An SAT Problem or Teaching/Learning Beneath the Surface

Ok, so I don't have another anecdote from my grandson today so I'll have to get back to mathematics -- problem-solving, teaching and learning.

The video below deals with an algebraic equation in 2 variables which should be straightforward for your stronger Algebra II or Precalculus student. But will it be? I invite you to predict how many in your classes will answer it correctly, then try it out. After all, it is multiple choice, so statistically some should get it right by some means or other!

Hopefully, the purpose of the problem and the video will become clear to all of you. If we want our students to demonstrate better reasoning and an understanding of important ideas in math, we need to feel comfortable in teaching for meaning and understanding. This doesn't mean we stop teaching algorithms and procedures, however. Exactly what all this means and how to do it is the reason for this blog. I certainly never claimed to know the answers or any other mystical secrets. I only know that I never gave up trying.  Sometimes my efforts failed miserably, but I hopefully learned from these attempts.

It would mean a lot to me if you share your thoughts here or on my You Tube channel, MathNotationsVids, where you will find my other videos.




Note: Another subtle point I should have made in the video---
y(x-4) = 0 → y = 0 OR x = 4
It is important for us to stress this point and distinguish it from "AND" logic. If the equation were in the form:  y² + (x-4)² = 0, we would have (y = 0) AND (x = 4), whose graph would be the single point (4,0). Another instance where an exercise on the board can lead to a rich, fruiful and profound discussion. If all of this is seen as taking too much time away from content, remember this is precisely the kind of change in curriculum and instruction that Prof. Schmidt has been trying to tell us about for over 15 years! Well, I'm preaching to the converted, aren't I...
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"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Same 2-Year Old, Different Day!

With all the material I'm getting from my 2-year old grandson, I may as well give up my math blog and start a new blog featuring "anecdotes about grandchildren"! I betcha' some of these would go viral like no math article ever would!

Anyway, while I am preparing my next "exciting" video, I decided to share with you what my grandson said to his grandmother (better known as "Mimi" and also known as my wife) this morning.

His mother (my daughter) and grandmother (my wife) took him this morning for his first blood test (for allergies). My daughter needed "Mimi" to hold her son as she thought she would pass out!  So, my wife had him sit on her lap. When he realized that the needle was for him, the screaming began and he tried to jump out of "Mimi's" arms. Fortunately the whole procedure took only 30 seconds. My grandson jumped off my wife's lap, wiped his tears away and in a serious tone stated to Mimi, "You're fired!" My wife was stunned, never expecting to be fired from being a grandmother! There's definitely some Donald Trump in that boy.

Of course, a few minutes later all was forgiven and they all lived happily ever after.

I realize grandparents think all their grandchildren are special and anything the children say must be a gem worth repeating a thousand times. But, I do believe this young man is quite a character.  By the way, my daughter has no idea where he got the expression "You're fired!" from, unless of course he is watching the Apprentice on the sly!

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"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Two-year olds say the 'darndest' things!

A respite from Pi, sorta...

Ok, so I'm a mad crazy proud grandpa (Pop-Pop to be exact) but I could not resist telling the following story (Reader's Digest fare!)...

My daughter is a PTA rep and had to take her two-year old to the budget meeting with the superintendent yesterday. There were about 50 people in the room with my daughter up front. After the superintendent painted a gloomy picture of the needed budget cuts for next year, he asked if there were any questions. Everyone was quiet except for my grandson who raised his hand. The superintendent looked whimsically at him and asked him what his question was. He stood up and asked the superintendent if he was going to eat the blueberry muffin he had in front of him!

The superintendent graciously gave it to him which my grandson devoured in short order. After a few minutes, the superintendent kindly looked back at my grandson and asked, "Do you have any more questions?"  Can you guess the reply?  That's right, he asked if there were any more muffins...

Can you guess how to motivate students like this!




"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

6 Divided By Pi^2 and Relatively Prime Integers - A Video Derivation

Jaime didn't just teach math. Like all great teachers, he changed lives

– Edward James OlmosSOURCE: THE SAN JOSE MERCURY NEWS5 DAYS AGO

The current post presents a non-rigorous video derivation of a formula in mathematics which might create 'shock and awe' in kids of all ages. Oh, alright --  in people like me! This 3-part video is the followup to the post from March `16th -- Pi Day, More Videos on Counting, etc...



The following description comes from the YouTube Channel, MathNotationsVids:

Designed for anyone who has a passion for mathematics, this derivation of a classical result in math is suitable for advanced middle schoolers through undergraduate math. Further, teachers may want to show this to Math Clubs/Teams. This 3-part video builds on results from previous videos, is related to a post on MathNotations and is dedicated to the "Greatest Teacher in America" -- Prof. Jaime Escalante who passed away a few days ago.

Part I




Part II



Part III




Comments, Notes:

• I hope you will see in these videos a central theme beyond the content involved -- a fundamental heuristic in teaching mathematics: When introducing an abstract concept or in deriving a formula or theorem or rule, avoid heavy symbolism and work with simple concrete numerical cases before generalizing results. I believe this has validity at all levels of math instruction.
• This topic ties together so many apparently unrelated topics in mathematics in a wondrous and surprising way. Perhaps it will inspire a budding mathematician as it did me...
• Visit my YouTube channel (see above) and please comment on these if you feel you want to see more. They are fairly labor-intensive (do you really think I'm speaking extemporaneously!) but they are worth it if someone enjoys them. The quality of the videos can still improve much more but this is just a beginning...

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

Thank You, Prof. Escalante...

Professor Escalante,
So many lives have changed course because of your influence and inspiration. You gave me reason to continue teaching when I was questioning my life's choices. You so affected a great actor like Edward James Olmos that he wanted to be there for you over two decades after the movie was released.



You showed all of us the difference one person can make -- when that person is a teacher who would not compromise his standards, who believed in the inherent potential of each of his students and whose dedication knew no bounds. You waged war against ignorance, racism and mediocrity.

I am retired now and I still well up when I see the movie that, for me, will remain timeless.  As exceptional as the acting performances were, I came to believe that this was art imitating life -- that the real person behind the stage was even more remarkable than even Mr. Olmos' brilliant portrayal. As I replayed that movie year after year following the AP Exam, I sensed that each new group of students were also watching my reaction. I would interject personal comments throughout and they must have realized that this was far more than a movie to me. They must have known that I responded to certain scenes on a very personal level.

I knew from the first viewing that the movie was not only about the "greatest teacher in America", but was also the quintessential story of the life of every dedicated member of our profession. Thank you for validating all of our efforts.

Just as in the classroom, you were surrounded in the end by those who loved you. I wish to express my deepest condolences to your wife, children, grandchildren and your extended family. I am also deeply touched by Mr. Olmos' tireless efforts to making your last few months more bearable, although it saddens me that lack of funds made it difficult for you to obtain all the care you needed.

If it is not too presumptuous, may I speak for all those of our profession who feel indebted to you. May we now Stand in your honor and Deliver you to eternity...

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"When the Power of Love overcomes the Love of Power, the world will know peace..."


"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

Pi-Squared Over 6: The Algebraic Genius of Euler

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes multiple choice, I/II/III case types and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

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Yes, Pi Day 2010 is "history" unless of course you celebrate July 22nd! Then again, π is so universal in our world that "All π, All The Time" seems appropriate to me. That was a long-winded way of motivating a post about one of the most famous formulas in mathematics:

1/1² + 1/2² + 1/3² + 1/4² + ... + 1/n² + ... = π²/6

The videos below were inspired by one of my most faithful readers, Prof. Jablow. He brilliantly outlined Euler's derivation of the above formula in one of his comments.  I decided to develop it in more detail and provide some background for the younger student. Advanced middle schoolers through undergraduates in college may find this interesting. You might also want to share it with your math team/club.  The 2-part video presumes strong algebraic background and some knowledge of calculus although the latter is not necessary if you simply accept the well-known series expansion for sin(x). You may also find background and details in the excellent Wikipedia article, The Basel Problem.

As always, I add my disclaimer that I am solely responsible for any errors. I know there are a couple of errors in Part II, towards the end. They're pretty obvious and not serious, so I hope they won't ruin it for you! I invite you to comment on these videos both here and on my new YouTube channel, MathNotationsVids.  Of course, as I am finishing this post on 3-25-10 in the AM, YouTubew is down apparently worldwide, so I cannot embed these videos yet!!

Part I of Euler Video



Part II of Euler Video





FURTHER INVESTIGATION
Although the material on infinite series seems quite advanced, middle schoolers can use their graphing or scientific calculators to compute the sum of the first 10, 20, or even 50 terms of the series above. A simple program can also be written on the graphing calculator for summing the first n terms up to, say, n = 500 or 1000. Challenge them to see how "close" they can get to the decimal value of π²/6...
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"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

PI Day, More Videos on Counting, "Odds and Evens"

Since pi day fell on a Sunday this year, we should still be celebrating it today. Besides, March should be declared pi-Month!


It is always fascinating to see how readership (or should I say one-time viewership) always picks up around March 14th every year! I feel obligated to add another pi Day activity or exploration in addition to those I've posted the past three years. By the way, the pi Day Scavenger Hunt is the most popular post by far and I'm not even the one who thought of that idea!


Despite the title of this post, I did not upload a video for this activity. However, there is another video on the MathNotationsVids Channel on YouTube.

Here is an investigation/exploration/activity for middle and secondary:

Part (A)
(i) List all ordered pairs of positive integers (m,n) such that
(1) 1 ≤ m ≤ 10 and 1 ≤ n ≤ 10
(2) m and n are divisible by the same prime p

For example, (m,n) could be (6,9) since 6 and 9 are each divisible by the prime 3.

(ii) Should (9,6) also be counted?

(iii) Another way of expressing Condition (2) is:
The _______________ of m and n is ________  one.
Answer: gcf; not equal to or greater than

(iv) If you listed and counted correctly, you should have found there are 37 ordered pairs which satisfy both conditions. If not, have a partner check your list. Each of you should be checking each other's lists routinely.

Part (B)
(i) Explain, using the multiplication principle, why there are 100 ordered pairs which satisfy Condition (1) above.

(ii) ) What % of all the possible ordered pairs from Condition (1) are relatively prime. If you have immediate access to the internet, research this term before asking your teacher what it means!

(iii) In probability terms, you could say:

If one of the 100 ordered pairs (m,n)  from Part (A) is selected at random, the probability that
m and n are relatively prime is ____%.

Part (C) (more advanced)

If you have access to a graphing calculator, such as the TI-84 or TI-Inspire, enter the following program into memory (call it RELPRIME):

:ClrHome
:Prompt N
:0 → K
:For (X,1,N)
:For (Y,1,N)
:If gcd(X,Y) ≠ 1
:K+1 → K
:End
:End
:Disp K
:Stop

Using this program, complete the following table:

N..........Total # ord. prs..........# of not rel prime prs........% rel prime prs

10.........100.............................37....................................63%

20.........400............................ 145.................................

30

40

50

100

Notes:
K represents the count of ordered pairs which are not relatively prime
N represents the greatest value for the integers
gcd is found by going to MATH, then NUM, then 9:gcd(
The program slows down considerably as N increases. For N = 10, it checks 100 ordered pairs which may take only 2-3 seconds. For N = 100, it checks 100^2 pairs, which could take up to 4-5 minutes. Be patient!!

Conclusion: So what does all of this have to do with π ?
Well, as N increases without bound in the program, the probability that a randomly chosen ordered pair of positive integers (with values up to an including N) will be relatively prime approaches 60.7% rounded.

From out of the blue, compute 6/π²...
Want to know why? Well, that requires some advanced machinery involving infinite products, infinite series, and the Riemann Zeta Function! Perhaps, I'll do an informal development in a video. I love this stuff...


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"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Counting, Multiplication Principle, Pigeonhole Principle and Reasoning for Middle Schoolers and Beyond

UPDATE: SEE THE NEW VIDEO BELOW EXPLAINING THE PROBLEMS IN THIS POST. PLS SUBSCRIBE TO THE NEW MathNotationsVids Channel and share your comments and ratings!




The following video is available on my new MathNotations Videos Channel.


This particular video is a 10 minute discussion of developing the Multiplication Principle of Counting. It is designed more for the instructor than the student although it may be helpful in clarifying this important concept. The focus is on using multiple representations to reach the widest variety of learning styles. It is appropriate for any teacher of mathematics but particularly for the middle school teacher or those who work with students who struggle with math concepts.


After watching the video (or skip it if you wish) scroll down to the two problems below. These are more sophisticated than the one in the video and they require application of other concepts as well. I believe they are appropriate for 8th graders through high school. A full investigation with questions is provided for each problem. Feel free to edit them to your own tastes or as needed for your students.

Problem I

Mr. M told his Period I 8th grade math class about the following imaginary scenario...

Before the first day of school, Mr. Serling noticed that the names of the 26 students in his 1st period class had an unusual property. All of their initials (First Initial, Last Initial) came from the letters A, B, C, D and E. Furthermore, some had duplicate initials like B.B.

Part (a)
He now asked his actual class to make a conjecture:

Do you think it's possible that all 26 students in this imaginary class could have different initials (from each other)?  Write down your "initial" prediction (Y or N) on a slip of paper and fold it over. 

Part (b) Ok, now that you've made a conjecture, get into your learning groups of 4 and individually make a list of all possible sets of initials using the letters A, B, C, D and E with repetitions like "B.B." allowed as I explained before. Make sure your lists agree - edit as needed. Are your lists easy to compare? Why or why not?

HOW MANY DIFFERENT SETS OF INITIALS DID YOUR GROUP AGREE ON? ________

Part (c) Show your predictions to your partners and, in pairs, explain your reasoning why you would stay with your original prediction or change. Then write your reasoning as follows:

I believe that it is/is not possible for the 26 students to have different initials because ___________________________________.


At this point, Mr. M reviewed the Multiplication Principle of Counting (see the video above).

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The following problem may be assigned for classwork or homework after Problem I has been discussed in class. You could also use it as an assessment.


Problem II
Mr. M decides to assign to each student in his 5 classes a unique code consisting of up to 5 colors in sequence. He has a total of 129 students and the codes will use only the colors Red, Yellow, Green, Blue and Purple. Mr. M explains that codes may have repeated colors (like GGG or GYG) and RYG is a different code from YGR.

Will Mr. M run out of different codes for his 129 students? Explain your answer carefully, using a method similar to Problem I. 

Comment: I haven't mentioned how the Pigeonhole Principle can be applied to these two problems. I'll leave it to my astute readers to comment on that!




Ok, here's another video explaining the two problems  above. I hope you will subscribe to my new channel on YouTube, MathNotationsVids. 



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Note: I've been asked why I'm using these signatures on my posts, particularly the 2nd one. Well...
"It's my party and I'll try what I want to!"
(Apologies to Lesley!)

I'm sure some of my devoted readers can figure out why I included Schopenhauer's quote and the 2nd one is really all about education, isn't it?


"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific