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

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.

If interested in purchasing my new 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 practice or Problems of the Day/Week.

Price is $9.99 and secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL FIRST SO THAT I CAN SEND THE ATTACHMENT!

"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

Murphy's Laws for Teachers/Students - A Murphy Wiki to Start the Year?

Sometimes levity is needed at the start of a new school year. In the past I have posted more serious "words of wisdom" but I'm in a more whimsical mood right now. Besides, I haven't posted anything for awhile, so here goes...  

Here are a couple of my own Murphyisms I just  posted on Twitter:

Murphy's Law for SAT Students: Running out of time on the last section of Math, you desperately guess C,C,C,C,C,C for last 6 answers. Of course, the correct answers turn out to be B,A,D,B,A,D.

Murphy's Law for Trig Students: You confidently apply the mnemonic "SAHCOHTAO" to your first major unit test!

A selection of my favorites from the wonderful Murphy's Laws site:

For Teachers

The problem child will be a school board member's son.

Students who are doing better are credited with working harder. If children start to do poorly, the teacher will be blamed

The school board will make a better pay offer before the teacher's union negotiates.

Personal note: Been there, done that! Here's my own version when I was non-tenured:

As we were picketing, my poster read "We've lowered our demands -- now up yours!" Which one would you guess got picked up by local newspapers...

Law of Universal Intelligence:

The most ill-behaved student in all of a teacher's classes is always one of the bright ones he can't flunk.

For Students

• 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.

The more studying you did for the exam, the less sure you are as to which answer they want

Eighty percent of the final exam will be based on the one lecture you missed about the one book you didn't read.

PLS PLS PLS ADD YOUR OWN TO THESE. MAKE THIS A REAL WIKI!

"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

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

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

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

Video Solutions to Two Twitter SAT Problems of the Day

Please note correction to 2nd problem in the video. The correct answer is 4096 "real" values. The original answer, 13, applies to rational solutions only. Thanks to Nick Hobson for pointing out my careless error. Haste makes waste!!


Please vote in the poll at the right. Be candid in your opinion of these videos. It will guide me in the future to improve. Don't hesitate to share your opinions on MathNotationsVids and rate each video there as well. If you subscribe to my feed, please vote directly on the site. Only a few days left...


The title says it all so here is the video as promised:

Note: See above correction to 2nd problem! The video has not been corrected so beware!




Comments on 2nd problem:


If x is greater than or equal to 0 and less than or equal to 3, for how many values of x will 16^x be an integer?

As mentioned above, Nick pointed out my error. I should have restricted x to be of the form a/b, where a and b are integers, b ≠ 0. Normally, SAT questions avoid use of the term rational so they would spell it out. This problem however is very questionable for SATs. If real solutions were sought, this question would be more appropriate for a math contest. Here's one way of explaining why the answer is 4096 for real solutions:

16^x = k, k an integer → 2^(4x) = k
3 ≥ x ≥ 0 → 12 ≥ 4x ≥ 0 →  4096 ≥ 2^(4x) ≥ 1 since the exponential function 2^(4x) is increasing. This argument is reversible, so there are 4096 solutions for x, one of each integer value of k from 1 to 4096 inclusive. This solution could be written more concisely using log base 16 or log base 2 as Nick did, but I wanted to show a method without the log symbol.

Again, the video solution is WRONG as it shows only rational solutions! Well, at least i was thinking "rationally!"

I fully realize that the school year is over for some and about to end for others but these SAT Problems will be around for you or your students in perpetuity! Let me know if you like the questions. They are now appearing in the right sidebar of my blog so you will need to visit the page to see them.
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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

A Little Birdie Tweeting the SAT Blues, Carnivals and Other Musings

Well, today's SAT Tweet comes too late for many students taking SATs this morning (unless you're in a much later time zone) but I posted it anyway since it can certainly be used to review for final exams in Algebra 2 or whatever"∫ -ated"  name you have for it in your district!

As you can see if you visit this site (rather than get the RSS feed), I'm now posting the Twitter Problems of the Day in the right sidebar.  I'm new at this, don't have too many Twitter followers yet and I am learning that you need to get the word out there any way you can. Those who have replied to me seem to really like the level of these questions. I do feel the need to explain some methods to students who want them. If they're following me, they can simply send me a Direct Message or, if not, they can reply with @dmarain. I've also placed these questions in the  #Math and #SAT categories on Twitter so more will be able to see them, but a lot of what's there is promotion, links and personal thoughts --  so who knows. I may also post a video or two here and on my YouTube channel, MathNotationsVids, to explain a couple of these problems using a variety of approaches both for teachers and students.

If I were a faithful math blogger I would have been  announcing Denise's latest Math Teachers at Play and latest Carnival of Math 66 over at Sol's Wild About Math sites. They are in my blogroll, but I am deeply ashamed I haven't been promoting them here. So, please please please go over to Let's Play Math and Wild About Math to view the latest and greatest Carnivals!  Also, look here for Denise's ranking of her most popular posts broken down by categories, a mammoth undertaking, but well worth it. Sorry for being so negligent...

Finally, I feel the need to say something that may be provocative but is absolutely necessary for my integrity and the raison d'etre for this blog:


While I have been advocating for a standardized math curriculum for the past 25 years, I know fully well that learning outcomes depend far more on teacher effectiveness than any other factor. Yes, algebra should  cover the same topics in every district, however, there's coverage and then there's teaching. It is my observation that most professionals who've been on the job for awhile, whether in education, medicine, engineering or whatever, are open to receiving new information about the latest research and technology, but when you start making suggestions about the actual practice of their profession, you're sure to provoke strong negative reactions in many.  


Bottom line, folks...
There are ways of introducing and developing ratio concepts, for example, that are more effective than other methods. I have never pretended to know what the best practices are in every case, but I sure know what has worked better for me and what has failed. We need to be open to these ideas and accept the truism that when students do not perform there is a myriad of reasons, one of which was our failure to reach these particular students. We have the obligation to vary our methods and be the researchers in the classroom. We have the obligation to learn from our students, our colleagues, our supervisors and others who have been there and done that.

Ok, I'm off the soapbox.

Also, I must say that I find it fascinating that recounting my grandchildren's latest observations on life seems to bring in far more my readership than any math post I have published! Surely this is the ultimate tribute to Art Linkletter.

Have a great "end of the year" and an even better summer!


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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

MathNotations Soaring With Eagles or Just For the Birds? Updates 5-28-10

NOTE: I added a new solution (see (e) below). Also, read the comments to see even more solutions. Thanks to Jonathan for pointing out my error in (d) of my results.

I'll get to that cryptic title in a moment (may be obvious to some)...

1.  Remember the challenge problem I posted in the tribute to Martin Gardner a few days ago? Well, we rec'd several excellent replies and I have an additional response from a very sharp high schooler as well. Here was the problem:

Can you form 95 using each of the digits 5-2-2-1-0 exactly once? No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc...  You may combine the digits to form numerals like 12 or 120.



Mr. Lomas: 5! - (2+2)! - 1 - 0   Perhaps the most elegant since it uses the individual digits in the given order.


Robot Guy: (21-2)*5+0


Nate (high schooler): 120-5^2   Oh, the simplicity of that one! Combining digits is not the first way I thought of...


Mine so far:


(a) 102 - (5+2)  Pretty simple but I wasn't thinking much of combining digits until I saw Nate's


(b) 120 -25 (Shameless plagiarism from Nate's but I couldn't resist!)


(c) (2^5)(2+1) - 0! (I posted this one already)


(d) 10^2 - 5 x (2 - 0!)   (I knew there had to be a way using 100 - 5)
NOTE: JONATHAN POINTED OUT MY ERROR HERE. SEE COMMENTS.


(e) A new one: (2 + 2)! x (5-1) - 0!  I felt I needed to atone for my error in (d)!


I suspect Mr. Lomas has even more! It was definitely the spirit of Martin Gardner at work here!

Keep these coming if you can find more. I'd like to see us get to 10 ways.

2. Remember the hens -a- layin' problem I posted a few days ago? The video on YouTube gave the answer for 6 hens in 6 days: 24 eggs.

The problem on the blog was:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Here the answer is: 6 eggs

Here's a black-box method, i.e., work shown but no explanation:

(2/3) egg per (hen⋅day) x 3 hens x 3 days = 6 eggs.

This is how most solutions are given online and in the literature. It has little to do with middle schoolers actually learning the underlying principles. See the video for details.

3. Now for something completely different as M.P would say!

I've decided for now to tweet a daily (SAT) Problem of the Day.  "SAT" is in quotes because you can use these in your class as regular warm-ups or students can try these on their own to prepare for the upcoming SAT on June 5th and beyond.

Answers to each question will generally appear the next day, just before I tweet the new question. I've posted two problems thus far and the answers are up there today. Today's question will appear shortly.

My Twitter address is naturally dmarain.
Get the RSS feed for this at Twitter/dmarain if you want to see the daily problems.
If you have a question about the problems or want more details about solutions, send me a Direct Message in Twitter or email me.

Follow me if you'd like. These questions will not appear on this blog, so you will need a Twitter account or subscribe to the RSS feed above. Let your students know about it as well if you'd like.

Let me know by commenting here or replying on Twitter (Direct Message) if you like these and want me to continue next fall. Last SAT Problem of the Day on Twitter for this school year will be 6-15-10.

Requiescant in Pacem, Martin...

"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

Martin Gardner - The Original 'Riddler'

Today and over the next few days, you will find, in the media and on many math and science blogs, many touching, almost reverential, tributes to the greatest puzzler of our generation. How I looked eagerly to the next edition of Scientific American when I was younger. We didn't have much money but my dad insisted on purchasing a subscription to this classic magazine, intended for those scientists and non-scientists who wanted to know what was happening in the forefront of modern science and mathematics. Of course, I turned immediately to the back page to tackle another set of Mr. Gardner's challenging puzzles. I was so proud of myself if I could solve even one of these! Many of his puzzles had an almost magical quality to them. Now you see it -- now you don't. My forte was the logic type of puzzle but I tried them all.

Martin Gardner died Saturday, 5-22-10, at the age of 95. (See the puzzle  created below in dedication to Mr. Gardner).
By the way, 95 = 19x5, 94 = 47x2, 93 = 31x3.
It is only fitting that he left us at an age which is the largest 2-digit number with exactly two prime factors.

For you puzzlers out there, here is my conundrum dedicated to Mr. Gardner. Feel free to submit your solution, but only one,  in the comments to this post. Our readers can choose which one they think is the most elegant. I found one way, but I'm certain there are others!

Can you form 95 using each of the digits 5-2-2-1-0 exactly once. No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc...  You may combine the digits to form numerals like 12 or 120.

He was not a mathematician, nor a professor, nor a scientist. Yet I feel strongly that he deeply influenced all of these groups as well as anyone who enjoyed the satisfaction of challenging the mind. Read about him in the Wikipedia article and in the many tributes. If you're too young to have experienced the sheer joy brought to so many of us then discover it for yourself by looking at the annals of Scientific American or reading one of Mr. Gardner's many books.

Martin Gardner was more than a maker of puzzles of course. He was also known as a debunker of quackery and pseudoscience. He was an amateur magician, a philosopher, a lover of knowledge, a true Renaissance Man - a man for the ages.

Dr. Gardner - thank you for making a difference in my life and the life of so many others. Now if only I could remember how to get the cherry out of the martini glass by moving two matches...

On behalf of all my fellow bloggers, my sincerest condolences to your family.

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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

Uh Oh -- The 2-yr old turns 3 -- or does he!

Hey, these are precious moments and, anyway, I can't do more than one video a week!  So to take a respite. here's another anecdote, and, by the way, his 6-yr old brother and 3-yr old cousin are now demanding equal time.

So, he turned three the other day and when he went to bed that night his mommy, aka my daughter, told him: "____, now that you're three, I'm just going to tell you a story, give you a hug and kiss and say goodnight" (rather than staying with him until he fell asleep).

Knowing how this young man's mind operates, what do you think his reply was?

"Mommy, I'm only three during the day, I'm still two at night!"


Good luck to my daughter and all of his teachers!


Ok, fair's fair...
When the 6-yr old brother who was then 5, got his new bed (full size like his parents) my daughter and son-in-law heard him pacing and sighing loudly outside their door after they had tucked him in. Finally, his mommy came out and asked him what was wrong. Here was his response:

"I don't want to sleep in that bed. I'm not ready to be married!" 

This young man definitely marches to a different drummer, perhaps an entirely different band! He does remind me so much of myself at that age. I'm not sure if that's good...

Finally, my son's beautiful 3-yr old daughter went to Disney World a month or so ago. My wife asked her what her favorite ride was and she replied, "the rollercoaster." Then my wife asked if she was afraid. This was her reply:


"I was not afraid. I just screamed the whole way."


May their innocence remain forever...

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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

Challenging Geometry Assumptions: Review for SAT I/II

The video below presents a more challenging 3-dimensional geometry problem which would be at the upper end of SAT I or SAT II - Subject Tests (Math I/II). The key here is to challenge students' assumptions about a quadrilateral being a square because it has 4 congruent sides, a common error. This question will also review a considerable amount of geometry: Pythagorean Theorem, Volume of cube, spatial reasoning, 45-45-90 triangles, area of a rhombus, etc.

As always, the focus is on the art of questioning, suggested instructional strategies and pedagogy, although this problem may be interesting enough to capture the attention of some students who are preparing for upcoming standardized tests. For students who need help with spatial visualization, a model could be provided or have enough empty boxes available (they don't have to be cubes!).  

I strongly urge using learning partners or pairs for the discussion. 

Benefits include:

(1) Students feel less tentative when offering ideas to one other person or in a small group.

(2) Instead of posing conceptual questions to individuals, receiving little or no response except from the most confident or capable, you can pose a question to a learning pair: "Julie and Jason, what is needed to insure that ABCD is a square?" They should be given a few moments to think and confer before responding. The stronger student will usually explain it to the other. If neither can respond, they can say, "Pass!"

(3) The biggest advantage of student dialog is that often our explanations simply don't click with several students, but they do make sense to others. Those who "get it" can usually explain it in terms that their peers understand better, a benefit to both the "explainer" and the "explainee"!



By the way, the question posed near the end of the video is worth pursuing if time permits:

"Without calculating the areas, is the area of the non-square rhombus less than or greater than the area of the square?"

The answer is less for many reasons, but we would hope they would recall the base x height formula for a rhombus. The height is maximized when the angle between the sides is 90°. Why? Interestingly, the areas are quite close: 19.6 vs. 20. I believe strongly that this is the type of higher-order question that not only reviews important concepts but promotes deeper thinking, or should I say, thinking more than one inch deep!

What are your thoughts? Would you give students the e√3 formula before a standardized test or ever?Are these videos helpful to you? If you respond both on this blog and on my YouTube Channel, MathNotationsVids, and also rate these videos, that gives me the guidance I need to improve them.


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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

If a hen and a half can lay an egg and a half in a day and a half...

The full version in one of its many many variations:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Putting aside the silliness of the riddle, there really is some serious mathematics going in these kinds of rate/ratio/proportion problems. Rather than solve the "hen" problem for you, I'll leave it to my readers to solve it by their own favorite methods. By the way, the answer to this riddle is in the description of the video below on my YouTube channel. Sorry 'bout that!!

Instead, the video below, which appears on my YouTube channel, MathNotationsVids, presents a developmental approach to a more complicated ratio problem for middle schoolers and beyond. I'm far more interested in your thoughts about the teaching strategies than I am about the problem itself. Please understand, further, that I am not suggesting the method shown in the video is efficient nor would it make much sense for the upper level math or science student. See comments below the video for further discussion of this.


The Problem in the Video Below:


If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days? Assume all workers build at the same rate.




More Advanced and Efficient Algorithms


(1) We assume from the "constant rate" assumption in  the problem that the number of houses (H) which can be built varies jointly as the number of workers (W) and the number of days (D).
Thus, H = kWD.

Substituting, H=3, W=10 and D=60, we obtain:
3 = k(10)(60) or k = 1/200. Note that the units of k are Houses/(Workers x Days).
We can interpret k to mean that 1/200 of a house can be built by 1 worker in 1 day. Thus, k is not only a constant but actually represents a rate. Another way of expressing this rate is
(1 House)/(200 Worker-Days) or the reciprocal version:
(200 Worker⋅Days)/(1 House)

Substituting the new set of values into the relationship H = (1/200)WD, we obtain:
5 = (1/200)(W)(40) or W = 25 workers.

(2) This can be made even more efficient using the "factor-label" (dimensional analysis, etc.) format:

(200 Worker⋅Days)/(1 House)) x (5 Houses)/(40 Days) = 25 Workers!

(3)  I could also exploit the inverse variation between W and D, but that's for my readers to bring up or for another video!

I see these efficient methods as "black box" methods for some students. Developing a deeper understanding of direct and inverse variation is far more important for the younger student.



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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)