A Recursively Defined Sequence to Challenge Your Algebra Students

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

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

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