Saturday, January 31, 2009

Was the First Super Bowl More or Less Than a BILLION Seconds Ago?

Hey, another kind of "over-under"!!
Just some food for thought to put the number one billion and history in perspective...

A billion seconds ago it was about 1976.
A billion minutes ago Jesus was alive.
A billion hours ago our ancestors were living in the Stone Age.
A billion days ago no creature walked the earth on two feet.
And a billion dollars lasts 8 hours and 20 minutes at the rate our
Government spends it.

There are many references for this on the web and I'm sure you've seen it before. You can check the accuracy (perhaps the last one is a little off!). I still like sharing this with students as it not only puts the concept of a billion in perspective but it does offer a wonderful application of 1-significant digit estimates, scientific notation and orders of magnitude. Can you imagine asking students to estimate that a billion seconds is roughly 32 years without a calculator!!

Where are these kinds of estimates currently in our math curriculum? More likely occurring in a science class? Do they belong somewhere in our classes or are they just amusing curiosities? You can guess where my thoughts lie!

Wednesday, January 28, 2009

Algebra 2 as a Graduation Requirement? Related Issues...

The deadline for registering for the First MathNotations Math Contest on Tue Feb 3rd is drawing to a close but you still have an opportunity to register! Look here for details and email me if you want a team of your students to participate!

As I've reported for some time (look here), here in NJ the Commissioner of Education has been promoting higher standards and more ambitious graduation requirements, choosing Algebra 2 as the cornerstone. I've had misgivings about this as a requirement for all students for several reasons although I'm a strong supporter of the American Diploma Project's Algebra 2 benchmarks and the End of Course Test for all students who choose to take the course because of their educational goals.

A recent article (1-27-09) on the website gives an excellent account of the debate raging over this topic at the State Legislative level. I will reprint a good portion of the article and then reprint the comment I posted on the site. I strongly encourage my readers to read the entire article and all of the comments posted thus far. It is a microcosm of much of the current debate in math education. Several of the commenters provided a commonsense view of these issues and gave me food for thought.

Education Commissioner Lucille Davy and a panel of education and business professionals appeared Monday before the Assembly Edu-cation Committee to discuss the Department of Education's High School Reform project.

The requirement that all students take algebra II has been controversial, and on Monday it dominated a discussion that attempted to identify just what students need to know to succeed and compete in the 21st century.

Davy insisted the algebra II requirement would not be so rigorous that it would lead to high rates of failure or students dropping out.

She said it would be a continuation of algebra I, but schools could offer more rigorous honors courses to those who would need them.

But Rutgers math professor Joseph Rosenstein, of the New Jersey Math and Science Coalition, wondered if the proposed courses might then get so watered down they would no longer really be algebra II.

"Most of our students don't need algebra II," said Rosenstein, who supports requiring more practical applied math courses.

Rosenstein said if courses were tailored just to meet state requirements, students who should take a true algebra II course might not get the higher level of work they need.

The algebra II issue has also frustrated vocational high school officials, who worry that too many requirements will make it impossible for students to complete programs in high school.

"These are students who benefit from applied learning," said Thomas Bistocchi, superintendent of the Union County Vocational School, adding that their goal is to have students graduate as industry-credentialed professionals. "We just want students who want to become plumbers have the time to do it," he said.

Davy said there will be flexibility in how the coursework is offered, so that it could be integrated into vocational coursework, but opponents wonder if that could be done and still teach what would be tested.

Stan Karp, of the Education Law Center, said reform is needed, but the state needs better education, not just more requirements. He said teachers and students will need better preparation to meet the new requirements.

"Less than half of the high schools now require those courses," he said. "What is it going to take to get there?"

Asked about the cost of reforms, Davy said the state already spends the most of any state and should not need more money, just a better reallocation of existing funds.

Business representatives said they just need students who can perform modern jobs.

Dennis Bone, president of Verizon, said students need the foundation of skills to be able to adapt to new and changing technology.

"We are being revolutionized by technology," he said. "Billboards now are electronic, run by someone sitting at a computer, not climbing a ladder."

"So what does algebra II have to do with that?" Education Committee Chairman Joseph Cryan, D-Union, asked.

Dana Egresky, of the New Jersey Chamber of Commerce, said that if taking algebra II can help a carpenter solve more problems on the job, then that is the carpenter who would get the job.

Assemblyman Joseph Malone, R-Ocean, Monmouth, Burlington, suggested asking professionals ranging from carpenters to doctors how they actually use algebra skills.

"We need to do a better job at finding out what people actually need to know, not what we think they should know," he said.

Here was my comment:

I thoroughly agree with Prof. Rosenstein that not all students will need the skills/concepts of a more advanced algebra class. While I admire Commissioner Davy's desire to significantly raise the bar for NJ students there are some underlying issues that must be addressed first. How many of you believe that the majority of NJ students have demonstrated proficiency in the foundational arithmetic and prealgebra skills needed to be successful in a legitimate Algebra 1 course, never mind Algebra 2? As a retired math supervisor, believe me, that question was rhetorical!
However, we must clearly distinguish between the issue of a graduation requirement for all and the need for consistent, clearly stated and rigorous standards for a 2nd year Algebra course. Despite opinions to the contrary, I believe the latter is necessary for most college-intending students. The American Diploma Project (NJ is a member of this consortium) has developed precisely those kinds of world-class standards and the result is the new End of Course Test in Algebra 2. This test, which many NJ students have already taken, requires a deeper conceptual understanding of topics such as mathematical modeling which separates the Algebra 2 of the 21st century from the Algebra 2 course many of us remember. And, yes, there are still some mechanical skills which students need to master away from the calculator!
I strongly advocate that NJ adopt these higher standards for those students who will go on to take more advanced math courses. Clearly, it isn't for everyone and therefore we should reexamine it as a grad requirement for all.
Dave Marain

I felt it was important to make a clear distinction between Algebra 2 as a high school graduation requirement and the need for a high-quality curriculum which should be uniform for all students who need to take the course. Many commenter ranted about the evils of testing, the "who really needs algebra anyway" argument, allowing politicians to make educational decisions they know little about (imagine acknowledging that it should be left to math education professionals!) , the skills needed for the 21st century, etc. Fascinating stuff...

Is this same discussion happening in your district or state? Your thoughts are important to me. Do you take strong exception to my comments? Do you agree with the NJ Commissioner of Education or has she gone too far? What do you say to the many adults who argue that, in their occupation, they haven't ever used any of the 'stuff 'they learned in Algebra 2?

Sunday, January 25, 2009

Math Contest Reminder and a Probability Paradox??

There's still time to register for MathNotation's First Math contest for Grades 7-12 to be held on Tue Feb 3rd. I've decided to extend the registration to Thu Jan 29th. We've had interest expressed from high schools, middle schools, homeschooling teams, even a chapter of an honorary math fraternity! I'd like to see 2-3 more teams compete but I understand that many students and teachers are overextended at this time of year and this was on short notice. Look here for how to register.

So what's the paradox in the title? To someone with a firm grasp of probability there won't be one, but the following series of questions may lead to a surprise for some students.

Overview of Problem
We have two scenarios in this investigation:
A set of five 4-choice multiple-choice questions and a set of five 5-choice multiple-choice questions. Of course the latter is typical of most standardized tests like SATs so this discussion may have relevance to many juniors right now!

Instructional Suggestion
For the following questions, ask students to first make educated guesses before attempting any calculations. The idea is to get them to trust their intuition which often is more accurate than their mathematical procedures!

We know that the probability of correctly guessing, at random, the answer to a 4-choice question is 1/4 which is greater than the chance (1/5) of correctly guessing, at random, the answer to a 5-choice question. That was easy, right? When we ask questions about more than one question the situation becomes more complicated and a deeper understanding of probability concepts is needed: Multiplication of probabilities of independent events, binomial probabilities, etc...

The Investigation

(a) Which of the following is more likely? Randomly guessing all 5 wrong on a 5-choice multiple choice quiz or randomly guessing all 5 wrong on a 4-choice multiple choice quiz?
By intuition (no calculation, respond in 10 sec or less): _________________
Explanation of Intuitive Guess (this may be worthy of class discussion):
Now compute each probability and compare result to your intuitive answer.

(b) Which is more likely? Randomly guessing at least one right out of five on a 5-choice multiple-choice quiz or on a 4-choice multiple-choice quiz?
By intuition: ______________
Explanation of intuitive guess:
By calculating:

(c) How's your intuition doing so far?
Let's try this one:
Which is more likely:
Randomly guessing exactly one right out of 5 on a 5-choice quiz or on a 4-choice quiz?
By intuition:
By calculating:

Any surprises? In case your results don't agree with mine, I will tell roughly you what I got (actual probabilities below). The probability of guessing exactly one right out of five on a 5-choice quiz is slightly more than the probability of guessing exactly one right on a 4-choice quiz! A paradox? An anomaly of the arithmetic involved? Logical? Can you explain it? Try!

(d) Back to normalcy? Compute the probabilities of getting exactly two right out of five on a 5-choice quiz and on a 4-choice quiz. Has the order of the universe been restored!

Selected Answers (not the norm for this blog):
(b) Approx 67.2% on a 5-choice quiz; 76.3% on a 4-choice
(c) Approx 40.96% on a 5-choice quiz; 39.55% on a 4-choice
(d) 20.48% on a 5-choice quiz; approx 26.37% on a 4-choice
Pls check these results for accuracy!!

What are the fundamental concepts in this investigation? What are the learning benefits of this series of questions? Please understand that my intent on this blog is to suggest instructional methods, never to impose. You may find far more effective ways to convey the essential concepts here but, from my experience, there's only sure way to perfect our craft. Keep experimenting and asking questions!!

Thursday, January 22, 2009

A Presidential Contest Digit Problem: Fours are Wild!

With MathNotation's First Math Contest less than two weeks away (look here for details), I wanted to provide another sample contest question (multi-part). By the way, we now have several middle schools, high schools and even homeschool teams registered from all overv the country! It takes only a few minutes to register and there's still time!
For President Obama, the number four has special significance. The most obvious is that he's the 44th president. You can ask your students to think of several other connections between our new president and the number four. But for now, we will focus on 44...

(a) Since 44 = 2^5 + 2^3 + 2^2, 44 equals 101100 in base 2 (binary representation).
Let S be the set of all base-10 numbers (positive integers) whose binary representation consists of six digits, exactly three of which are 1's. Find the sum of these base-10 numbers to reveal part of the mystery behind the title of this post!
Note that the leftmost binary digit must be "1".

Comment: This is a fairly straightforward 'counting' problem accessible to middle schoolers as well as older students. One could simply make a list of the numbers and add them. However, there's a more systematic way to count the 'combinations' and a "different" way of adding here that may help you solve the next problem. Can you find it?

(b) Consider the set of all base-10 numbers (positive integers) whose binary representation consists of ten digits, exactly three of which are 1's. Show that the sum of these base-10 numbers can be written 44(2^9) - 4 - 4. "Fours are wild!"

Note: This seems like a tedious generalization of Part I, but, again, if you find the right way to count and add it won't take long!

BTW, if you're wondering how I came to find all these 4's, well, it might have been serendipity. After all, serendipity has 11 letters and 11 is a factor of 44 and... (Twilight Zone music playing in the background...). Also, if you're wondering what my outside sources for these kinds of problems are, do you really think there's anyone else out there whose mind could be this warped!

Monday, January 19, 2009

MLK, Inauguration, Math Contest and A Radical Investigation!

Don't miss registering for MathNotation's First Math Contest. Registration is as simple as emailing me (dmarain "at" "gmail dot com") to request a form and the Rules. The contest is team-based (up to 6 students), is designed for both middle and high school students and should take 45 minutes or less (extra time is provided for students to enter their answers/solutions on the official answer form in Word). Look here for further info.

I would also like to thank the following blogs and/or webmasters for their graciousness in spreading the word about our first math contest:
Let's Play Math!
Wild About Math
Note: Take a look at jd2718 to see the latest Carnival of Mathematics. Another excellent job by Jonathan!
Homeschool Math Blog

While we're waiting for the Inauguration on 1-20-09 (12,009 = 3 x 4003 of course), today is Dr. King's birthday, 1-19-09 and 11,909 is prime as it should be! How appropriate it is that we should be honoring today the man who paved the way for our new President...
The title of this post reminds me of an old Johnny Carson routine: Which one doesn't belong with the others! In fact, we can probably make connections among all of these if you're willing to play with words...
In case you thought that the Math Contest would lead to a hiatus in publishing investigations and instructional strategy articles, fear not! Today we will once again examine the raison d'etre of this blog:

Part I

Consider the equation

To reinforce multiple representations (Rule of Four) we can ask students to:

Explain or show why this equation has no real solutions
(a) Graphically
(b) Numerically (TABLE)
(c) Algebraically

At this point I am including some ScreenShots from the TI-84. The bold graph is Y1:

Part II - The Extension!
Consider the equation

(a) For what value(s) of k will the above equation have one real solution? In this case, also determine an expression for that solution in terms of k. Show method clearly.

(b) For what value(s) of k will the above equation have no real solutions. Show method clearly.

(c) Demonstrate your results in (a) and (b) by choosing specific values of k for each case. Use both a graph and a TABLE to support your argument. [Use of the graphing calculator makes sense here.]

Which do you think is more helpful to students -- the graph or the TABLE? From my experience I find that both are important for comprehension and concept. They not only complement each other but each contributes something by itself. The graph not only suggests (not prove!) that the two graphs in part I do not intersect but it leads to a natural questions like: Why is the graph of y = √x + 2 above the the graph of Y1? What do the graphs suggest about the domain of each function? Explain the ERR messages!
Note: I used the word "suggest" because we want our students to understand that graphs do not prove mathematical truth.

When is it appropriate to use this approach?
After you've taught the algebraic procedures of solving radical equations? Of course, part (c) of the activity asks for the algebraic explanation, but I've often used the graphical and numerical approach BEFORE teaching the procedure. I believe that it developed meaning for the traditional procedure but, in no way, did it replace the need for carefully explained instruction with a variety of examples! (The "balanced" approach!).

Further, the common reaction I've heard to this kind of instruction is that it is too time-consuming and appropriate only for the honors students. I couldn't disagree more. Developing meaning does take time and is absolutely worth it. It's all part of the "less is more" philosophy and, that, if the foundation is properly put into place, students can develop both the skills of solving radical equations and an understanding of the underlying mathematics. Enough preaching to the choir...

I hope you find this useful when building your next exploration in mathematics! Let me know...

Thursday, January 15, 2009

A Preview of the Contest: Probability Investigation with Replacement

Don't forget to email me if you want your students to participate in the first MathNotations online math contest on Tue Feb 3rd. There is still time! Look here for info.

There may not be a probability question on the first contest but the following gives you a flavor of the type of multi-part question I'm talking about -- an investigation in more depth.

You will find many variations of the following problem in texts. From experience we know that the student needs to have numerous experiences with these. How do many students do on this topic when the exam question is slightly different from the ones reviewed in class!

Five cards are numbered 1 through 5 (different number on each card). Typical scenario, right?
George chooses cards randomly one at a time. After he selects a card, he marks a dot on the card, then puts it back (replacement!) in the pile of 5 cards, reshuffles them and draws the next card and so on. The game continues until he selects one of the "marked" cards.

Before a technical analysis of this experiment (sample space, random variable, specific probabilities, expected value), I would typically ask students a broad intuitive question or ask them to suggest questions one might ask about this "game".

Intuitively, I might ask:
In the long run, how many draws would you "expect" it to take for the game to end?

With five cards, what do you think most students would guess? Draw three? Draw four? I think asking this initial question is crucial. In most cases, we want the mathematical result to be reasonable and to roughly agree with our intuition (not always of course, there are paradoxes in math which are counterintuitive!).


Part I
What is the probability that George chooses a "marked" card on his second draw for the first time? On the 3rd draw for the first time? 4th draw? 5th draw? 6th draw?
Another way to ask these are: What is the probability that the game "ends" after 2 draws? 3 draws, etc.

Part II
"On average", how many cards would George need to draw to get one of the marked cards for the first time?

Note: In more technical language we are asking for the expected number of draws before the game ends?

Normally, I don't publish answers to these questions but, in this case I will give partial results. Please check for accuracy.

The probability the game ends after 3 draws is 8/25 or 32%.
The expected value for the number of draws for the game to end is approximately 3.51. What does this mean!

Sunday, January 11, 2009

First MathNotations Math Contest Offically Announced and It's FREE!

Important Updates:

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

After getting several helpful comments and suggestions, I have now made an "official" decision (always subject to last minute changes of course!) regarding the date and details of our first contest. I chose this date to accommodate schools' exam weeks. The date is also a week before AMC-10 and -12. I will run this event if I get at least 6 schools participating. Pls spread the word to your friends in other schools. I understand there is not much time to consider this but the registration process and administration of the test should not be too burdensome.


  • SUITABLE GRADE LEVELS: 7-12 (Some questions can be handled by Middle School students)
  • CONTENT: Up to and including precalculus; emphasis on Algebra II
  • FEE: NONE!

What makes this contest different?
  • Team event - Up to 6 participants may work together!
  • All answers/solutions must be submitted electronically
  • Some multistep and open-ended questions
  • FREE! (At least this first one is!)
  • Separate acknowledgments on MathNotations given to Middle and High School teams
  • All students will receive a Certificate of Participation and top-scoring schools and students will receive a Certificate of Merit via email.
All interested sponsors should email me at dmarain "at" "geemail dot com" ASAP for the official registration form and detailed rules and procedures. After receiving your registration form, the contest problems and official answer form will be emailed before or on Jan 30th. Don't hesitate to contact me if you have any questions (the rules/procedures form should answer most questions).

Thursday, January 8, 2009

Odds and Evens: 1-8-09...An online math contest and other passing thoughts...

Many thoughts are running through my mind right now...

Projects I'd like to move forward, some changes in this blog, perhaps a different website altogether...

Working on the MathAnagram for the first quarter of 2009...
I've already selected the mathematician. Writing an anagram that has embedded clues is labor-intensive...

My reactions to a Commentary in this week's Education Week authored by the esteemed Steven Leinwand who is calling for a fascinating new concept heretofore unheard of -- a national K-12 Math Curriculum. I wish I had thought of that!

Thoughts about an online math competition for high schools...
Yes, that's right, I've already written the rough draft of the first six questions. I need to get the word out to high schools who might want to pilot this a few weeks from now. No cost for this first contest, but I would like to have at least a dozen schools express some interest in this before I formally announce it. If any of you reading this might be interested or know the math supervisor in your district, please spread the word. High schools can field one team of up to 6 students and will have a short window of time (from the moment I publish the questions) to submit their answers/solutions electronically. The contest will differ from others in that some questions may be multi-part and some parts will require explanation. Not just short answer! In other words, while there will be traditional contest problems, there will also be questions that reflect the investigations on this blog. Calculators will be permitted and a faculty sponsor would be needed to proctor the contest. Clearly there are major logistic problems (registering teams, different time zones, international participation, etc.). I need to work out many issues here. Questions will not at this time go past precalculus.
I might need to have my head examined for considering this since I will have to read and evaluate every one of the responses!
I may also need a separate website for all of this but I need to get a sense of interest out there before I jump in deeply. If this blog elicits few responses, I will probably have to disseminate this in some other way. I'd really appreciate suggestions/reactions to this both in the comments section and via email. As always you can email me at dmarain at geeeeemaillll dot com...

Have embarked on a collaboration with a University math professor who is developing problem-solving experiences for his students. Some of these problems will be based on investigations published in MathNotations...

So many wonderful math blogs out there not only from our regulars but new ones entering the math blogosphere every day. Exciting stuff...

Sunday, January 4, 2009

Using Algebra to Enhance Numerical Accuracy On Your Calculator: When is ZERO really ZERO!

When a calculator displays zero as a result should students assume that is exact or only accurate to the precision the machine can store and/or display?

The next time students ask you why we use the conjugate method to rationalize denominators, here's an example of why we sometimes use the method in "reverse". This happens more frequently in calculus but the following is an apparently trivial numerical computation your students can try on their graphing calculators. The results of this computation depend heavily on the specific technology used (e.g., expect different results between the TI-89 and the TI-84), but hopefully they will get the idea. This numerical issue came up as I was solving an applied problem which required finding the difference between two very large numbers (the difference between distances from the center of the earth to a point slightly above its surface and the radius of the earth). This numerical issue has come up more before on this blog. Look here if you want to see another application.

Here's the computation:
Let R = 2.0916 x 10^7
We need to compute the following expression (denoted by **)


For the Student

(a) Do the calculation directly on your calculator. You will want to store this value of R as a variable for later use:
2.0916x10^7 STO> ALPHA R
Does your calculator display zero? If so, explain this "error."
Note: This display depends on the calculator being used. I experimented with the -84 and -83. Let me know how the display appears on other machines. Of course, one would expect a very different outcome if using Mathematica!

(b) Rewrite the above expression ** by multiplying the numerator and denominator by the conjugate of the expression. (Hint: Put the original expression "over 1").

(c) Recalculate the value of ** using the modified but equivalent form from part (b).
What result do you see this time? Can you explain what may be going on?

(d) Find other numerical expressions that produce an incorrectly displayed result on your calculator! Post these in the comments section pls!

Friday, January 2, 2009

Odds and Evens - First Week of 2009

The end of 2008 and the beginning of a new year has ushered in many excellent posts from some of the top math bloggers out there.

1. I plead guilty to an error of omission -- not contributing to the 46th Carnival of Mathematics over at Mike's Walking Randomly. Then again I've missed the last several so I need to make another New Year's "Re-Solution"! Mike did an excellent job of putting together the last Carnival of the year. In particular, he featured his own choices for articles for each month of the year, introducing readers to some excellent sites. Great job, Mike.

2. Naturally, the new calendar year has sparked a flurry of posts about the number 2009:
(i) The least "interesting" such post, "Get Ready for Happy 41*7^2" was probably mine.
(ii) This was followed in quick order by Mike again with his "What is interesting about the number 2009?" post over at Walking Randomly. Mike suggested representing 2009 as sums of powers leading to extensions from some excellent commenters, Sol in particular.
(iii) 360's "The number 2009" post adds a different perspective to this game -- some clever identities involving sums of fractions, not to mention some demographic info about the 2009th largest city and using extrapolation to surmise the 2009th richest person in the world!
(iv) Denise has followed her "annual" tradition of a number game with the "2009 Mathematics Game," challenging her readers and students everywhere to represent as many as possible of the integers from 1 to 100 using only the digits 2, 0, 0 and 9 and standard arithmetic operations (which she clearly defines) and grouping symbols. I particularly like her use of the convention 0^0 = 1, a controversial definition to say the least. This game is addictive and will keep Denise's readers busy for the next 12 months or so!
(v) Other 2009 posts and curiosities I've omitted?? Let me know in the comments...

3. Speaking of challenges, I asked my readers to solve a silly little riddle:
"What do call solving an equation twice on Jan 1st?"
We had three "first responders" so I will close the contest down now and announce our winners in the order in which I rec'd their email solutions. By the way the answer can be found "hidden" near the top of this post! And the winners are...
SEAN HENDERSON (and his wife!)

4. Finally, I discovered by accident that MathNotations and several of the math blogs I enjoy reading are featured in a new aggregator of sorts, Alltop, All the top Math News. The developers liken it to a virtual magazine rack, in which the titles of the latest 5 posts or articles from the selected web sites are listed (you can see more detail by rolling over the titles). I am finding it useful for getting current information from some sites I had not seen before. We'll see where this goes. Ars Mathematica, Sol's Wild About Math and MathNotations are all ranked in the Top Ten whatever the significance of that ranking may be...