As of 7-1-07, I will be retiring from public education after 0.037 millennia. I intend to maintain this blog for now and in the future. I have enjoyed sharing classroom investigations and challenge problems, which I have personally developed. I want to personally thank all of my faithful readers and those who find this site by searching. Your comments and support make my efforts worthwhile.
If any educational institution, state education department, publisher, etc., is interested in using the kinds of instructional materials I have been publishing on this blog or using my services in a consultancy capacity, you may contact me at: email@example.com.
I will shortly post this announcement as a link in a sidebar. Thank you.
Monday, April 30, 2007
As of 7-1-07, I will be retiring from public education after 0.037 millennia. I intend to maintain this blog for now and in the future. I have enjoyed sharing classroom investigations and challenge problems, which I have personally developed. I want to personally thank all of my faithful readers and those who find this site by searching. Your comments and support make my efforts worthwhile.
Posted by Dave Marain at 4:40 PM
[Update - The answers to the questions below appear in the comments section.]
I am continually amazed by some of the search phrases that lead to this blog. Although many are math topics about which readers are looking for more information, some are actual math problems that intrigue me. Here is one for today that led me to probe more deeply. On the surface it doesn't seems to require that much analysis, just an understanding of the rule for finding the number of factors of a positive integer from its prime factorization (see my earlier post on Fun with Factoring) and a good list of primes, but you may see something deeper here. At the least, it looks like an interesting investigation for middle schoolers and beyond with a webquest built in. I am indebted to the searcher whoever she or he may be!
Here's the actual search phrase I found:
1. What is the largest 3-digit integer with exactly four factors?
Before revealing the answer, I decided to expand this a bit:
2. (Easier but still worth doing) What are the largest and smallest 2-digit integers with exactly four factors?
3. Ok, so naturally, we would also ask: What is the smallest 3-digit integer with exactly four factors?
5. Keep going... What are the largest and smallest 4-digit integers with exactly four factors?
Of course, a simple factoring program written on a graphing calculator or in C++, etc., would suffice, but see how long it takes you to search and how logical reasoning and analysis can save some time. Of course it always helps to have a list of primes handy so don't forget the ultimate primes list from the U. of Utah.
Before you decide this is just a way to keep kids busy, try it. If you see a pattern or wish to expand this, go for it!
Sunday, April 29, 2007
[You may want to read the comments for this post. Some useful devices to help students recall important rules/facts from trig & calculus.]
Regardless of whether one approves of giving students mnemonics to help them recall various math facts or terms, students do use some of these and, in fact, don't we all! I know many math teachers detest PEMDAS because it can mislead students but the 'positives may outweigh the negatives'!
Here are a few of my favorites, some of which I've devised and some I've learned from creative teachers and students. I know some of you have your own pet phrases - pls share!!
With the May SATs only a few days away, perhaps one of these will stick in a student's head and help...
1. Zero is a WEIRDO (last 2 letters need to have a strikethrough)
Ok, here's how this works: Each letter helps students recall an important fact about the number ZERO which many students seem to forget almost daily! I'll start you off - try to guess the rest:
W: Whole (i.e., Zero belongs to the set of whole numbers)
2. Spell the word 'WHOLE'. The middle letter reminds us that ZERO is WHOLE and (E)VEN.
3. INTEGER (underline the N, E, and G) - to help students recall that integers can be NEGative.
4. PRIME (strikethrough the letter I, circle the last letter 'E.')
This may help students recall that 1 (the letter I) is not defined to be a prime number; further, there is only one (E)ven prime. Lame yes, but the lamer the better.
5. F)M Some students still listen to their favorite FM station.
This is to help them recall that a (F)ACTOR 'goes into' a (M)ULTIPLE. Thus, 4)12 suggests that 4 is a factor of 12, while 12 is a multiple of 4. Ok, stop groaning!
6. (From one of my outstanding Algebra teachers E.S.):
Permutations are Picky
Combinations don't Care (about order).
7. 'If you're Y's, you go to the top' or RISE rhymes with Y's (and things that RISE always end up on TOP).
These silly statements may help them recall that, in the formula for slope, the y's are in the numerator.
Now it won't be hard to top these, so go ahead...
Saturday, April 28, 2007
Pyramid Power: An Investigation that Develops Spatial Reasoning with Pyramids, Nets, Constructions,...
This investigation focuses on regular square pyramids, i.e., those with a square base and whose vertex is directly above the center of the base (informally stated).
The questions below are designed to further students 3-D visualization by constructing and 'deconstructing' several of these pyramids. Younger students in upper elementary or middle school should have had several experiences building and manipulating these kinds of solids, long before quantitative considerations of lengths of segments, angles, areas or volumes. Middle schoolers and high school students can always benefit from a hands-on approach to review the basic ideas, and geometry software like Geometer's Sketchpad is also very helpful to explore lengths, angles, surface areas, etc.
1. Draw a regular square pyramid, the kind you would see in Egypt. Give it a 3-D perspective. Each base edge should be 10 units for the rest of this investigation.
2. Draw a net for your pyramid.
3. Based on these drawings, answer the following:
(a) The triangular faces are always __________ triangles.
(b) If the lateral edges (segments from the vertex of the pyramid to a base vertex), are also 10 units, then each triangular face is a(n) _____________ triangle.
(c) Explain why the lateral edges cannot each be 5 units.
(d) Many students would guess that the minimum lateral edge is 10, but in fact it could be less. Finish this statement: The lateral edges must be greater than x. The greatest possible value of x is ________. (Mathematicians would call this greatest lower bound).
(e) If the lateral edges are each 10 units (all faces are equilateral), determine the height of the pyramid.
(f) If the height of the pyramid is 5 (half the base edge), it is easy to show, using a formula, that the volume of this pyramid is one-sixth the volume of the smallest cube containing the pyramid (the base of the cube coincides with the base of the pyramid). However, your task is to explain this visually without any formulas! You could 'build' a few of these pyramids and show that six of them will fit in the cube but is that necessary?
To be continued...
Thursday, April 26, 2007
[Update: Read Eric Jablow's profound comments on this post and some general discussion of graphing calculators...]
As promised, here's another installment of a piecewise function development of the absolute value function, suitable for advanced Algebra 1 students but more appropriate for Algebra 2. You may not agree with the target audience or the approach, but I have used it with mixed effectiveness. Of course you can redesign it to meet the needs of your students but the key ingredient is the use of function tables. Do you see the Rule of Four being utilized? I apologixe in advance for the klutzy formatting of the tables and the inequality symbols. I will eventually clean this up.
1. Consider the functions, f(x) = |x|, g(x) = x, and h(x) = -x.
(a) Complete the following function table.
x ............... Y1=f(x)=|x|...............Y2=g(x)=x...............Y3=h(x)=-x
-3.............. 3 ................................ -3 .......................... 3
-2............... ___ ............................ ___ ..................... ___
-1............... ___ ............................ ___ .................... ___
0............... ___ ........................... ___ .................... ___
1................. ___ ........................... ___ ................... ___
2................ ___ ........................... ___ ................... ___
3................ ___ ........................... ___ ................... ___
(b) Sketch the graphs of f(x), g(x) and h(x) on the same set of axes in THREE different colors on the domain [-3,3].
(c) Answer the following based on the table and graphs:
f(x) = g(x) when x is _________
f(x) = h(x) when x is _________
Now, rewrite this symbolically as:
|x| = x when x is __________ and
|x| = -x when x is _________.
[Note: This could easily have been handled on a graphing calculator, which is why the functions are labeled Y1 and Y2. This is one of the best uses of this technology. However, I'm a believer in doing it by hand the first time around - your choice! Also, note the heuristic of repeating the function on each line rather than the standard braces used for piecewise definition. Later on the student can abbreviate the format. ]
2. Consider the function f(x) = |x| - x
(a) Complete the table:
(b) Sketch the graph of f(x) on the domain [-3,3].
(c) From the table and/or the graph we conclude that
f(x) = _____ for x < 0;
f(x) = _____ for x ≥ 0
3. [More difficult] Consider the function f(x) = |x-2| + |x-4| + |x-6|
(a) Make a table of values for f using the ten integer values from x = -2 to x = 7 inclusive.
(b) Sketch the graph of f.
(c) Define f piecewise, similar to 2(c).
(d) Determine the coordinates of the minimum point of f. Justify.
4. [The Generalization] Consider the function f(x) = |x-a| + |x-b| + |x-c|,
where a < b < c
(a) Define f piecewise as in 3(c).
(b) Determine the coordinates of the minimum point of f. Justify.
Tuesday, April 24, 2007
[If you absolutely can't wait for a challenge problem, go to the bottom, but it might be worth reading through this first...]
Absolute value equations and inequalities, in particular, are notoriously difficult for most students. If you are a math educator, is this a topic you relish?
So what method of solution usually works best for the student? What method of presentation is most effective for the instructor? If the equations are straightforward such as |x+3| = 7, most students seem comfortable with expressing the equation as a disjunction: x+3 = 7 or
x+3 = -7. Some instructors, in preparing students for the technical definition (using cases), require the student to express this as x+3 = 7 or -x-3 = 7. Rarely have I observed instructors introduce the full-blown piecewise definition using cases early on in algebra:
x+3 = 7 if x>=-3
-x-3= 7 if x<-3. This is generally believed to be too sophisticated for an introductory treatment. Motivating the technical definition of |x| usually comes later on in Algebra 2. However, I have always been a bit uncomfortable teaching the traditional algorithm for absolute value inequality problems such as |x+3| "<" 7 which leads to the conjunction x+3 ">" -7 AND x+3 "<" 7 or, in combined form, -7 "<" x+3 "<" 7. Math instructors devise creative mnemonics to help students recall the procedure. This all begins when prealgebra students are exposed to the verbal description of the piecewise definition of the absolute value function:
The absolute value of a positive number is that number and the absolute value of a negative number is its opposite. The absolute value of zero is zero.
This is immediately followed by a number of numerical examples, guided practice with some more complicated variations involving mixed operations and an assignment. The student sees this as another example of something they're supposed to learn in math without much meaning attached. Most catch on to the idea by repetition and errors are generally caused by weaknesses with signed numbers or order of operations.
However, some educators prefer the 'distance' interpretation of absolute value to make this notion more meaningful. Technically one has to distinguish between the real number, x, and the graph of x on a number line, but this distinction is often sacrificed for clarity:
The absolute value of x is its distance from zero.
Thus, both 8 and -8 have the same absolute value because they are the same distance from zero on a number line. Since distance cannot be negative, we have a powerful visual model that students can use to make sense of this idea, although it seems limited when solving more complicated equations or inequalities later on.
Stay tuned for more on this topic (including a 2-dimensional graphing approach using functions), but, since some readers are disappointed if there is no challenge problem, here's one for you. It's not that difficult, a version of this has recently been tested on the SAT and it can be approached in a variety of ways (formal algorithm, guess-test, etc.), but I will challenge you to solve it using the distance model! Don't hesitate to take strong exception to my comments above!
For how many positive integer values of x is
|x-2007| > x?
Friday, April 20, 2007
Update: I've added another 'paradox' in the comments. With the AP Calculus (BC) Exam looming, AP teachers may want to share this with their students for review.
[The following AP level question is designed for upper level students.]
Why do mathematicians have to be so rigid, um, I mean, rigorous?
Here's an AP Calculus problem brought to my attention this morning by one of our outstanding Calculus teachers, Mr. D. He found this in an AP Review book.
[Rather than play with the symbols, I'll 'write it out']:
The definite integral of sec2(x) from x = 0 to x = 3pi/4 is?
It was multiple choice and the answer given was -1.
I shared this problem with my AP group later on in the morning and I asked them why that answer makes no sense. R.J. immediately replied, "The answer can't be negative since sec2(x) is never negative."
Of course, but let's work it out!
[I intentionally did it incorrectly at first]:
By the Fundamental Theorem, the integral equals tan(3pi/4) - tan(0) = -1!
What's going on here! I was gratified that one of my students recognized that the function sec2(x) has an infinite discontinuity at x = pi/2, so the original integral is improper. When we integrate from 0 to pi/2, then from pi/2 to 3pi/4, and apply the rigorous limit definition of an improper integral, we see that the integral diverges! If anything, the 'area' is infinite or unbounded.
Using these kinds of 'paradoxical' examples and asking students to 'FIND THE ERROR' is a wonderful device many educators use to deepen student understanding of mathematics and demonstrate the need to be rigorous!
Now why isn't the definite integral of 1/x from -1 to 1 equal to zero, since the region in the first quadrant 'clearly cancels' the part in the 3rd quadrant?? Hmmm... I'll bet some of you could explain this and find many other such 'paradoxes'!
Thursday, April 19, 2007
[Update: The answer, thanks to tc, and an in-depth treatment of this problem now appear in the comments. There are also some thoughts about geometry curriculum and how I develop some of these problems. I would be very interested in reader reactions to this and other problems I have written. Are they of any use for math teachers in the classroom or just curiosities to think about for the moment? Sometimes I feel that many educators just don't have the time in a packed curriculum to be able to give any of these 'enrichment' experiences. I guess I am looking for some validation here to continue writing these...]
A recently released SAT question (for copyright reasons I avoid posting exact SAT questions) motivated me to generalize the result of the problem and provide a challenge for the stronger geometry or algebra student. This problem can be solved several ways, some of which involve some 'messy' algebra. I invite our readers to find a Euclidean method that requires very little algebra and can be done mentally! When giving this type of question to our students, it is natural to want to provide hints when they become frustrated. From my own experience, I've learned to allow them to play around with it for awhile and discuss it in their groups before 'steering' them. An algebraic approach using equations of lines is certainly a worthwhile experience. The 'elegant' method I'm suggesting may not be the most desirable to show them at first. Besides, someone may devise an ingenious approach none of us would imagine if we didn't allow them to explore! Isn't that what teaching really is all about - leading the student to find her/his own path?
Ok, here's the question:
Refer to the above diagram. Lines j and n are perpendicular and contain point P(a,b) in quadrant I. Express the ratio of the area of triangle OPC to the area of triangle OPD in terms of a and b.
Wednesday, April 18, 2007
[Important Note: Thanks to 'e', Question 1 below has been modified to 'five' 2-digit numbers with 12 factors. Please read the comments for follow-up.]
[The ideas and problems today are suitable for grades 6-12.]
Those into number theory know many basic principles that help them solve problems involving factors that seem arduous at first. One extremely useful formula that mathletes are taught early on and SAT students should know is the following (the abstract form hides how easy it is, so get to the example quickly!):
The FUN-damental FACTORING RULE: (I coined this silly name so don't quote me!!):
If the positive integer N = p1e1p2e2p3e3...pnen then
N has (e1+1)(e2+1)(e3+1)...(en+1) positive integer factors!
The pi's are distinct primes and the ei's are positive integer exponents.
Note: From this point on, whenever the term factor is used, it refers to a positive integer factor.
Ok, we need an example fast!
Example: How many positive integer factors does 48 have?
Solution: First we need to write the prime factorization of 48 = 2431
[For larger numbers, writing the prime factorization is more problematic and a computer program or a calculator like the TI-89 would be useful].
ADD ONE to each of the exponents and MULTIPLY: (4+1)(1+1) = 5 x 2 = 10. Voila!
Verify: 1,48; 2,24; 3,16; 4,12; 6,8. Ten, indeed!
The explanation of this very handy rule involves some basic combinatorial thinking since EVERY factor of 48 (similar argument for N, in general) can be written in the form 2a3b where a could be 0,1,2,3, or 4 and b could be 0 or 1. Thus, there are FIVE (4+1) possibilities for a and TWO (1+1) possibilities for b. By the multiplication principle, there would be 5 x 2 ways to form different factors of 48.
Ok, so here are some examples (not very challenging) for middle schoolers and on up:
1) Using our FUN-damental Rule above, find the five 2-digit positive integers which have exactly TWELVE distinct factors.
The object is not to list every number from 10-99 and count factors!
Extras: Explain why a 2-digit number cannot have more than 12 factors. What would be the smallest integer that has more than 12 factors?
2) How many factors does 2007 have?
[Easy, once you have the prime factors, but it's always fun finding them for each new year or showing it is prime. Students better know why 2007 is NOT prime!!].
3) SAT-type (easy using above rule): If N = pqr, where p, q and r are distinct primes, explain, without listing or plugging in numbers, why N has exactly eight factors.
Then list the eight factors in terms of p, q and r.
There are endless variations and applications of the FUN-damental Rule. I'll leave it my readers to suggest really 'wicked' ones!
Sunday, April 15, 2007
NJ Commissioner of Education, Lucille E. Davy, announced on 4-10-07, that New Jersey is one of nine states that will administer a common exam for Algebra 2 students in May 2008. A link to the actual Algebra standards that will be used is given near the end of this post.
Here is an excerpt of her statement:
New Jersey Joins Nine-State Partnership to Administer New Algebra II Exam
New Jersey has agreed to join a partnership of nine states in the American Diploma Project Secondary Math Partnership to administer a common exam with common standards for Algebra II students beginning in May 2008. The eight states joining New Jersey in the partnership are Arkansas, Indiana, Kentucky, Maryland, Massachusetts, Ohio, Pennsylvania and Rhode Island. The project is an initiative of the ADP Network, a group of 29 states that educate more than 60 percent of all American public school students.
“This new exam will help to ensure that our children are learning the math skills that are becoming more and more essential in an increasingly competitive job and secondary education marketplace,” said Commissioner of Education Lucille E. Davy. “Our work in the American Diploma Project (ADP) Secondary Math Partnership complements our efforts in New Jersey to re-design our schools to meet the challenges of preparing our young people for the demands of the 21st century.”---------------------------------------------------------------------------------------------------------
Those of you who have been reading my posts for the past four months know that I am elated at this news. I had mentioned the possibilities of this happening a couple of months ago and now it's gaining momentum. Common exam, common standards, common curriculum,...
Of course, we will need to see how this plays out. When it becomes mandatory, will all NJ high schoolers be required to pass this in order to get credit for the course? Does this mean that all NJ students will eventually be expected to take Algebra 2 in high school and does this imply that those students taking more basic level mathematics will have to reach much higher than now? Will it follow a Regents model as in NY? If a student does not pass the exam, what appears on their transcript? NYS educators can probably provide insight into how this is handled in their state. From my understanding of the American Diploma Project, developed by Achieve.org, ALL youngsters need to and will be strongly urged to take Algebra 2. For those intending to continue their education, this is not an option.
I do believe this is a big step in the right direction, however, Commissioner Davy must surely recognize that there is a demographic of youngsters who, at this time, do not yet have the skills to tackle this course. Perhaps she and the others in the consortium believe that this will force the math curricula in Grades K-8 to be significantly upgraded and compel high schools to begin to phase out low-level math courses for most students. Not that there's anything wrong with that but we're certainly not there yet!
For those interested in the Algebra benchmarks (standards) developed by Achieve.org, read this. I looked it over and I like its structure, clarity, examples and content. I need to consider more carefully whether it encompasses all of the important topics in Algebra 2, but, on first glance, it looks good. Their goal and mine is to make our children more competitive and to upgrade the quality of education for ALL of our students, particularly underrepresented groups in our society. After looking these over, let me know if you see any omissions in these benchmarks.
Update on my view of the benchmarks: After reading both the geometry and algebra 'standards' and some of the sample postsecondary problems more carefully, I detect a more traditional flavor with some newer content sprinkled throughout. Those who remember learning from or teaching from the Houghton Mifflin Dolciani series for Algebra 1/2 (Structure and Method if I recall correctly) may feel nostalgic. This is more of an instinct than a careful analysis, so feel free to correct me....
MAA members will likely recognize the following challenge that appeared on the outside of the envelope in the mailing to members or prospective members. I plan on giving this to my AP Calculus students as review for the exam or afterwards. As usual I will modify it for the student, place it in the context of an activity, broken into several parts with some hints. The original problem comes with a helpful diagram, however, unless I scan it, it would be difficult to reproduce. The problem involves a property of a point on an ellipse and requires basic understanding of the parametric form of this curve and some basic calculus and trig. The last part of the activity suggests a possible significance of this property but I'll leave the details to our astute readers.
Consider a standard ellipse, center at (0,0), with major axis of length 2a on the x-axis and minor axis of length 2b.
Let P(x,y) be a generic point on this ellipse with the restriction that P is not one of the endpoints of the major or minor axes. Consider the tangent and normal lines at P. Let P denote the point of intersection of the normal line with the x-axis and Q, the point of intersection of the tangent line with the x-axis.
Prove that (OP)(OQ) = a2-b2, where OP represents the distance between the origin and P and similarly for OQ.
Here is an outline with several parts for the student:
(a) Show that x = acos(t), y = bsin(t), 0<=t<2pi,>2-b2)/a)cos(t).
(f) Use (d) and (e) to derive the desired result: (OP)(OQ) = a2-b2
(g) Explain why we did not allow P to be an endpoint of the major or minor axes.
(h) What does the expression a2-b2 have to do with the foci of the ellipse? For EXTRA CREDIT, investigate this 'focal' property further.
Friday, April 13, 2007
To continue our discussion of infinite series, I usually show students the famous proof that the harmonic series 1+1/2+1/3+1/4+... diverges. This series is paradoxical to students because, in their minds,there is convergence, since the terms themselves approach zero. With some exploration they can begin to appreciate that convergence of the sum of the terms depends on how fast the terms approach zero! Most of the content of the student investigation below can be found in MathWorld or Wikipedia but my intent, as it almost always is on this blog, is to produce a classroom experience for students and an activity for teachers to use, not just an expository piece of writing.
Consider the following "S-series":
1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 +...
(a) Continuing this pattern (of repeating groups of reciprocals of powers of 2), what would the 16th term be?
(b) If Sn represents the sum of the first n terms of this series (where n is a positive integer), what is the value of S16? No calculator!
(c) Develop a formula for S2n and verify your formula for S1024. Here, n = 0,1,2,...
(d) What conclusion can you draw about the convergence of the "S-series?" Explain.
(e) Consider the harmonic series (which we will call the "H-series"):
1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n + ...
Let Hn represent the sum of the first n terms of this series.
Show that H16 > 3, H1024 > 6 and H65536 > 9 by comparing the "H-series" to the "S-series" term by term.
(f) Based on the above, what conclusion can you draw regarding the limit of the sequence of partial sums, Hn? What does this imply about the convergence or divergence of the harmonic series? How would you describe the rate at which this series converges or diverges?
(g) Research the harmonic series online. Be prepared to answer the following question:
What does the harmonic series have to do with overtones in music?
(h) Consider generalizations of the harmonic series, such as replacing 1/n by 1/(kn+j). Make two such generalizations and examine convergence in each case.
The possibilities are endless. If two roads diverged in the woods, which one would you take?
Thursday, April 12, 2007
You may want to visit Math Concepts Explained, a new site that is designed for students looking for easily understood conversation-like explanations of important math concepts and procedures. It has a pleasant tutorial feel. I've added this to my blogroll listed as sk19.
MathNotations is designed more for educators and, from an analysis of the searches leading to my site, I know there are many many students out there who are looking for help with homework or reinforcement of classroom explanations. The site has nice visuals of graphs and I particularly like the idea that students can leave their questions in the comments section. I wish this new blogger the best of luck!
Posted by Dave Marain at 6:17 AM
Tuesday, April 10, 2007
The following may drive away most casual readers but it does describe what I try to do every day. One of my goals in starting this blog was to enable a dialogue for effective instructional strategies. My focus has generally been on middle and secondary school curriculum up to Algebra 2, bordering on Precalculus. Today I am sharing a different experience. I hope some of you will appreciate it beyond its technical aspects. Similar developments can be found in some textbooks and experienced teachers already do most of this but as this scenario is fresh in my mind, I thought I'd re-play it for you...
Although most Advanced Placement Calculus (BC) teachers are completing or have already completed the unit on infinite series, I would like to offer a view that I hope brings a sense of 'shock and awe' to the student of the 21st century who rarely has the time to stop and appreciate the beauty of our subject. To those who have been teaching this for a while, you may not quite feel this. However, I still get goosebumps when I observe student reactions as this unfolds in front of their eyes...
Assume that students already have a basic understanding of infinite series, the infinite geometric series in particular.
Consider the following three infinite geometric series:
1+1/2+1/4+1/8+... = 1/(1-1/2) = 2
1+1/3+1/9+1/27+... = 1/(1-1/3) = 3/2
1-1/4+1/16-1/64+... = 1/(1-(-1/4)) = 4/5
Just a collection of simple geometric series, boys and girls?
Genius is looking at an ordinary collection of objects and seeing something different. Some mathematician or mathematicians (research this and report back with their bios!) may have considered a reverse view of these series. Instead of the goal being a formula for the sum of the series, perhaps the goal was to represent a function in a different way. Step back into history...
Consider the general formula for the sum of all these series: 1/(1-r) provided r is between -1 and 1. Replace r by x, the variable we usually use for functions, and we can write:
1 + x + x2 + x3 +... = 1/(1-x) provided x is between -1 and 1.
The 'polynomial' of infinite degree on the left is known as a power series in x. As long as x is between -1 and 1 (the interval of convergence), this 'equation' makes sense and allows us to use algebraic and calculus operations to represent other related functions. Think of how one might have felt when 'discovering' this and I'm just speculating here. The rational function 1/(1-x) is being represented by some kind of polynomial that never ends. Even though x= 1 is not in the interval of convergence, substituting leads to 1/0 = 1 + 1 + 1 + 1 +.... Hmm....
Let's try substitution on this representation.
Replace x by -x2:
(You can show the domain is unchanged)
1/(1 - (-x2)) = 1 + (-x2) + (-x2)2 + (-x2)3 + ... OR
1/(1 + x2) = 1 - x2 + x4 - x6 +...
Ok, let's integrate both sides (assuming it's legal to do so):
tan-1(x) = x - (1/3)x3 + (1/5)x5 - (1/7)x7 + ... + C
Replacing x by 0, we see that C = 0.
Now, you'll have to accept this for the moment (to be proved later), equality holds for x = 1, even though 1 was not in the original interval of convergence! It is not unusual when integrating a power series to see the domain include one or both endpoints even though the original function excluded them!
Thus, tan-1(1) = 1 - 1/3 + 1/5 - 1/7 + ...
Anyone recognize the left-hand side?
The bell rings...
Saturday, April 7, 2007
A recent post in MathNotations received a favorable comment by Charles Daney in the 5th edition of the Carnival of Mathematics. This is particularly gratifying because the post relates to one of my students.
This week's Carnival pays special tribute to one of the 20th century's most outstanding mathematicians, Fields Medal recipient, Paul J. Cohen, who passed away on March 23rd, the date of the previous Carnival. Dr. Daney's tribute to the 'beautiful mind' of Dr. Cohen is sincere, passionate and touching. This is one mathematician's way of expressing gratitude to someone whose inspirational teaching affected his life personally. I am willing to wager that each of us who has pursued mathematics as a career in some capacity has had their life altered in some way by an inspiring educator. In addition to the major influence of my father, a great teacher in his own way, I was blessed to have many other outstanding math teachers, but my watershed experience came in Dr. Silvio Aurora's general topology class at Rutgers over forty years ago. He showed me what it means to think mathematically and to appreciate its beauty. Thank you, Dr. Daney, for bringing that flood of memories back to me.
Thursday, April 5, 2007
Update: Answers, solutions have now been posted in the comments.
I thought of these variations on the well-known combinatorial problems involving 3-digit numbers that pop up frequently as I was teaching arithmetic and geometric series yesterday.
These questions are appropriate for grades 6-12 provided students are given definitions and some practice with arithmetic and geometric sequences, topics that are well within the abilities of middle schoolers. A quick intro to these sequences is all that is really needed OR, as I did below, they can be defined in the problem itself. Thus, these questions provide both practice in arithmetic skills and in combinatorial thinking. Of course, all the experienced or budding programmers out there can write simple code to have their graphing calculators count these, but that should only complement and verify their results, not replace the reasoning needed to solve them, unless these are used for a computer science class (even then, programmers should independently verify their code by solving the problems!).
These are not highly challenging and therefore can be used as Problems of the Day, for extra credit, or enrichment. Our readers will hopefully suggest other extensions and further variations (some are suggested below).
1. The digits of 246 form an arithmetic sequence from left to right because 4-2= 6-4. How many positive 3-digit integers satisfy this condition?
2. The digits of 248 form a geometric sequence from left to right because 4/2 = 8/4. How many positive 3-digit integers satisfy this condition?
Now, how could we make these more challenging? 4-digit numbers or will that make one or both easier, i.e., fewer possibilities? What if the digits were allowed to form these sequences in any order? BTW, I apologize for the music pun in the title. I hope you will respond to that with a positive tone!
Tuesday, April 3, 2007
I normally don't have time to become engrossed in new books but, yesterday, one of my students, J.B., presented me with this 265 page paperback whose title is the same as that of this post. It has already received recognition from the BookLovers Review.
The authors, Edward B. Burger and Michael Starbird are esteemed mathematicians who have each been recognized as distinguished teachers. After reading part of their book, I can understand why! The ability to communicate difficult concepts for all to understand is rare indeed.
I started reading it last night, worked my way through the mesmerizing explanation of public key cryptography which explains in the simplest terms how sensitive information can be transmitted in encoded form, then decoded. While the encoding method is completely revealed to the public, there's a key factor (figuratively and literally in the mathematical sense) in the decoding process that no person and no computer could determine in real time except for the person holding the key! If that sounds too cryptic (sorry, I couldn't resist that), read the chapter 'Secrets Held, Secrets Revealed.' The secret lies in the realm of Number Theory (congruences and residues) but the ideas are explained using a simple numerical example that all can verify on their calculators. Considering that a large security firm purchased an encryption company for $400,000,000 to own the rights to this method, I guess I can now tell my students there is another reason to study math beyond the stories of Bill Gates and Paul Allen!
The chapter on chaos (Understanding Uncertainty - Chaos Reigns) is brilliant and again brought to a level everyone can understand. The following quote about the source of chaos is profound and embodies the essence and spirit of this exceptional work:
"Even mathematics, complete and precise, is subject to the perils of tiny variations in initial conditions, which, when multiplied and magnified by the tyrant of repeated application, end by leading us far astray..."
I've only scratched the surface of this book. I want to personally thank my student for this special gift to me. She took the time to find the perfect way (for a math nerd like me) to express her gratitude for my writing a college recommendation that, she perceived, helped her gain admission into a prestigious university. Trust me, this exceptional young lady needed no help from me!
Monday, April 2, 2007
Algebra teachers, like myself, are always looking for ways to help students make sense of exponents. We look through copies of the Mathematics Teacher, we go to the Math Forum and now we Google, Google ad infinitum (or some other search engine to be fair!). Here's an approach that I have found helpful. I assume the student has had some basic introduction to exponents and their properties. I call it the exponential function approach which sounds too challenging for middle schoolers but you decide if they can handle this. Students will use pattern-based thinking and graphs to make conjectures about extending powers of 2 to include zero, negative and even fractional exponents. Properties of exponents will then be used to 'justify' the conjectures. The juxtaposition of the numerical, symbolic, graphical and verbal descriptions are consistent with the Rule of Four that is now regarded as the most powerful heuristic in teaching mathematics.
Begin by making an x-y table - this is the critical piece.
Exponent (x)...........................Power (y = 2x)
3 ..................................................23 = 8
2 ..................................................22 = 4
1 ..................................................21 = 2
0 .................................................20 = ??
The instructor of course is prompting the students for the powers while they are taking careful notes. At the same time the instructor is plotting these results as ordered pairs and the students do likewise. It might be helpful to let 2 or 4 boxes represent one unit on the y-axis since, at some point, the y-values will be fractional. Similarly for the x-axis (play with it first).
At this point, the instructor asks a key verbal question (you may phrase it much differently depending on the level of the group and your preference):
[While pointing to the left and right columns]
"When the exponent decreases from 3 to 2, the corresponding power of 2 is divided by ___.
Repeat this phrase a couple of more times until you reach an exponent of 0, then -1 and voila! Keep going until x = -3, plot the points and the students are seeing an exponential curve in grade 7? 8? 9?
Motivating zero and negative exponents using a function model (tables!) seems to make sense to me because it begins to create a 'function' mind-set that can be carried through all subsequent math courses. It may also help students to 'see' that the range of the function consists of positive real numbers. If you're wondering why I didn't mention turning on the graphing calculators to make the TABLE and GRAPH, I hope you can guess why. It was important for me to have students do this by hand first, then I will turn on the overhead viewscreen and we can explore with technology. Just my opinion of course but students in my classes seem to make sense of this. Of course, I don't kid myself that this approach will lead to better grades on tests of this unit! Facility with the properties of exponents only comes from considerable skill practice with paper and pencil.
For fractional exponents, I'll begin the discussion but I will have to explore further on another post or leave it to your imagination. "Ok, boys and girls, if mathematicians believed exponents could be zero or negative integers, would you be surprised if they wondered about 21/2? From the table and the graph, 21/2 should fall between ___ and ___? Do you think it will be exactly 1.5? Why or why not?
I know many of you use the exponent properties to develop this topic, but I wanted to suggest an alternative. I usually follow this discussion with arguments like: " Hmmm, I wonder what
21/2 times 21/2 would be?" etc...