In continued tribute to Dr. Mandelbrot, here is a challenge problem for your Algebra 2 students which develops the ideas of iteration and recursively-defined sequences while providing technical skill practice. From my own experience, even some of the strongest will trip over the details so don't be surprised if you get many different answers for the 5th term in part (c) below! We all know that current texts do not provide enough mechanical practice and this becomes more evident as our top students move into the advanced classes.
THE CHALLENGE
A sequence is defined as follows. Each term after the first is two less than three times the preceding term.
(a) If the first term is 2, determine the 2nd through 5th terms.
(b) If the first term is 1, determine the 100th term. Explain.
(c) If the first term is x, determine simplified expressions in terms of x for the 2nd through 5th terms. To help you verify your answers, the 5th term is 81x - 80. Show all steps clearly. Compare your results with others in your group and resolve any discrepancies.
(d) Write a general expression for the nth term if the 1st term is x. It should work for all terms including the first! Explain your method. Proving your formula works for all n is optional.
Answer: 3^(n-1)x - (3^(n-1) - 1)
NOTE: Students who have learned the formula for the nth term of a geometric sequence should recognize the first term in this answer! Help them to make the connection...
(e) Extension: Change the recursive relationship to: Each term after the first is three less than twice the preceding term. Redo part (d) for this new sequence. The pattern is more challenging!
Ans: 2^(n-1)x - 3(2^(n-1) - 1)
NOTE: For the more advanced students, have them prove their "formula" by induction.
Final Comment: In what form do you think this kind of question would appear on the SATs and, yes, this topic is tested and has appeared!
"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
Thursday, October 21, 2010
A Recursively Defined Sequence to Challenge Your Algebra Students
Monday, October 18, 2010
Odds and Evens Week of 10-18-10
Much has been happening in the world of mathematics and mathematics education. I'm only scratching the surface here.
- The passing of Professor Mandelbrot -- There is no question that this man has left an eternal "singularity" in the profession. Who among us has not been mesmerized by the computer images generated by one of his creations. He dared to think different and was not always recognized or lauded for his uncanny knack of seeing patterns no one else could. When asked to look back on his career, Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s.
“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”
- I read so many tweets/posts on Twitter, Facebook and in blogs discussing whether or not to continue teaching particular traditional topics in an algebra, geometry or precalculus class. Curriculum supervisors, math department chairs/supervisors and therefore teachers should now be guided by two key documents, particularly for Algebra 2 content:
- The Common Core State Standards Initiative for Mathematics
- ADP/Achieve Algebra II Test Overview
- You may also want to download the document from Glencoe which does a nice comparison of required content to the their text.
"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
Monday, October 4, 2010
Odds and Evens- October 2010
The following is excerpted from the essay, "When Pedagogic Fads Trump Priorities" in the 9-29-10 edition of Ed Week. The author is Mike Schmoker, an author, speaker and education consultant.
"First we need coherent, content-rich guaranteed curriculum - that is, a curriculum which ensures that the actual intellectual skills and subject matter of a course don't depend on which teacher a student happens to get...
Second - and just as important - we need to ensure that that students read, write and discuss, in the analytic and argumentative modes, for hundreds of hours per school year, across the curriculum...
Third, we need to honor, beyond lip service, the nearly half-century-old model for good lessons that all of us know, but so few consistently implement:
Good lessons start with a clear curriculum-based objective and assessment, followed by multiple cycles of instruction, guided practice, checks for understanding (the soul of a good lesson) and ongoing adjustments to instruction... multiple checks for understanding may be the most powerful, cost-effective action we can take to ensure learning. Solid research demonstrates that students learn as much as four times as quickly from such lessons.
For decades we have put novelty and the false god of innovation above our most obvious, proven priorities"...I've been in touch with Mr. Schmoker to congratulate him for the courage to speak the truth. I hope to continue the dialog. He also takes on "differentiated instruction" and mindlessly incorporating technology into lessons as if "that will rescue poor instructional plans from failure."
I rarely say this, but, if you disagree with him, you are either wrong or hypocritical! Yup, dems fighting' words!
And now for something completely different...
"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
Saturday, August 28, 2010
Video Solution and Discussion of Twitter SAT Probability Question from 8-25-10
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 both multiple choice 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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I decided to post a video solution of the Twitter problem I posted on 8-25-10:
4 red, 2 blue cards; 4 are chosen at random. What is the probability that 2 of the cards will be red?
Because of the 140 character restriction on Twitter, the questions are often highly abbreviated and I actually consider it a "fun" challenge to write the question both concisely and clearly. Of course, as we all know about human interpretation of word problems, "clear" is in the eye of the beholder!
There's no doubt that the question above needs some fleshing out and might appear on the SAT and other standardized tests something like this:
A set of six cards contains four red and two blue cards. If four cards are chosen at random, what is the probability that exactly two of these cards will be red?
I'm sure my astute readers can improve on this wording but we'll leave it at this.
A few questions naturally pop up:
(1) Could this really be an SAT/Standardized Test question? Well, as I state in the video below, a question quite similar to this appeared on the College Board website the other day as the Question of the Day.
(2) For whom is the video intended? Everyone who happens upon it! I certainly wrote it to be helpful to students who will be taking the PSAT/SAT in the near future. Rather than simply presenting a single quick efficient solution, I demo'd 2-3 methods and indicated some important strategies and reviewed key pieces of knowledge to be successful on these harder probability questions. By the way, someone who is comfortable with probability will surely not find this question so formidable, but we're talking here about high school students or even undergraduates who struggle mightily with these.
(3) I'm hoping that the video will also serve as a catalyst for dialog in your math department. From the inception of this blog, I've never even intimated that a suggested way of explaining a concept, skill or a problem solution is in any way prescriptive. I encourage you to continue using whatever instructional methods have worked for you and to share these with our readers! However, for novice teachers or those who wish to see other approaches, I hope it will have some benefit. Of course, the video is not in a classroom. There are no students asking or being asked questions. There are no interruptions and I have a captive audience (except for my dogs who bark incessantly!).
SOME KEY STRATEGIES/TIPS/FACTS FOR PROBABILITY QUESTIONS
(1) It is highly recommended that students begin by listing 2-3 possible outcomes and to include at least one that is NOT one of the desired outcomes! This will help you to decide on a plan: organized list vs more advanced counting/probability methods. Further, you can ask yourself the key question in all counting/probability problems: DOES ORDER COUNT!
(2) Although it appears difficult for most test-takers to be systematic when making a list under test-taking conditions, preparation is critical here. If one practices several of these in the weeks leading up to the test, the chances of success improve dramatically. Did I just suggest preparation and practice could make a difference!
Where do you find these problems? Any SAT/ACT review book or my Twitter Problems of the Day or my upcoming SAT Challenge Quiz book to name a few sources...
(3) The basic definition of probability should always be in the forefront of your mind:
P(an event) = TOTAL NUMBER OF WAYS FOR THAT EVENT TO OCCUR DIVIDED BY TOTAL NUMBER OF OUTCOMES.
As indicated in the video, one can and should think of this ratio as TWO SEPARATE COUNTING PROBLEMS! Do the denominator first, i.e., the TOTAL number of possible outcomes. In the Twitter problem it is 15 if order is disregarded. Whether you arrive at 15 by listing/counting or by combinations methods, the denominator is 15 and is a completely separate question from "How many ways are there to get 2 red and 2 blue cards?"
(4) Finally, there are other methods for solving this probability question using Laws of Probabilities and/or permutation methods. I was going to make a 2nd video but I'm not so sure about that now.
An important point about the video below: I used 4 Blue and 2 Red cards, the opposite of the original Twitter problem but that won't change the final result!
Look for my other videos on my YouTube channel MathNotationsVids. Look for all of my Twitter SAT Problems on twitter.com/dmarain.
As I develop my Facebook page further, I may start posting these questions there as well as my videos. Facebook allows up to 20 minutes videos, much less restrictive than YouTube's 10 minute limit.
"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
Friday, August 20, 2010
Murphy's Laws for Teachers/Students - A Murphy Wiki to Start the Year?
- If you study hard for that important examination, the focus of the exam will be 'thinking-based' and 'analytical'.
Corollary: If you memorized information, it will be useless. - If you don't study for that important examination, the paper will be content-based.
Corollary: If you don't study, every question will appear to be something you remember reading on your textbooks from a month ago, hence will appear (deceptively of course) easy, although you will not recall the exact phrasing of an answer.
"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
Friday, July 30, 2010
Where in the World is MathNotations?
Have you been wondering about the question in the title or just assuming that I'm on hiatus?
No videos?
No provocative comments about standardized curriculum?
No interviews with the movers and shakers in math education on the national front?
Most importantly, no anecdotes from my 3-yr old grandson? Is the world coming to an end?
Seriously, I suspect the world is still spinning on its axis, perhaps just a tad more tilted. And my faithful readers/subscribers have far more important concerns in their life than waiting for my next post! On the other hand, some of you know that I've discovered Twitterdom or should I say Tweedledum and Tweedledee!
What exactly have I been doing other than coping with the joys and trials of keeping my wife, 3 teenagers and a 21-year old happy? Not to mention 4 grandchildren and 3 other older children not "officially" living at home...
I. Twittering a brand-new SAT Twitter Math Problem of the Day virtually EVERY day since May 26th! Answers were provided up until about the middle of June. You can see these problems in the right sidebar (limited view) of this blog or follow me on Twitter here. I've had very interesting reactions from 8th and 9th graders in Indonesia who seem to find these problems intriguing although I really don't have a very good translation of their tweets. I think they keep calling me Papa and some are probably afraid of me!
II. Completing Volume I of SAT Math Quiz Problems. It's still in draft mode but when completed it will have 150-200 challenging math questions not previously published on my blog. Many of the Twitter Problems will be included, answers will be given for all questions and selected hints/solutions will be provided. The book will be available for download as a pdf with some copyright limitations. I will post more on this here, on Twitter and on my new MathNotations Facebook page (still under construction!).
III. I've been in contact with K.C. Yan from Singapore who has been enlightening me re the model method in Singapore Math. I strongly encourage you to check out his website Singapore Math and follow him on Twitter here. He is a remarkable individual and possesses a profound understanding of mathematics and pedagogy. He is a math coach, writer and editor. He currently conducts recreational and competition math courses and workshops for schools and enrichment centers, and educates the public against innumeracy and pseudoscience. I strongly encourage you to read his blog and, in particular, his demonstration of several non-algebraic "model" methods for solving the following question:
A farmer has twice as many ducks as chickens. After the farmer has sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?
Who knows? Perhaps, we will one day collaborate!
IV. I've been in touch with Professor William Schmidt of TIMSS renown. We were scheduled to have a Skype conversation back in May but our schedules were at cross purposes. I'm hoping to contact him again and reschedule. Some of you know how much respect I have for his knowledge, dedication and tireless efforts to improve the math education of our children.
Talk to you soon!!
"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
Sunday, July 4, 2010
Happy 4th! SAT Problems on Twitter, SAT Math Quiz Book, Updates?
Well, let's see...
I haven't posted in almost a month, I haven't been promoting our wonderful Math Carnivals, I haven't brought you any updates or controversial material, I haven't produced any new and exciting videos,...
So what have I been doing? Enjoying the heat wave here in the Northeast?
1. Finishing up my SAT Math Quiz Book Volume One which will hopefully be done before the world ends in 2012. I haven't decided yet how I will make these available to my readers or schools or students or whomever but that will all be worked out. One possibility is to send the book electronically upon payment.
2. Continuing to post a Twitter SAT Problem of the Day despite the fact that I said I would take a respite for the summer. Further, many of these problems will appear in the Quiz Book. Can't stop writing these -- please help me! These problems also appear in the right sidebar of this blog but they may be truncated. If you only get the feed for this blog then you may want to subscribe to the RSS feed for my Twitter posts.
3. Exciting new trends in math education? Actually other than states racing to the top and continued movement toward standardization of math curriculum, it's really the same old, same old. Technology will always evolve and influence math education -- that's a given -- however the nuts and bolts of what makes for effective math teaching, well, that's still the ten trillion dollar question and that's still the reason for this blog.
Stay tuned and enjoy the summer hiatus!
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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
Friday, June 11, 2010
SAT Videos: Twitter Problems of the Day 6-9 and 6-10-10
As we wind down toward the summer my SAT Problems and Videos continue to pick up steam! Below is the latest video from you YouTube channel, MathNotationsVids. I want to thank those who voted in my survey of these videos. I am gratified but I really need more specific suggestions on how to improve these. Your comments on YouTube or here are welcome!
Note: Because I am explaining two problems on one video, I am omitting details and multiple solution paths. Therefore these videos may be useful for your students who want to practice over the summer or revisit in the fall.
The percent increase problem could be asked in a variety of ways and demonstrated using multiple representations, aka The Rule of Four. The visualization suggested in the description of the video has students physically demonstrating that doubling the edges of a rectangular solid, a cube in this case, will allow placing not only the original box inside of the bigger box, but SEVEN MORE! There's your percent increase, hands on!
I will be stopping the posted SAT Problem on Twitter on Tue 6-15-10. If I am able to sustain it, I will try to keep this up for the entire 2010-11 school year but who knows...
Finally, as posted on Twitter, I will be offering an individual or small group online course (using Skype) for the SAT or ACT Math this summer on a very limited basis. If you know of any student who might benefit from individualized instruction just email me at dmarain@gmail.com and I will provide details. This must be done ASAP however, as I will be closing this out very quickly.
"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
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