Friday, May 29, 2009

Geometry Challenge for SAT Prep or Review for Final Exam

In the coordinate plane, what is the area of ΔPQR given the coordinates P(4.5,4.5), Q(8.5,8.5), and R(6,0)?

Comments

  • Example of "Grid-In" or student-constructed response question on the SAT
  • This question seems more difficult than it really is. Students often give up on questions near the end of a section. DON'T!!
  • Hopefully you will view the utility of questions like this as I do:
    Students become more competent problem solvers only when challenged with nonroutine problems which are not always to be found in the textbooks. Questions like these should become more routine in our texts and in our classes (not only honors!).
For further strategies, answer/solution, discussion, click Read more...


Answer: 12
Solution (no explanation, details omitted):
(1/2)(6)(8.5) - (1/2)(6)(4.5) = 12


Discussion Points
  • What are some problem-solving strategies we need to review with our students here? Draw a diagram for sure but what are some other general attack strategies students should employ in triangle area problems?
  • Although advanced theorems could be used here, the actual solution given above is efficient and fairly basic. But what insights are needed to use that approach? What geometry or algebra standards are being tested here?
  • I chose this problem because coordinate geometry problems connect many important ideas in geometry and algebra. Not to mention that they are becoming more common on standardized tests like SATs, ACTs and state assessments. Besides, I enjoyed writing the question! Sometimes I'll get the germ of an idea, re-work it many times and then the question takes on a life of its own.
  • If you find an error in my work or want to share your thoughts, please add a comment!


...Read more

Tuesday, May 19, 2009

Light Humor and More on a Math Puzzle

We all need a break from the seriousness of math, so here are some (actual) amusing headlines. Before I get nasty comments, remember these are real!

1. Teachers Strike Idle Kids
2. Eye Drops Off Shelf
3. Miners Refuse To Work After Death
4. Hospitals Are Sued By 7 Foot Doctors
5. British Left Waffles On Falkland Islands

Ok, enough of the silliness...

I received several emails and comments about the math puzzle from the other day. Essentially, they all indicated that the problem was readily solved using technology.

First, here is the solution:
123456789 = 10821 x 11409

Aniket S., a computer science major from Solapur University, India, sent me the solution and the code she used in C++ to find the factors.

Both Mike Croucher and Pat Ballew, who emailed me, used the powerful new computing engine Wolfram Alpha to find the factors. As Pat pointed out, "the times they are a changin'...

In fact, I found the prime factors of 123456789 on the TI-89:

factor (123456789)
3^2 x 3607 x 3803

A few additional thoughts...
Is there anything special about the number 123456789? If we randomly permute the digits to, say, 586421739, we obtain the prime factorization 3^2 x 107 x 608953, which cannot be rearranged to form two 5-digit factors due to the 6-digit prime. Can you find a permutation of 123456789 which can also be factored into two 5-digit numbers?

If we make one change to the original number we can solve the puzzle:
123456784 = 10406 x 11864.

Have fun computing...



Monday, May 18, 2009

A Puzzle To Start the Week

Number puzzles always intrigued me and, perhaps, they are one way we can invite our students into the wonderful and exciting world of mathematics. Oh, alright, maybe that's a bit of a stretch, but, I suspect that if you give the following famous puzzle to your students in Grades 5 and up, they will try it even if you don't offer food or a 10 point bonus! Yes, calculators are allowed but after a few minutes of frustration they will be begging for a hint.

(Oh, and if you give them this problem at the beginning of class, you may as well forget the lesson!)

Find two 5-digit numbers whose product is 123456789.


If you solve it, don't post your answer immediately. I will probably publish a hint or the answer in a day or so. You can always email me with your solution at "dmarain at gmail dot com."

Click Read more for a hint and comments...


HINT: Rather than pressing random numbers into the calculator as some would do, encourage them to find the prime factors of 123456789. It's easy to show that this number is divisible by 3 and 9, but find finding the other factors will be challenging. I'll post another hint if you request it...

COMMENT: This beautiful puzzle was invented by Y. Yamamoto and has intrigued many puzzle enthusiasts for awhile now. Is there some profound meaning behind the solution or is it just a curiosity? Perhaps we'll have to wait for Dan Brown's next book to unlock the mystery! I will probably post the answer if I don't get a response within 24 hours. Probably...

If any of your students solve it, email me at "dmarain at gmail dot com" and let me know if I can post their names.

...Read more

Saturday, May 16, 2009

The "POWER" of Circles Part I - An Open-Ended Geometry Challenge and CONTEST Update

Well, registration for MathNotation's 2nd Online Contest has now closed. Not as many participants this time but we do have schools representing several states and one high school from Japan! The questions have been emailed to advisors but results will not be available for a couple of weeks. I also plan on publishing at least one of the contest problems in June. I'm still in the planning stages for running 2-3 of these contests for the 2009-10 school year. Stay tuned...

If you're interested in signing up for upcoming contests, just drop me an email at "dmarain at gmail dot com" and I will put you in my database.

For those of you who haven't been reading about these contests it's the team approach, open-endedness and multi-part nature of some of the problems which separates these contests from most others out there. In other words, these questions reflect the types of investigations I've been publishing since 2007.

Also, after I write 5-10 of these contests, I plan to publish these in a book with detailed solutions and comments. As my wife would say, "I'll believe that when I see it!"
**************************************************************************

Speaking of investigations, here is Part I of an open-ended geometry problem that starts out relatively simply but eventually will lead to deeper results. This investigation will review basic circle concepts involving tangents and secants but will connect to the more advanced ideas of power of a point and the inverse of a point in a circle in later posts.




In the diagram, O is the center of the circle PT is a tangent segment, segment AB is a diameter.







(a) If the radius is 6 and PA = 4, show that the length of tangent segment PT is 8.

Note that (PA)(PB) = (4)(16) = 64 and, from part (a), (PT)2 also equals 64. This is a special case of the secant-tangent power theorem you may recall from geometry. Your job in the next part is to demonstrate a particular case of this theorem this using the above diagram.

(b) If the radius of the circle is r and PA = x, show that
(PA)(PB) = (PT)2.

Note: Using the secant-tangent power theorem here trivializes this problem. The idea is to demonstrate the result without using that theorem, in effect, proving a special case of this rule!

Click on Read More for further comments and a hint for part (b)...




Comments
(i) Neither of the above parts was intended to be highly challenging. Part (a) is definitely an SAT-type question. A review of geometry never hurts!

(ii) Here's a hint to get started on part (b):
From Pythagorean we know that (PT)2 = (PO)2 - r2. Factor this expression...

(iii) Note that my approach as always is to introduce or develop a theorem or concept such as the secant-tangent power theorem or "power of a point" by looking at particular or special cases (the secant in the diagram contains a diameter) and starting with numerical values rather than variables. This sequence (numerical values, particular case, generalization) forms the basis of the investigations I've been writing for this blog, but, more importantly the underlying foundation for the lessons I planned when I taught. I believed then and now that this approach may be time-consuming (both in planning and implementation) but the payoff is deeper conceptual understanding. Of course I needed to modify the presentation according to the backgrounds and ability levels of my students but I never assumed only my honors and AP classes were up to the challenge.

(iv) In Part II which I will post in a few days, we will discuss the "power of a point" and add a second circle, in which segment PT will be a radius. This will produce a another point, P', the inverse of point P.

...Read more

Monday, May 11, 2009

A Sample Contest Problem (Open-Ended), Odds and Evens,...

First, some reminders and updates re the upcoming MathNotations (FREE) Online Math Contest:

  • I am still accepting registration up to this Fri 5-15-09 and that may be extended. Just email me at "dmarain at gmail dot com" and I will email you the registration and information forms in short order!
  • Because a few schools have expressed concern that some students are still taking AP's next week (makeups?) or their brains will be fried after this weeks AP's, I am willing to allow sponsors to administer the contest either the week of the 18th or the week of the 25th (after Memorial Day of course here in the US).
  • Participating students should review their trig identities, infinite geometric series and probability. However, there are other questions or parts that do not involve these more advanced topics.
  • Several questions are multi-part with later parts of increasing difficulty.
  • At least one question requires a detailed explanation, i.e., showing one's method clearly.
ODDS and EVENS...
  • Have you been keeping up with Burt's insightful comments, clear explanations and advocacy for balancing concept and procedure in our classrooms, K-12? Read Burt's comments to this post...
  • I've been remiss in keeping up with all the carnivals. I will get caught up in a few days.
  • Been thinking about the AP issues I raised in a recent post (from the NYT article)? I will have more to say about this, particularly based on the reader comments to that article. Link to the Times article from my post and skim through the 60 or so reader comments. Fascinating stuff...


Here's a sample contest question that demonstrates showing all work. Some of you may recognize a similar question posted earlier on MathNotations.

(i) Consider the circle of radius 1 centered at (0,0). Let L be the line tangent to this circle at the point (a,b) in the first quadrant. If P and Q are the x- and y-intercepts of L, respectively, show that the length of segment PQ equals 1/(ab). All work must be shown clearly.

(ii) Same as part (i) except the radius of the circle is now r. Show that the length of segment PQ can be expressed as r3/(ab). All work must be shown clearly.











Saturday, May 2, 2009

A Tale of Two Equations: Balancing Procedures and Conceptual Understanding

WHAT WILL YOUR STUDENTS BE DOING AFTER THE AP'S?
TAKING MATHNOTATIONS 2ND ONLINE (FREE) MATH CONTEST!

UPDATE
-- Registration deadline extended until Fri May 15th!

FOR MORE INFO, LINK HERE. We already have several schools registered but there's room for more!

The following is intended for all students in 2nd year algebra. Your stronger students should not find these overly challenging but there is more here than meets the eye. The purpose here is to demonstrate how we can review procedures AND develop deeper understanding of important mathematical ideas in the same lesson. The graphing calculator can be used to enhance the lesson by employing multiple representations (Rule of Four) to reinforce the essential ideas.

Note: Finding the solutions is only the tip of the iceberg. Understanding WHY one equation must have finitely many solutions and the other must have infinitely many solutions is the bigger idea here...

SOLVE EACH OF THE FOLLOWING

Equation 1:
(x-1)(x-2) = (1-x)(2-x)


Equation 2:
(x-1)(x-2)(x-3) = (1-x)(2-x)(3-x)


Click on Read More... for solutions and further discussion.




ANSWERS
Equation 1: All real numbers
Equation 2: {1,2,3}


DISCUSSION

Would most of your students eliminate parentheses in the first equation and solve by traditional methods? Even though the left and right sides of the equation appear similar, it is reasonable to expect they will distribute and solve since that is what they're used to doing. This is fine and the standard procedure should be reviewed.

Assuming students will not make "careless" mechanical errors in distributing, they should obtain:
x2 - 3x + 2 = 2 - 3x + x2.
This generates a nice discussion of an "identical equation" or identity since the left and right sides are mathematically equivalent (if they recognize that!). The instructor may or may not want to continue the mechanical approach of moving all terms to one side producing 0 = 0 to reinforce that the equation is satisfied by all real numbers. Your stronger student will not have much difficulty with this.

Before moving on to the 2nd equation, we can develop a
deeper conceptual understanding by asking students to approach the problem another way. We know that some students will wonder about the form of the original equation. Could we have predicted that the two sides would be identical without removing parentheses? Could we also have determined by inspection that both x = 1 and x = 2 are solutions? Asking them to revisit the original equation to see this is critical. Now what about trying some other real number, say x = 5. This should strongly suggest that all real numbers will satisfy the equation. Using the graphing calculator will also drive this point home visually. Store the left side of the equation Y1 and the right side of the equation in Y2. Change the appearance of Y2 (make it bold for example) and have them observe on the viewscreen that the graphs are identical.

So, how come the 2nd equation only has 3 solutions! I'll leave that to my readers to elaborate on...
How can we generalize this?

These kinds of lessons seem to involve way too much overhead, stealing so much valuable time away from other content. BUT these are precisely the kinds of problems students are expected to grapple with in Japan and other countries. Do you really believe "Less is More?"


...Read more

Wednesday, April 29, 2009

NY Times Article on AP Courses Raises Critical Issues

The article published 4-29-09 is entitled, Many Teachers in Advanced Placement Voice Concern at Its Rapid Growth. Look here for the complete text. With the AP Tests only a few days away this is certainly timely and will surely generate many reader comments on the NYT website. I'm sure it will also appear in Education Week and many other print and online media. The article refers to a study done by the prestigious Thomas B. Fordham Institute and the opening paragraph really says it all:

A survey of more than 1,000 teachers of Advanced Placement courses in American high schools has found that more than half are concerned that the program’s effectiveness is being threatened as districts loosen restrictions on who can take such rigorous courses and as students flock to them to polish their résumés.
Having taught AP classes for a few decades I have strong feelings about the equity vs. excellence issues surrounding these classes but I would rather raise a series of questions and ask my readers to provide some data as well as voice their opinions.

Trevor Packer, the Vice President of the College Board, was very pleased by the "questions the report asks" and welcomes the dialog. I'm not so sure, however, that he is pleased by the majority sentiment of AP teachers that the quality of these classes has been compromised by encouraging more and more students to take classes for which they may not be prepared.

My sense is that like most complex issues we need to first determine who is doing what out there. I believe that there is wide disparity among districts regarding entrance into these classes and whether AP tests are required or not. The College Board has attempted to address some of these issues with their new AP Audit process but I wonder how effective that really is.

These questions are being posed to AP Calculus teachers or those familiar with how their district handles these decisions.

1) What stated prerequisites (e.g., in a Program of Studies), if any, are there in your district for entrance into an AP class?
For example:
"Students must earn a grade of B or better in Precalculus and/or be recommended by their teacher."

2) Is the test mandatory for all students in the class? If a student does not take the test do they still receive AP credit for the class? (This is more relevant of course, transcript-wise, for Juniors, Sophs and Freshmen).

3) Who pays the $86 for each AP test?
Note: I'm guessing that parents in most cases pick up the tab but, if a hardship can be documented, the district would pay. I've also read that in some states the cost is covered by the state or district.

Click Read more... for questions regarding your opinions.



Some more questions/thoughts...

4) Do you believe that strict entrance requirements are needed to insure the quality of the course or that enrollment will ultimately be affected by the reputation (quality, expectations and grading standards) of the teacher?


5) Your other thoughts...




...Read more

Tuesday, April 28, 2009

Strong Opinions from Curmudgeon Re Math Education...

DON'T FORGET -- YOU STILL HAVE ABOUT TWO WEEKS TO REGISTER FOR THE 2ND ONLINE MATHNOTATIONS (FREE) ONLINE CONTEST. TO LEARN MORE, LINK HERE. We already have several schools registered but there's room for more!



An innocent recent post, Two SAT-Type Percent Problems Appropriate for Middle School as well..., generated some intense and passionate opinions about the current state of US education, math in particular, from one of our readers, affectionately known as Curmudgeon (I'll abbreviate his name as CM).
Unless you subscribe to my Comments feed (see link in sidebar), you might not have followed this dialog so I've decided to post a link to these comments and to copy some of the more provocative ones in this post.

Based on his comments I find CM to be highly informed, articulate and not intimidated by anyone. He eloquently challenged my difficulty rating system for the SAT-type problems I have posted and he made some excellent points. Re curriculum/instruction, his perspective is somewhat similar to mine although I lean a bit more toward a balanced view. A direct quote describing his blog summarizes his beliefs very well.

"I believe that mathematics should be taught, not collaboratively explored; algebra and geometry are better than a vague course of Integrated Math; spiraling doesn't work nearly as well as learning it properly the first time; "I don't DO math" should be an incentive rather than an excuse. "I don't DO English" should be treated the same way."


I encourage you to visit his blog, Curmudgeon.

Below is CM's long response to my remarks about the teaching of the part:part construct which, IMO, receives far less emphasis in classrooms than part:whole approaches.


Rarely taught? Probably, for three reasons. The first is that teachers are under pressure to "finish" things, to check off the standard and move on, to "get through" the material. As soon as the kids "know" the material, it's time to move on. There is rarely time for the extensive exploration that I seem to recall from my own days. Standards-based educational theory says that you need to pick and choose your topics until you get each kid to the understanding point but says nothing about total mastery. "Drill and kill" is an epithet. "Drill and Practice" is unknown. Many people have also fallen for the "spiraling" fad and never quite complete a thought before they're on to the next one. "We'll spiral around to this again in that module in next year's course" is probably the dumbest thing ever to come out of edumacation colleges. I can't tell you how many times (because I've lost count) I and the other teachers have been told that we needed to get our "bubble" kids over the line into the passing zone. "To hell with knowing math, just know enough for the test" seems to be the rallying cry. The upshot is that you can mention these things to the better students who will get it easily and gain an even better understanding. The weaker students just trundle along. The second reason is that weaker students are resistant to trying a second or third idea. They want to understand THIS one. A second approach is confusing. It takes a while before they get comfortable with multiple approaches and some never get much beyond "I'm not a good math student. I'll learn this but only to a point." It takes a determined teacher to ease them into this, but she can't have anyone breathing down her neck to do test-prep. The worst reason is that a fairly large percentage of our teachers don't really understand math to the level required. They've either bought into failed and worthless education theory or they simply are stupid. I've related the story of the fourth grade teacher in the in-service this year ... Me: "You say you want to be a guide on the side not a sage on the stage, yet you're teaching 4th graders. You still need to teach them things. They have to memorize 4*3 for example." Her: "No. We should be teaching them how to look that up on Google. Did you know that Google will give you the answer to that? It's true." She leaned back, satisfied that she had put one over on me. Is it any wonder that her kids arrive with no understanding of fractions, decimals, percents, operations? "Where's the fraction key?" "Sorry, that calculator doesn't have one, the problem you're doing doesn't need one and you wasted more time than if you just looked at it and solved it." And some elementary and middle-school teachers "just don't DO math, tee-hee-hee." They SEEM to do math - they have lots of test-prep bubbling exercises from the publisher of the math book, but they fundamentally don't understand the nuances of the material. "Let's see what the calculator says." In a different in-service this year, the presenter was showing how pre-schoolers and elementary kids learn math. She had lots of visuals and was "teaching" the teachers as if they were students. The aides were getting questions wrong and the elementary teachers weren't doing too much better. This explains a lot.


To read my response (aka, "rant'), click on Read more...
To read all the comments to this post and/or post a comment, look here.



May I call you C.M.??
I'm not going to say something inane like "I feel your pain." Most of the angst you expressed is a reflection of what I was feeling when I retired. You're describing a system that is in dire need of systemic change, not tweaking.


I will not apologize however for my advocacy of national standards in math. It is unconscionable that students across the country are not learning the same content - concepts, skills, procedures, terms, definitions,... This is truly inequitable.

However, I have also been preaching "LESS IS MORE" for the same period of time. Finally, NCTM has taken this position with their Curriculum Focal Points document. William Schmidt (of TIMSS renown) stated this obvious fact 15 years ago when he described our math curriculum as "one inch deep, one mile wide."

We cannot expect students to learn math well if we fill our textbooks each year with every topic under the sun. If, for example, our teachers could focus on the essential ideas of ratios and ALL students were required to solve a range of problems from the basic to the more challenging, then most students would eventually learn ratios and be able to handle fractions. BUT facility with ratios and fractions requires facility with division which requires mastery of multiplication, etc.

"Jack, you will learn the times tables. The facts you got wrong in class today, you will have to write five times each for homework tonight. Tomorrow, I will ask you to answer just those." I know there are some teachers out there who are doing this. Is everyone?

While base ten blocks, unifix cubes and learning software have their place, we all know that nothing replaces repetition. Some students need far less than others but all students need some.

Yes, C.M., there are serious teacher preparation issues out there. Read my comments at the bottom about Finland. Yes, C.M., I share your feelings about spiraling, although there are aspects of spiraling which make sense.

I am also concerned as you are that standards-based learning has devolved into learning only for the state assessment. While we are moving toward national standards, we have to rethink how we will evaluate student learning and the bottom line is: "WHAT IS THE REAL PURPOSE OF TESTING?" What you and I and millions of other teachers see is that testing has little to do with helping children improve. It has everything to do with POLITICS:

"MY DISTRICT IS BETTER THAN YOURS; MY STATE IS BETTER THAN YOUR STATE; LET'S HOLD THOSE D*** TEACHERS ACCOUNTABLE FOR THOSE 'HIGH' SALARIES THEY'RE GETTING PAID"; "WE HAVE TO JUSTIFY ALL THAT TAX MONEY WE PAY FOR EDUCATION."

Remember that quote:
"In other countries, education is seen as an investment; in the US, it is seen as an expense."

Perhaps this administration will "see" it differently. I truly hope so. In Finland, teachers, have to take additional years of training beyond college before they officially are certified. This additional 2-3 years culminates in a year of working in a laboratory school with real students. A true internship. And, by the way, THE GOVERNMENT PAYS EVERY PENNY FOR ALL THIS ADDITIONAL TRAINING. This is how Finland has turned around its system in the past 20 years.

INVEST IN EDUCATION, INVEST IN OUR CHILDREN, INVEST IN OUR FUTURE. Don't look for short-cuts, folks. There are none. Expedient solutions lead to students who only care about the bottom line, the grade, not about learning. This "get results without really earning it" mentality is pervasive in our society. In the worst case, this mentality produces the AIGS, the Enrons and the Madoffs of the world.

Ok, now you got me to rant too. I guess I needed that catharsis. otherwise it sounds like I'm just pontificating about challenging our best and brightest with all these problems I'm writing. But there's so much more to it, C.M...

P.S. I have a feeling that your comments and mine are not being read by most of my readers. I'm thinking I should copy them into a separate post and really incite a riot! I think I will do that unless you state an objection! THANK YOU!



...Read more