Changes in the Wind...

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: dmarain@gmail.com.

I will shortly post this announcement as a link in a sidebar. Thank you.

More Prime Searches...

[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!

sec tan cos sin 3.14159 - Math Team Cheers and Math Mnemonics

[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...

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...

Absolute Zero Part II: Applying Piecewise Function Approach for Algebra 2

[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:

x..............Y1=f(x)=|x|-x
-3
-2
-1
0
1
2
3

(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.

Absolute Zero Part I: Do Absolute Values Leave Some Out in the Cold?

[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
OR
-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?

'Rigor Mathis' - A Calculus Paradox or...

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'!

The 'Power' of Geometry - Ratios of Areas

[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.