Odds and Evens Week of 12-1-10

• Here's my most recent Twitter Problem of the Day:

How many 3-digit positive integers are there in which the absolute value of the difference of their hundreds' and units' digits equals 4?


For students: Reply on Twitter, Facebook or my email (dmarain@gmail.com) by 12-6-10.
For everyone else: Comments are always welcome but please hold off on solutions until 12-6-10. Thanks!

• I've been contacted again by the Education Editor of Parent Paper magazine, a well-known publication here in North Jersey.  I was asked to write a piece on helping parents to help their children with schoolwork, particularly in math. I'm reprinting here in full since it will be most likely edited down to a few sentences. Most of the general suggestions are obvious but sometimes I feel that the obvious needs to be stated. I'm basing this on my experience with 7 children, 4 grandkids and over 30 foster children.

General Suggestions for Parents Helping Children With Assignments

• TV, radio, music, any other distractions turned off when your child comes home after school.
• Establish a consistent location where they will do their homework every day -- dining room table, coffee table -- preferably in the same room as parent until they are older
• Establish a routine where the child takes out the assignment book, folders, etc., before their snack.  If you do it for them, they will come to depend on you for this.  Have them hand you their parent folder with all papers you're supposed to read, sign, etc.
• It's up to you but I would allow the child to have their snack while they start their homework.  Be less concerned about the mess and remember, if they're not allowed to start homework until they 've finished their snack, I guarantee you that snack time will extend for longer and longer periods of time (even if you say the have to finish in 15 minutes!).
• DO NOT OFFER TO HELP THEM WITH THEIR WORK UNLESS THEY ASK!  DO NOT HOVER OVER THEM - JUST BE IN  THE VICINITY!  ONCE YOU'VE MADE THEM DEPENDENT ON YOU, IT'S HARD TO BREAK THE HABIT!
• If they ask for help, ask them to read the directions out loud. If you then ask them what it means or what they are supposed to do, many children will reply something like, "I don't know. I don't get it. I can't do it!"  You know your child best. If you believe they are capable of the assignment, you can help them get started and then say you have to do something, but you'll be around if needed.
• If you cannot make sense of what the assignment is, then ask them to explain it. If they can't, the issue may be they are not yet ready to neatly/clearly copy the assignment form the board. Address this with the teacher the following day.
• Ask the teacher whether they prefer voicemail, email or face-to-face questions after or before school.  Ask them if it's ok if they occasionally email concerns.
• Establish a "social" network of parents in the class - take the initiative!  Set up a class group on Facebook so that parents can help each other with clarifying assignments. Parents can routinely check in.  If electronic networking is not feasible, go back to the tried-and-true getting phone numbers from 2-3 other parents thus making a smaller network.  Trust me, you will need to use this often unless your child is mature, organized and responsible/independent, in which case you will be helping others! 
• Keep repeating to yourself the Golden Rule of Parenting: THE MORE YOU DO FOR YOUR CHILD, THE LESS HE/SHE WILL LEARN TO DO FOR HIM/HERSELF !!

Specific Suggestions for Math

• Most children have more difficulty with the wording of the directions or of the problem than the math itself!  Try to break it down for them.
• Don't be too quick to correct their mistakes. When checking over their work, try "I'm not sure about #5. Would you tell me what you did?" Most of the time they can correct their own errors!
• It is important to become familiar with your child's math program.  You will probably already have heard about it through the grapevine, but you can find out what it is even before school starts by asking the office or leaving a message for the math specialist in the district.  Go to any meeting the school offers to introduce parents to the math program. 
• All new math programs come with extensive parent resource materials. You should receive these regularly but don't hesitate to go online and find them for yourself!  
• Be prepared to ask questions, but don't start tearing the program down b/c you've heard there are problems with it.  The program will not be changed in the current year no matter how parents may feel.  
• Recognize that every math program, whether more traditionally skill-based or reform-oriented (more problem-solving, projects, less drill) has its merits and its weaknesses. Whether you believe there is too much emphasis on basic facts (less likely!), or not enough, you can supplement with the myriad of resources on the web.
• Don't be shy about asking the teacher for guidance with your child or with the math program itself.
• Remember: MATH IS ALL AROUND US ALL THE TIME!  Ask your children lots of questions involving numbers and shapes around them. For example, "I need to cut up this square into two equal parts. I know an easy way (like this) but I think there's more than one way. Can you help me?"OR  "I have a riddle. What movie comes before Toy Story 1000?" OR Place four quarters on the table. "Can you give me a dollar?" Put coins back. "Can you give me a half dollar?" etc... 
• Never assume a concept is too hard for them. If simplified, they can often find a way.    

SOME OF THE BEST MATH RESOURCES ON THE WEB

• Home School Math (http://www.homeschoolmath.net/) - Incredible resource for ANY parent of ANY child!  Free worksheets, math ebooks for elementary grades, extensive link list of games, interactive tutorials & quizzes, curriculum guide, andmath teaching help articles/lessons. The resources emphasize understanding of concepts instead of mechanical memorization of rules.
• Let's Play Math (http://letsplaymath.net/) - Outstanding site by an exceptional educator who knows her stuff and hasd done her homework! Check out her Free (Mostly) Math Resources
• Math.com (http://www.math.com/) - Excellent resource with numerous electronic tools for mastering basics and beyond.  Start with Basic Math and Everyday Math (not the well-known math program) for K-8.
• 10 Best Web Sites to Enrich Your 
  Elementary Math Curriculum
    (http://www.am.dodea.edu/acss/AES/sip/documents/WebSitesforProblemSolving.pdf)
• PurpleMath (http://www.purplemath.com/) - The best resource for algebra! 

• And now for the latest offerings from my 3-year old grandson. The last time I posted his "muffin" comments, I had more views than from any math post in 3 years!
• My daughter has been trying to get him to go to sleep without her staying in the room. She told him that his 3-yr old cousin, with whom he is very close, is getting big now. My daughter commented, "Her mommy reads her a story, gives her a goodnight hug and leaves." My grandson replied, "Do you think I could do that, mommy?" "Of course", my daughter replied, to which my grandson immediately came back with, "Ok, but not tonight!"
• He is all boy, all the time.  Aggressive, loves contact sports and is becoming a rabid NY Giants football fan like his daddy.  He wears his Giants shirt on game day and can throw his little football with velocity.  After seeing him throw the football a couple of times like a pro the other day, she said, "Wow, you threw the football really well, twice." "No, mommy, only once", he replied. "Are you sure? I saw you throw it twice", my daughter asked.  "Yes, mommy, the other time was the highlights!"

"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

Another Cone in a Sphere Problem? - A Guide for the rest of us...

Students who have been out of geometry for a year or so and are preparing for standardized test like Math I Subject Test or SATs/ACTS need occasional review. The following is similar to several other cone problems I've posed before but even our strongest Algebra 2 through Calculus students lose their "edge" when it comes to "solid" geometry questions (yes, believe it or not, my terminal course in high school was called Sold Geometry and we covered topics like spherical trigonometry!).


A right circular cone of height 16 is inscribed in a sphere of diameter 20. What is the diameter of the base of the cone?


Reflections....

1)  Are these kinds of problems somewhat hard merely because students forget? I can think of several more reasons:

• The problem itself is somewhat challenging, however it's far from over their heads!
• The student never experienced a question like this in Geometry; perhaps questions like these were in the B or C or D exercises in the text and were never assigned or only for the "honors" students? Do you recall seeing a problem similar to this in the textbook from which you taught?
• The student did not take a formal course in geometry
• The topic was covered in a cursory manner or perhaps not at all because of time crunch. That's the whole point of a standardized curriculum, isn't it? To know what is needed to be covered and plan accordingly. Of course, I'm  a realist enough to know the myriad of reasons why the best laid plans oft go .........
• Students don't remember how to start because key geometry strategies were not explicitly stated and reiterated ad nauseam. Were your students asked daily to begin by reciting the key strategies such as those for circle and sphere problems? Were they placed on index cards or blocked out in a particular section of their notebook?:
• DRAW THE BEST DIAGRAM YOU CAN (and believe me, I'm no artist!)
• Always locate the CENTER of circles, spheres and label the point
• Label the measurements of all segments (angles) - I know, everyone does that!
• Successful problem-solving in mathematics is based on finding relationships! Were guiding/leading questions asked 
• What do the cone and sphere have in common? 
• TRUE  FALSE  The height of the cone is the same as the diameter of the sphere.  EXPLAIN!
• Was the student exposed to the strategy of comparing the 2-dimensional analogue of the 3-D problem? Would it be a right triangle in a circle? Equilateral triangle inscribed in a circl or???  
• Oh and yes... 
• Draw the radius of the sphere (or circle) so that it is the hypotenuse of some right triangle!

"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

11-22- A Remembrance - Soon It Will Be Half A Century

And the night comes again to the circle studded sky
The stars settle slowly, in lonliness they lie
'Till the universe expodes as a falling star is raised
Planets are paralyzed, mountains are amazed
But they all glow brighter from the briliance of the blaze
With the speed of insanity, then he died.


From Crucifixion, Phil Ochs




"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

CONTEST! Just Another "Rate-Time-Distance" Problem?

CONTEST IS OFFICIALLY OVER AND THE WINNER IS ----- NO ONE! Guess I should have offered a 64GB 3G IPad! to be awarded on Black Friday...


The floor is now open for David, Curmudgeon, and my other faithful readers to offer their own solutions.  


And the next contest is...




This is a contest so students must work alone and this needs to be verified by a teacher or parent. No answer will be posted at this time. Deadline is Wed 11-17-10 at 4 PM EST.






Here's a variation on the classic motion-type problems we don't see as often in Algebra I/II but still appear on the SATs. I found this in some long-forgotten source of excellent word problems to challenge NINTH graders! 

Barry walks barefoot in the snow to school in the AM and back over the same route in the PM.  The trip to school first goes uphill for a distance, then on level ground for a distance and finally a distance downhill.  Barry's rate on any uphill slope is 2 mi/hr, any downhill slope is 6 mi/hr and 3 mi/hr on level ground.  If the round trip took 6 hours (hey, these are the old days in the 'outback'), what was the total number of miles walked?


First five correct answers  with complete detailed solutions emailed to me at dmarain@gmail.com will receive a downloaded copy of my new book of Challenge Problems for the SATs and Beyond when it becomes available. Both the student and teacher(s) will receive this.  (Illegal to reproduce or send electronically!). Read further...

Submission by email must include (Number these in your email and copy the validation as well).


1.  Answer and complete detailed solution. If answer is correct but method is sketchy or flawed,      the submission will be rejected.
2.  Full name of student
3.  Grade of student
4.  Math course(s) currently taking
5.  Math teacher's name(s) and parent's name(s)
6.  Name, Complete Address of School; Principal's Name & Email address (if known) 
7.  Email addresses of teacher(s),  parents, student 
8.  Phone number (in case I need to call you) - Optional
9.  How your or your teacher or parent became aware of MathNotations.




VALIDATION


I certify that my student (child) did the work independently.




--------------------------------------------------------------------------------


Name of Teacher or Parent (if work done at home)



"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

Algebra 2/Precalculus "Extended" Activity Based on an SAT-Type Question

Consider the following problem:

If -5 ≤ x ≤ 4, and f(x) = 2x² - 3, how many integer values are possible for f(x)? 


One can simply view this as a more challenging question to pose to your honors/accelerated students, but, for me, it's an opportunity for all your students to think more deeply about important concepts. I feel strongly that our role here is to ask the key questions which will guide them toward understanding the "big ideas" underlying this problem. In fact, we can turn this question into an extended activity: 15-20 minutes).

Here is one idea for creating the environment currently being recommended. Please keep an open mind before concluding that there is simply not enough time for these explorations...


WITH YOUR LEARNING PARTNER(S):

1.  Sketch the graph of the function on the given domain from recognition of quadratic functions and by making an x-y table with 4-5 points. WRITE YOUR INFERENCES FROM THIS. For example, from the sketch we believe that the greatest y-value on this domain is ___.

WRITE your conjecture for the answer to the problem: ____

2.  Using the TABLE feature of your graphing calculator, with TblStart = -5 and ΔTbl = 1, display the Table.  Now turn TRACE on and analyze the graph on this domain. Does this alter or confirm your conjecture from Step 1?  YES   NO

3.  The following statement is plausible but FALSE.

The domain consists of 10 integer values. Therefore there are also 10 integer values for f(x), so the answer is 10.

Explain why this is wrong. There is more than one error! 

4.  The correct answer is 51. Depending on the class, a few, if not several,  students should be able to come up with the correct answer and provide a thorough explanation.

5.  Group Discussion:

• Ask students how they might have approached this question if it appeared on a standardized test? Plug in x-values? Use the graphing calculator? Guess? Skip it?
• Ask the group what made this questionable formidable for some students? How important was understanding what was asked for?
• Review one successful approach to solving the problem by calling on individual students to give the "next" step.

NOTE: This  problem also presents a highly teachable moment for students to see an application of the Intermediate Value Theorem in Precalculus (or more intuitively in Algebra 2).  Help them make the connection! Is this easy for us to do?

 Your thoughts?



"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

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