep all conditions, except side 'c' will now be a parameter).
Show that, in general, 2c/√3 ≥ b ≥ c.

(j) Surely, we can't go further other than searching for a general solution for the 60° problem. Of course, we can: Come up with similar questions and solutions to all parts above if ∠C = 120°. I'll start you off: a=3, b=5, c=7. Show that ∠C = 120°, etc...

I'm sure our astute and talented readers will pick up on my usual errors or omissions or make suggestions to improve the flow of the activity. I'm counting on you! Enjoy...

Poe, E.: Near a Raven - "Poe"-tic Justice and a Tribute to Pi and Mike Keith

To kick off Pi Week, I cannot ignore the most famous Pi Poem I have ever read - Mike Keith's web site is worth visiting. He is a unique and talented individual who has written several excellent books. On his home page, he refers to his creations as "mostly original diversions in mathematics and word play." I agree!

Here are the first couple of verses of his poem (he refers to the style as constrained writing) he wrote over a dozen years ago. Again, go here to see the entire opus! I'm sure most of you are thoroughly familiar with this, but it's always worth seeing it again. If you're not familiar with the code beneath this, just count the number of letters in each word, starting with the title:
Poe:3
E:1
Near:4
a:1
Raven:5

Poe, E.
Near a Raven

Midnights so dreary, tired and weary.
Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap - the weirdest tap!
An ominous vibrating sound disturbing my chamber's antedoor.
"This", I whispered quietly, "I ignore".

Perfectly, the intellect remembers: the ghostly fires, a glittering ember.
Inflamed by lightning's outbursts, windows cast penumbras upon this floor.
Sorrowful, as one mistreated, unhappy thoughts I heeded:
That inimitable lesson in elegance - Lenore -
Is delighting, exciting...nevermore.

...

My 2nd favorite Pi-Poem is from Liz:
She has an excellent website -visit it!
Here is an excerpt of her ode to π:

Why, π! Stop, π! Weird anomalies do behave badly!
You, madly conjured, imperfect, strange, numerical,
Why do you maintain this facade?
In finite time you are barbaric!
You do wonders, mesmerize minds!
....

You may want to start your own Pi Poem contest this week, although this idea is coming very late. Somewhere out there is a hidden talent, perhaps another genius like Mr. Keith.
If you do this, please share some of your favorites with MathNotations....

Geometry WarmUp - A Simpler Integer Triangle Problem

While we're waiting for the 60° integer triangle problem, here's an easier one for both middle schoolers and secondary students. The only fact from geometry that is needed is the all-important triangle inequality:
Any side (in particular, the largest side) of a triangle is less than the sum of the other two sides.
Of course this refers to the lengths of the sides and one can express this in other forms, but I'll leave it at that.
This type of question has become a favorite on the SATs and other standardized tests but, more importantly, it develops clear systematic thinking - the organized list....

How many different triangles have integer side lengths and a perimeter of 5? 10? 15? 20? 25?

COMMENTS/INSTRUCTIONAL HINTS:

• There are really five separate questions here. The instructor can give some or all of these depending on the time allotted. To help the group get started and for clarification, it may be helpful to demonstrate the first question for the group: For a perimeter of 5, there is only one possible triangle, which we can symbolize as {2,2,1}. If these are older students who are comfortable with the triangle inequality, you do not necessarily have to model this one, but that's your call. By modeling the first one, you eliminate some of the ambiguity of ordering the sides.
  
• Since a primary objective here is to make an organized list, you may want to stop after the perimeter of 10 and discuss it at the board. Depending on the ability level of the group, I usually have students work independently, then check each other's work in pairs after they do a couple of these questions. Sort of a think-pair-share approach. Also, don't be afraid to provoke their thinking with questions as they begin to develop their systematic lists (which can get boring for some): "So, do you expect more triangles for a perimeter of 10? Twice as many?"
  
• As each question is reviewed, encourage students to record their results in a table:
  Perimeter..................Number of Triangles
  ......5........................................... 1 ................
  ....10.......................................... 2 ................
  This is critical for middle schoolers in particular, since tables are a basic model for functions! At some point, you can use n or p for the perimeter and symbolize the number of triangles having perimeter n or p as T(n) or T(p).
• Naturally, some students will assume there is a pattern and guess there are 3 possible triangles with a perimeter of 15 - NOT! However, it is natural for all of us to ask: "WHAT'S THE FORMULA?" Well, there is one. It's fairly sophisticated and related to partitions of numbers, but I'll let our readers do their own research for this...
  

Odds and Evens: Week of 3-10-08

Don't forget to submit your solution to the Mystery Mathematician of the Week! So far, one correct solution has been emailed to me. Remember to email me at "dmarain at gee-mail dot com." Slight hint: Our star this week delved into many areas of mathematics including those special types of equations we have been
discussing...

• Yes, Pi Day is coming. I hope to have something special up for that occasion. It would have been 'pi-fect' had our next Carnival of Mathematics coincided with that event, but I'll leave it to our crack math team out there to project in what year the next Pi Day will match with one of our Carnivals!
  
• Unfortunately I missed submitting to the perfect 28th edition of the Carnival over at Tyler and Foxy's Scientific and Mathematical Adventure Land. That title by itself should win an award!
• Have you been keeping up with the comment thread to our recent post regarding an integer parallelogram with sides of 39 and 25 and with a diagonal of 34? TC contributed a couple of beautiful variations, one of which was easily solved by Joshua. Joshua also gave a thorough solution to the original problem. Eric provided the theoretical background for solving these kinds of Diophantine equations. I figure if I compile Eric's comments for the past year, I would have one amazing textbook for algebraic number theory!
  
• Stay tuned for an investigation that builds on the integer parallelogram problem as mentioned in the comment section. This one involves finding a limited number of solutions to integer-sided triangles which have an angle of 60 degrees. Of course the trivial solution would be the equilateral triangle but we're looking for scalene triangles here. A little bit of trig and some algebra and geometry will be required. The general solution requires, you guessed it, Pell's equation, but we will not go there!
  

A Parallelogram Has Sides of Lengths 39 and 25 and a Diagonal of Length 34. So, What Makes It So Special!

Thanks to TC's inspired challenge to our readers in a comment on the Medians of a Triangle post, I've decided to expand it into an investigation for our readers and students (geometry with some trig needed).

Consider a parallelogram whose sides have lengths 39 and 25 and with one diagonal of length 34.

(a) Explain why this parallelogram is unique, i.e., all parallelograms with these characteristics are congruent. Why was it not necessary to specify that the 'shorter' diagonal was given?

(b) A parallelogram has sides of lengths a and b and diagonals of lengths c and d. Use the Law of Cosines to show that
c² + d² = 2(a² + b²).

(c) Determine the length of the other diagonal. As an alternative, how would you do it without the formula in (b)?

(d) Determine the area of this parallelogram.

(e) So what makes this parallelogram unusual?

Comments:
(1) From the comments on the Medians post (Cotton Blossom and others), we know that we can construct rectangles and rhombuses whose sides and diagonals have integer lengths, but the above demonstrates a parallelogram that is neither of these special cases.
(2) The formula in (b) is not too difficult to prove, however, finding solutions to this Diophantine equation or a general solution is far more challenging!
(3) Note that the parallelogram in this challenge also has integral area. Finding other such parallelograms is not a simple exercise!

Calculus Humor?

From Savage Research, Humor...

Math Knowledge

Two mathematicians were having dinner in a restaurant, arguing about the average mathematical knowledge of the American public. One mathematician claimed that this average was woefully inadequate, the other maintained that it was surprisingly high.

"I'll tell you what," said the cynic. "Ask that waitress a simple math question. If she gets it right, I'll pick up dinner. If not, you do." He then excused himself to visit the men's room, and the other called the waitress over.

"When my friend comes back," he told her, "I'm going to ask you a question, and I want you to respond `one-third x cubed.' There's twenty bucks in it for you." She agreed.

The cynic returned from the bathroom and called the waitress over. "The food was wonderful, thank you," the mathematician started. "Incidentally, do you know what the integral of x squared is?"

The waitress looked pensive; almost pained. She looked around the room, at her feet, made gurgling noises, and finally said, "Um, one-third x cubed?"

So the cynic paid the check. The waitress wheeled around, walked a few paces away, looked back at the two men, and muttered under her breath, "...plus a constant."


I'm sorry, but that did make me smile! Reminds me of when I was teaching calc, I would tell my students that if they forgot the +C in an indefinite integral, their grade would be C+! Actually, I wasn't kidding...

BTW, there are many more of these at the above web site. Many are one-liners with that twisted sense of humor characteristic of Steven Wright or Jackie Vernon. I will not apologize for laughing!

Here are a few more...

1) Save the Whales -- collect the whole set.

2) If you believe in telekinesis, raise my hand...

3) The early bird may catch the worm, but it's the 2nd mouse that gets the cheese.

Ok, enuf' already (for now)...

Mystery Mathematician Week of 3-3-08

It's that time again! I suspect several of our talented readers will identify this transcendent mathematician but remember the following:
(i) Don't name him in a comment! Please email me at "dmarain at gee-mail dot com".
(ii) Include interesting facts, contributions or personal anecdotes about our mathematician. I chose him for several reasons. Can you guess one or two of them?

Another Math League Challenge - Medians of a Triangle...

As promised...

In a triangle, two sides and the median to the third side have respective lengths 5, 13, and x. List all possible integer values of x.

“Copyright Mathematics Leagues Inc 2007. May not be reproduced without permission of the copyright holder.”


Comments:

• There is an elegant solution/proof here! I found a far more complicated coordinate argument.
  
• Some students might make an educated guess based on insight but the real challenge is to explain one's conjecture!
  
• Challenge yourselves with this - if you see how to do it, suggest an approach but don't give it all away for at least a few hours!
• We generally think of these as only appropriate for our best and brightest but there's no harm in throwing it out to all of our geometry students- amazing things happen when we open up these kinds of problems to everyone!
  

SAT-Type Algebra Challenges - How Would You or your Students do on these?

Of course this comes a bit late for all those juniors who took their SATs today but the following questions can be used to prepare for the next one OR for anyone who wants to challenge themselves or their students...


-4 ≤ P ≤ 3 and -5 ≤ Q ≤ 4


(i) What is the greatest possible value of (Q+P)(P-Q)? Explain your reasoning.

(ii) What is the least possible value of (Q+P)(P-Q)? Explain your reasoning.

Comments:
(a) SAT questions don't ask for explanations but this goes beyond that.
(b) These would be known as 'grid-ins' or student-constructed response questions. On a real SAT, answers to these must be greater than or equal to zero.
(c) The intent here is to go beyond the typical 'plug-in' methods most students use. One can apply actual skills and reasoning!
(d) Even the strongest students fall into a 'trap' set in these questions. Try them in your classes!
(e) Do these kinds of questions develop algebraic reasoning and mathematical power OR are they just your typical 'tricky' SAT-type that has little value outside the test?
(f) Make up your own version of one of these or, even better, encourage your students to invent their own!