MathNotations

There is still time to register for the upcoming MathNotations Third Online Math Team Contest, which should be administered on one of the days from Mon October 12th through Fri October 16th in a 45-minute time period.

Registration could not be easier this time around. Just email me at dmarain "at" "gamil dot com" and include your full name, title, name and full address of your school (indicate if Middle or Secondary School).

Be sure to include THIRD MATHNOTATIONS ONLINE CONTEST in the subject/title of the email. I will accept registrations up to Fri October 9th (exceptions can always be made!).

BASIC RULES
* Your school can field up to two teams with from two to six members on each. (A team of one requires special approval).
* Schools can be from anywhere on our planet and we encourage homeschooling teams as well.
* The contest includes topics from 2nd year algebra (including sequences, series), geometry, number theory and middle school math. I did not include any advanced math topics this time around, so don't worry about trig or logs.
* Questions may be multi-part and at least one is open-ended requiring careful justification (see example below).
* Few teams are expected to be able to finish all questions in the time allotted. Teams generally need to divide up the labor in order to have the best chance of completing the test.
* Calculators are permitted (no restrictions) but no computer mathematical software like Mathematica can be used.
* Computers can be used (no internet access) to type solutions in Microsoft Word. Answers and solutions can also be written by hand and scanned (preferred). A pdf file is also fine.


Ok, here's another sample contest problem, this time a "counting" question that is equally appropriate for middle schoolers and high schoolers:

How many 4-digit positive integers have distinct digits and the property that the product of their thousands' and hundreds' digits equals the product of their tens' and units' digits?

Comments
The math background here may be middle school but the reading comprehension level and specific knowledge of math terminology is quite high. This more than counting strategies is often an impediment. If this were an SAT-type question, an example would be given of such a number to give access to students who cannot decipher the problem, thereby testing the math more than the verbal side. On most contests, however, anything is fair game!

Beyond understanding what the question is asking, I believe there are some worthwhile counting strategies and combinatorial thinking involved here. Enjoy it!

Click More to see the result I came up with (although you may find an error and want to correct it!)




My Unofficial Answer: 40
(Please feel free to challenge that in your comments!!_

...Read more

Posted by Dave Marain at 6:29 AM 3 comments

Labels: combinatorial math, math contest problems, MathNotations Contest, middle school, more

At the same instant of time, trains A and B enter the opposite ends of a tunnel which is 1/5 mile long. Don't worry -- they are on parallel tracks and no collision occurs!

Train A is traveling at 75 mi/hr and is 1/3 mile long.
Train B is traveling at 100 mi/hr and is 1/4 mile long.

When the rear of train B just emerges from the tunnel, in exactly how many more seconds will it take the rear of train A to emerge?

Click on More to see answer (Feed subscribers should see answer immediately).

Comments
1. Appropriate for middle schoolers even before algebra? Exactly when are middle schoolers in your district introduced to the fundamental Rate_Time_Distance relationship?
2. What benefits do you think result from tackling this kind of exercise? If it's not going to be tested on your standardized tests, is it worth all the time and effort?
3. How much "trackwork" needs to be laid before students are ready for this level of problem-solving?
4. As an instructional strategy, would you have the problem acted out with models in the room or use actual students to represent the trains and the tunnel? OR just have them draw a diagram and go from there? Do a simulation on the TI-Inspire or TI-84 using graphics and parametric equations for the older students?
5. If you believe there is still a place for this type of problem-solving, should it be given only to the advanced classes and depicted as a math contest challenge?
6. I'm dating myself but I remember seeing problems like this in my old yellow Algebra 2 textbook? Uh, I believe this was B.C. -- before calculators! Can you imagine! Do you recall these kinds of problems? Do you recall the author or publisher?
7. Of course, the proverbial "two trains and tunnel" problems are frequently parodied and used as emblematic of the "old math"! They've been replaced by "real-world" applications. "Progress makes perfect!"

YOUR THOUGHTS...





Answer: 9.4 seconds (challenge this if you think I erred!)

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Posted by Dave Marain at 6:16 AM 9 comments

Labels: algebra, more, prealgebra, problem-solving, two trains in the tunnel classic, word problems

UPDATE #2: Answers to the quiz are now provided at the bottom. If you disagree with any answers or would like clarification, don't hesitate to post a comment or send an email to dmarain "at gmail dot com".

UPDATE: No comments from my faithful readers yet -- I suspect they are giving students a chance to try these! I will post answers on Friday 9-25. However, students or any readers who would like to check their answers against mine need only email me at dmarain "at" gmail "dot" com and I will let them know how they did!


With the SAT/PSAT coming in a few weeks, I thought it would be helpful to your students to try a challenging "quiz". Most of these questions represent the high end level of difficulty and some are intentionally above the level of these tests. Then again, difficulty is very subjective. A student taking Honors Precalculus would have a very different perspective from the student starting Algebra 2!

Also, these questions can also be used to prepare for some math contests such as the THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST! Yes, another shameless plug, but time is running out for your registration...

A Few Reminders For Students

(1) Do not worry about the time these take although I would suggest about 30 minutes. The idea is to try these, then correct mistakes and/or learn methods/strategies. It's what you do after this quiz that will be of most benefit!

(2) I added strategies and comments after the quiz. I suggest trying as many as you can without looking at these. Then go back, read the comments and re-try some. I will not provide answers yet!

(3) Don't forget these problems are copyrighted and cannot be reproduced for commercial use. See the Creative Commons License in the sidebar. Thank you...


PRACTICE PSAT/SAT QUIZ
1. If n is an even positive integer, how many digits of 100²ⁿ - 100^2n-2 will be equal to 9 when the expression is expanded?

(A) 2 (B) 4 (C) 8 (E) 2n (E) 2n - 4


2. The sides of a triangle have lengths a, b and c. Let S represent (a+b+c)/2. Which of the following could be true?

I. S is less than c
II. S> c
III. S = c

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III


3. The mean, median and mode of 3 numbers are x, x+1 and x+1 respectively. Which of the following represents the least of the 3 numbers?

(A) x (B) x - 1 (C) x - 2 (D) x-3 (E) 2x - 2


4. (10/√5)⁵⁰⁰ (1/(2√5))⁵⁰⁰ = _________


5. A point P(x,y) lies on the graph of the equation x²y² = 64. If x and y are both integers, how many such points are there?

(A) 4 (B) 8 (C) 16 (D) 32 (E 64


6. Each side of a parallelogram is increased by 50% while the shape is preserved. By what percent is the area of the parallelogram increased? __________


7.


AB is parallel to CD , AB = 3, CD = 5, AD = BC = 4. If segments AD and BC are extended to form a triangle ABE (not shown), what would be the length of AE?
Ans_________

Figure not drawn to scale



-----------------------------------------------------------------------------------------------
STRATEGIES/COMMENTS

1. Most students learn to substitute numbers for n here although it can be done algebraically by factoring. However, the real issue here is figuring out what the question is asking. Reading interpretation - ugh!!

2. When you are not given any information about what type of triangle it is, just choose a few special cases and draw a conclusion. O course, if one recalls a key inequality theorem from geometry, this problem can be done in short order.

3. If you don't feel comfortable setting this up algebraically (preferred method), PLUG IN A VALUE FOR x...

4. Your calculator may not be able to handle the exponent so skills are needed. The large exponent also suggests a Make it Simpler strategy. This is a "Grid-In" question so if one is guessing remember that most answers are simple whole numbers! Finally, if one knows their basic exponent rules and basic radical simplification, none of the above strategies are needed!

5. Possibilities should be listed carefully. It is possible to count these efficiently by recognizing the effect of reversals and signs. Easy to get this one wrong if not careful.

6. For those who do not remember or want to apply a key geometry concept about ratios in similar figures, there are a couple of essential test-taking strategies which all students should be aware of of:
(a) Consider a special case of a parallelogram
(b) choose particular values for the sides.
In the end, even strong students often make a different error, however. That darn ol' percent increase idea!

7. Should you skip this if you have no idea how to start? Absolutely not! Draw a complete diagram and even if you don't recognize the similar triangles, make an educated guess! It's a grid-in and there's no penalty for guessing. Further, answers tend to be positivc integers!!

-----------------------------------------------------------------------------------------------------
ANSWERS

1. B

2. B

3. C

4. 1

5. C

6. 125

7. 6

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Posted by Dave Marain at 6:18 AM 1 comments

Labels: math contest problems, MathNotations Contest, more, PSAT, SAT strategies, SAT-type problems

Have you forgotten to register for MathNotation's Third FREE Online Math Contest coming in mid-October? We already have several schools (from around the world!) registered. For details, link here or check the first item in the right sidebar!!

Before tackling a more challenging problem in the classroom, I would typically begin with one or more simpler examples. My objective was to review essential concepts and skills and demonstrate key ideas in the harder problem. This incremental approach (sometimes referred to as scaffolding) enabled some students to solve the problem or at least get started. Usually within each group of 3-4 students, there was at least one who could help the others. Some groups or classes might still not be ready after one example, so more would be needed. I never felt that this expense of time was too costly since my goal was to develop both skill and understanding.


SIMPLER EXAMPLE
Consider the following two statements about positive numbers A and B:

(1) A is 80% of B.
(2) A is 20% less than B .

Are these equivalent, that is, if values of A and B satisfy (1), will they also hold true for (2) and conversely?

How would you get this idea across to your students?

Again, depending on the students, I would often allow them to discuss it first in small groups for two minutes, then open up the discussion.

Note: If the group lacks the skills, confidence or background (note that I left ability out, intentionally!), I might first start with concrete values before giving them the 2 statements above: E.g., What is 80% of 100?

How would I summarize the methods of solution to this question. Here's what I attempted to do in each lesson. I didn't reach everyone but I found from further questioning and subsequent assessment that this multi-pronged approach was more successful than previous methods I had used. Most of these methods came from the students themselves!

INSTRUCTIONAL STRATEGIES
I. Choose a particular value for one of the numbers, say B = 100. Ask WHY it makes sense to start with B first and why does it make sense to use 100. Calculate the value of A and discuss.

II. Draw a pie chart (circle graph) showing the relationship between A and B. Stress that B would represent the whole or 100%.

III. Write out the sentence:
80% of B is the same as 100% of B - 20% of B
In other words:
80% of B is the same as 20% less than B.

IV. Express algebraically (as appropriate):
0.8B = 1B - 0.2B

Numerical (concrete values)
Visual (Pie chart)
Verbal (using natural language)
Symbolic (algebra)

Yes, it's Multiple Representations! The Rule of Four!

To me, it's all about accessing different modes of how students process. Call it learning styles, brain-based learning, etc., it still comes down to:
RARELY DOES ONE METHOD OF EXPLANATION, NO MATTER HOW CLEAR OR STRUCTURED, REACH A MAJORITY OF STUDENTS. YOUR FAVORITE EXPLANATION WILL MAKE THE MOST SENSE TO THE STUDENTS WHO THINK LIKE YOU!!


Now for today's challenge.
(Assume all variables represent positive numbers)

M is x% less than P and N is x% less than Q. If MN is 36% less than PQ, what is the value of x?

Can you think of several methods?
I will suggest one of the favorite of many successful students on standardized assessments:

Choose P = 10, Q = 10. Then...

Click on More (subscribers do not need to do this) to see the answer without details.




Answer: x = 20

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Posted by Dave Marain at 6:02 AM 1 comments

Labels: conceptual understanding, instructional strategies, more, percent, percent word problem, SAT strategies, SAT-type problems

How many integers from -1001 ro 1001 inclusive are not equal to the cube of an integer?

Hint: This could be a real 'Thriller'!

Click Read more for comments...


Comments
1) Do you think daily exposure to these kinds of problems as early as 7th grade will improve student thinking, careful attention to details (reading!) and ultimately performance on assessments? I think you can guess my answer!

2) I've published many similar questions on my blog but I couldn't resist this tribute to MJ.

3) I strongly believe we must occasionally remove the calculator to force their thinking. The stronger student recognizes immediately that 1000 and -1000 are perfect cubes and that one does not need to count the cubes but rather the integers which are being cubed (aka, their cube roots). The student with less number sense and weaker basics will feel lost at first but eventually their minds will develop as well if challenged regularly.

4) I added some complications to this fairly common 'counting' problem, similar to many SAT problems. This type of question is also typical of 8th grade math contests. Where do you think the common errors would occur assuming the student has some idea of how to approach this? Is understanding the language the primary barrier or not?

5) Let me know if you use this in September to set the tone for the year!

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Posted by Dave Marain at 6:12 AM 12 comments

Labels: math contest problems, middle school, more, SAT-type problems, warmup

Typical Classroom Scenario?
We're introducing the idea of least common multiple of two positive integers and after defining the terminology and illustrating several examples most students are catching on to some procedural method of which there are many:
Listing common multiples of each
Prime Factorization
The "upside down division method" you saw at a conference...

Yes, we are all very good at demonstrating step by step procedures and having students practice repetitively until they catch on and can reproduce this with some speed and accuracy. We feel this is a worthwhile skill (they'll need it for common denominators, clearing denominators in rational equations, useful for solving certain types of word problems, etc), it's in the curriculum and the standards, it will be tested in various places and the lesson plays out. Some students pick up the method(s) quickly, while others struggle, particularly those who haven't learned their basic facts.

BUT how can we raise the bar to stretch their minds? Can the above scenario be restructured to enable students to gain a deeper understanding of the concepts of lcm and gcf? Perhaps we can start the class off with a more open-ended type of question and ask them to work in small groups to solve it. Perhaps, we can ask a different type of question after teaching some standard procedure. A nonroutine, higher-order question that is not in the text...

What resources are available for more open-ended or nonroutine questions to enable our students to delve beneath the surface and actually think about what they are doing? Well, I can't answer all these questions but here are a few thoughts...


1) Write two examples for which the lcm of two numbers is their product.
2) Write two examples for which the lcm of two numbers is not their product. The numbers in each example must be distinct (different).

3) The lcm of 12 and N is 24.
a) What is the greatest possible integer value of N?
b) What is the least positive integer value of N?


These are just a few samples to start you off. You could probably come up with better ones or you've read some excellent ideas in some publication. Please share...

To see a more challenging version of the examples above, click Read more...



You might want to give the following for homework or an extra practice problem in class. Do you think students will require a calculator? How about telling them they cannot use it!

The lcm of 100 and N is 500. What is the least positive integer value of N?

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Posted by Dave Marain at 7:30 AM 16 comments

Labels: higher-order questions, lcm, middle school, more, number theory, teaching for understanding

As my devoted readers know, Totally Clueless, affectionately known as TC, has contributed many insightful comments and profound ideas for us to think about. His sobriquet belies a brilliant creative mind of course. He recently sent me a geometry problem which was motivated by his own experiences driving to work. The problem itself is accessible to advanced middle and secondary students but the result is interesting in its own right and should generate rich discussion in class. I recommend giving this as a group activity, allowing about 15 minutes for students to work on, then another 15 minutes to discuss it. Save it for an end-of-year activity or bookmark it for the future. Beyond the problem, there are important pedagogical issues here:

• How to introduce this
• Asking questions to provoke deeper thought
• Drawing conclusions and further generalizations
• Connecting this problem to other circle or geometry problems
  
• Maximizing student involvement
  

I told TC I would need some time to rework the original problem for the younger students so here goes...





Diagram for Parts I and II







Part I (middle and secondary students)
In my city, there are two circular roads "around the center" of the city, of radii 6 and 4. There are a number of radial roads that connect the two loops. Points A and B in the diagram above are at opposite ends of a diameter of the outer loop and the dashed segment is a diameter of the inner loop.

If I have to go from point A to point B on
the outer loop, I have two options:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially (blue) to the inner loop, drive along the inner loop (red), and then drive radially out (blue). (Assume that there are radial roads that end at point A and point B).

Show that Option 2 is shorter than Option 1.

Part II (middle and secondary students)
Same diagram but now the radii are R and r with R> r.
Show algebraically that Option 2 is shorter.


Part III (secondary students)












To generalize even further, points A and B are distinct arbitrary points on the circle, central angle AOB has radian measure θ where θ ≤ π. OC and OD are radii of the inner loop; OA and OB are radii of the outer loop. Again the radii of the two circles are R and r, where R> r.

As before, there are two options in going from A to B:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially from A to C (blue), then along the inner loop from C to D (red), then radially outward from D to B (blue).

Show that Option 2 will be shorter provided π ≥ θ> 2.

Click Read More for further discussion...


Further Comments
(1) TC's original problem was Part III. I decided to add Parts I and II to provide 'scaffolding' for students. Was this really necessary in your opinion?
(2) The results of these questions are independent of the actual radii. TC felt this was an interesting aspect of this problem and I agree. Do you think students will be surprised by this? Do we need to point this out to them? Are there other circle problems you can recall which have a similar feature?

Thanks TC for providing us with another stimulating challenge!

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Posted by Dave Marain at 8:15 AM 7 comments

Labels: circles, geometry, middle school, more, real-world applications, tc'sTotal Challenges

Well, the June SATs have arrived today so these problems come too late for that, but these kinds of questions can be used to review basic ideas while strengthening thinking skills. Both questions below are appropriate for both middle and secondary students, although the second requires knowledge of a fundamental geometry principle regarding the sides of triangles.

There are other important principles embedded in these problems as well. In the end, I believe that students need to be exposed to many of these "contest-type" challenges to improve reading skill, learn how to pay attention to detail and think clearly. As a separate issue, performing well under testing conditions requires extensive training. You may not feel this is an important objective for math teaching in the classroom, but testing is a reality for the student...

These questions may appear fairly straightforward at first but be careful! I believe the second is more challenging than the first. These are not so different from the "gotcha" problem on our latest online contest.

1) The dimensions of a rectangle are odd integers and its perimeter is 100. How many different values are possible for its area?

2) The perimeter of an isosceles triangle is 96 and the lengths of its sides are even integers. How many noncongruent triangles satisfy these conditions?

For my "unofficial" answers, click on Read more...


Unofficial Answers (no solutions):
1) 13
2) 11
Feel free to challenge these answers or express agreement!

Comments

Which of the following do you believe would cause the most difficulty for students?

• The wording/terminology (e.g., noncongruent); general reading comprehension issues
  
• The sheer number of details (e.g., odd vs. even, perimeter vs. area, integer values)
• A precise counting/listing strategy vs. an abstract or commonsense approach
• The "square is also a rectangle", "equilateral is also isosceles" traps
  
• The issue of different areas for #1
  
• The triangle inequality for #2
• Other concerns?
  

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Posted by Dave Marain at 7:48 AM 9 comments

Labels: critical thinking, geometry, math contest problems, more, SAT strategies, SAT-type problems

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!
  

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Posted by Dave Marain at 8:16 AM 9 comments

Labels: area, coordinate problems, geometry, more, SAT strategies, SAT-type problems, triangles

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.

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Posted by Dave Marain at 6:31 AM 3 comments

Labels: math challenge, middle school, more, puzzle

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.

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Posted by Dave Marain at 8:02 AM 3 comments

Labels: geometry, investigations, more, power of a point, SAT-type problems, secant-tangent power theorem

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?"

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Posted by Dave Marain at 6:02 AM 4 comments

Labels: algebra 2, conceptual understanding, instructional strategies, more

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ドル 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

Posted by Dave Marain at 6:17 AM 6 comments

Labels: AP Calculus, AP Courses/Policies, more

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

Posted by Dave Marain at 7:54 AM 11 comments

Labels: curriculum, issues in education, more

• Don't miss the latest Math Teachers at Play #5 . Denise somehow is pulling this together and producing a delectable and high-quality carnival of K-12 Math every two weeks and this one is no exception. I particularly enjoyed reading her intriguing quotes. One of my favorites came from Paul Halmos who quoted a cynic on student learning. Here's an excerpt...
  "...since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick"

• I've already received several requests from teachers for the registration form for MathNotation's 2nd FREE Online Contest. One request has made me re-think my rule about one team per school. So here is the latest revision:
  Schools may field up to TWO teams.
  
  A couple of other points about the contest...
  (1) Even if you're not sure you will be able to field a team, don't hesitate to email me (dmarain at gmail dot com) to request a registration form - no obligation here!
  (2) I want to reiterate that this contest is designed for the student who has completed Algebra 2 and knows some trigonometry.
  (3) Please include the phrase 2nd MathNotations Contest in the subject line of your email to me. Just copy and paste that to make it easier for me to organize all of my emails and not miss any.

• Walter Isaacson, former managing editor of TIME magazine has written a definitive piece on the National Standards movement and the need for improved quality of standards in this week's edition of TIME. It's entitled How to Raise The Standard In America's Schools and I strongly urge all my readers to read the online version at TIME.com.

Click Read more for further comments on the TIME article...


This one article justifies purchasing the print version of the magazine in my opinion. Mr. Isaacson documents the history of the Standards movement but more importantly makes two compelling points which my readers will surely recognize from the dozens of essays I've written on the subject:
(1) The existing standards are not working (most are of poor quality and there is considerable inconsistency of expectations from state to state).
(2) There is, not surprisingly, a significant gap between the proficiency levels being reported by many states (based on their own assessments) and the NAEP results which set a higher and more uniform standard.

How do I feel about my passionate beliefs and advocacy for uniform standards beginning to come to fruition two decades after I ranted on every forum for these changes? Relief, not self-satisfaction. It's long overdue and we better move quickly now. Right, Mr. Duncan?
...Read more

Posted by Dave Marain at 9:19 AM 2 comments

Labels: more, national standards, update

Don't forget to register for the upcoming 2nd MathNotations Free Online Contest for secondary students. Click here for more info.



The first 4 terms of a sequence are 2, 6, 18, and 54.
Each term after that is three times the preceding term.
If the sum of the 49th, 50th and 51st terms of this sequence is expressed as k⋅3⁴⁹, then k = ?

Click Read more to see the answer, solution, discussion...


Answer: 26/3

Suggested Solution
The first three terms can be written as
2(3⁰), 2(3¹) and 2(3²). (***)
In general, the nth term is 2(3^n-1).
The sum of the 3 desired terms would then be 2(3⁴⁸) + 2(3⁴⁹) +2(3⁵⁰). Factoring out 3⁴⁹, we obtain 3⁴⁹(2/3 + 2 + 6) = (26/3)(3⁴⁹), so k = 26/3.

Comments
(1) Too hard for SATs? Similar (but slightly easier) problems have appeared on the test.
(2) Could students use the "Make it simpler strategy" here to reduce the problem to the sum of just the first three terms? But this is the essence of geometric sequences (or exponential functions):
From (***) above, this sum would be
2(3⁰) + 2(3¹) + 2(3²) = 2 + 6 + 18 = 3(2/3 + 2 +6) = 3(26/3). The coefficient 26/3 would be the same for any three consecutive terms! Is this concept/technique worth developing?

...Read more

Posted by Dave Marain at 7:37 PM 4 comments

Labels: advanced algebra, exponents, geometric sequence, math contest problems, more, SAT-type problems

If interested in participating, please send an email as soon as possible to "dmarain at gmail dot com." I will then email a registration form. Your initial email expresses only your interest. You are under no obligation.

There will be several changes from the first contest.

1) Team advisers may administer the contest at any time during the week of May 18th-22nd.
2) Advisers must email the registration form no later than Fri 5-15-09.
3) The contest is designed for secondary students who have completed Algebra 2. Some trigonometry may be necessary. I would not recommend middle schoolers take the contest unless they have completed or are completing Algebra 2.
4) Questions include topics from geometry, algebra, trigonometry, discrete math, etc. No calculus...
5) Teams must consist of from two to six members. Homeschooling and international teams are welcome!

For more details, click Read more.




FORMAT OF TEST
Questions types include
(a) Short constructed response (students enter only numerical answers)
(b) Open-ended requiring detailed work and explanations
(c) Multi-part questions


ADMINISTRATION OF CONTEST
1) Team members must complete the contest within 45 minutes on the same day.
2) Advisers must email the official answer form the same day the contest is administered. Scanned solutions will be accepted.
3) Any scientific or graphing calculator is allowed.

...Read more

Posted by Dave Marain at 3:38 PM 0 comments

Labels: MathNotations Contest, more

HAPPY HOLIDAYS!

The following is a series of apparently straightforward arithmetic problems for middle schoolers. However, the objective is to have students justify their reasoning beyond "guess and test" methods. Proving there is only one solution or none requires more careful logic using algebra as needed. Students will need some basic algebra for the "proofs." For the younger student, modify these questions to have them find the squares in questions 1,2,3 and 5. Take this as far as you wish...


In the following, square refers to the square of an integer. Justify your reasoning or prove each of the following.

(1) There is only one square which is 1 more than a prime.

(2) There is only one square which is 4 more than a prime.

(3) There is only one square which is 9 more than a prime.

(4) There is no square which is 16 more than a prime.

(5) There is only one square which is 25 more than a prime.

(6) Can one generalize this or not??

Click Read More for selected answers, solutions...



Selected Answers, Solutions

(2) If n² is 4 more than some prime, p, then we can write
p = n² - 4 = (n-2)(n+2). Since p is prime, the smaller factor must be 1, so
n-2 = 1 or n = 3. Thus, there is only one square, 9, which is 4 more than the prime, 5.

(4) p = n² - 16 = (n-4)(n+4). There n would have to equal 5, n² would equal 25 but 25 - 16 = 9 is not prime.

(6) If there were a general rule would that mean we'd have a formula for primes?

Your thoughts about these questions...

...Read more

Posted by Dave Marain at 8:04 AM 4 comments

Labels: algebra, middle school, more, number theory, proof

I'm sure many of my readers out there are aware of Tanya Khovanova. Including her blog in my blogroll is long overdue and was a result of ignorance on my part. I've corrected that oversight and, now that I have discovered and enjoyed some of her essays, I want to share this with the rest of you. Brilliant (winning a Gold Medal in the IMO might suggest that!) and witty, her writing style is original and provocative. BTW, we share a passion for linguistic puzzles although I don't consider that unusual for those with a math bent. I strongly recommend you visit her website, Tanya Khovanova's Home Page as well as her blog. Her site contains a rich mine of creative and challenging problems. One of my favorite 'puzzles' is called Fly Droppings On Your Pizza. Click on Read more for further details...




Her most recent article, Multiple Choice Proofs, is a fascinating discussion of how to improve assessments (math contests in particular) which tend to be multiple choice here in the US and proof-oriented in her native Russia. Tanya suggests a compromise between those two extremes ("balance!") offering some alternatives one of which would involve artificial intelligence (AI) to evaluate proofs. I've always believed it might be possible to evaluate one's writing using an algorithmic approach but the obstacles certainly seem formidable at this time.

I also posted a couple of comments on her "About" page. I included a challenge mental math problem for her talented son, Sergei, to attempt. It may not appear for awhile as her comments are moderated. I may post it here as well...

...Read more

Posted by Dave Marain at 7:18 AM 0 comments

Labels: featured blogs, more

SAT "Grid-In" Type
Level of Difficulty: 5 (High)
Content: Algebra 2, precalculus

The function F satisfies the condition
F(N + 6) = F(N) + 8, for all integers N.
If F(7) = -2, what is the value of F(25)?

Click on Read more to see the answer, solution, discussion.


Answer: 22

Suggested Solution:

Replace N by 7 since F(7) is known:
Therefore, F(7 + 6) = F(7) + 8 or F(13) = -2 + 8 = 6

Next, replace N by 13 since F(13) is known:
F(19) = F(13) + 8 = 6 + 8 = 14

Finally, F(25) = F(19) + 8 = 14 + 8 = 22.

Comments
(1) Too difficult for the SATs? Not really! A similar problem recently appeared. There aren't that many "hard" questions (Level 5) on the SAT but, if a student wants to score over 700 they will need exposure to these types in practice.

(2) Consider writing some variations of these function-type problems for additional practice. At first, change the constants, then consider changing the operations (from addition to multiplication for example). One could raise the bar even higher by asking the question in reverse:
If F(25) = 22, what is the value of F(7)?

(3) There is considerable advanced theory in functional equations and recurrence relations underlying these problems. However, the student needs only to feel comfortable with the function symbolism (or should I call it "Math Notations!"). Starting by "plugging in' N = 7 seems simple in retrospect but most students are too intimidated to consider it. Even the precalculus student may be able to get started, but, without experience, they will often get lost. This is all about exposure, but isn't it always?

(4) One could rewrite this problem using sequence notation:
a_N+6 = a_N + 8. By expressing the problem in the context of the Nth term of a sequence, students may grasp it a bit better, but, in the end, it's all about interpreting function notation.

...Read more

Posted by Dave Marain at 1:51 PM 0 comments

Labels: algebra 2, functional relationships, functions, more, SAT strategies, SAT-type problems

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