The Limit

Deltas and Epsilons

2010-08-22T15:48:00.000-05:00

Here is a way to help your students understand deltas and epsilons. As I've said before I'm not that sure that being able to find the d-e relationship alone does that. So lets assume students are able to compute d-e in general and look at the lim(x-->3) (2x-5) = 1 where the usual approach will give d = e/2

Begin by drawing the "box," that is the rectangle whose vertical sides are x = a - d and x = a + d and whose horizontal sides are y = L+ e and y = L - e. Use a calculator or better yet a good graphing program like Winplot (or Geogebra, Nspire, etc.) Use sliders for the actual values of d and e.

CASE 1: With d = e/2 the box will be twice as high as it is wide. The graph of y = 2x - 5 will go exactly through the top left and lower right corners of the box. For all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. The requirements of the definition are met and we have proven that the limit is 1.

CASE 2: Let d = e/4. The box is now 4 times as high as it is wide. The graph enters and leaves through the sides of the box. For all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. The requirements of the definition are met and we have proven that the limit is 1.

CASE 3: Let d = e, the box is now a square. The graph enters the bottom of the box and leaves through the top. It is NOT true that for all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. There are some points outside the box top and bottom that should be between the horizontal lines but are not. The requirements of the definition are NOT met and we have NOT proven that the limit is 1.

It makes no difference what value of e is used. And this is important too. The fact that the box looks the same (technically a similar figure) for a particular delta is the graphical way of saying "for any e" ("for all e's", "for every e"). Large e, small e, the d = e/2 box is always twice as high has it is wide and the graph goes through the corners. This makes it easy to believe that as you go smaller and smaller the picture will not change ever.

With the sliders you can play with different d-e relationships for this or any function even if you cannot find algebraically the largest such relationship.

For polynomial and other functions where the relationship is something like d = min(1, e/3) the CASE 1 graph is not possible. The function may enter or leave at one corner but not the opposite corner; it will leave through the opposite vertical side meeting the requirements; it's more like Case 2. But with small values of d an e local linearity takes over and things look like Case 1. Taking e large enough so that d = 1 is instructive as well. As long as the graph enters and leaves through the sides of the box everything is wonderful.

Of course the box is the same box that we've all drawn on the board and is in all the textbooks. But the dynamic role played by the sliders is far more instructive than the dynamic hand waving with a static picture.

Hope this helps

Grading

2009-12-26T09:45:00.002-06:00

Let's think about grades for a minute. The percentage of the total points on a test, quiz, project, etc. seems to be what everyone thinks they should give their students for a grade. But this carries a lot of assumptions, which IMHO make little or no sense. Percentage grading assumes:

You can write a test or judge another type evaluation so that a grade of ,say, 80% means the same thing each and every time, and that the minimum "passing" grade is that much better than one percent less than that grade.

Each test has the same difficulty, so that 80% always means the same thing through the year.

That the teacher across the hall teaching the same course, will give percentages that mean the same thing as yours, and the teacher in the next school district, the next county, the next state ...

Then what happens? All your grades for a particular test are low. Does this mean the test was too difficult or that the students did not learn all they should? What to do? Some folks "scale" or "curve" the these tests (but not the ones where everyone does well), or they make sure the next test is "easy." Then of course you have to justify scaling only the "hard" tests and not the ones where everyone does well. (This in fact assumes that the test was too hard -- else why reward kids who didn't learn what they should have -- and proves you can't judge the difficulty of your own tests.)

Percentage grading only seems to work for very experienced teachers who use the same tests from year to year and have adjusted them so the grades come out the way they would like.

There are various ways to scale or curve tests.

One that has been discussed recently is the "square root curve" where the grade is 10 times the square root of the percentage. This raises everyone's grade; those in the middle a lot, those on either end a little. (Max/min question: which percentage gets raised the most? Which the least?). I can see even less justification for this approach -- except to raise everyone's grade.

Other schemes are to list the grades in order and look for natural breaks. Highest group gets A or 90s, next get B or 80 etc. Somewhat arbitrary, but may make more sense than anything above.

Find the z-scores and force the grades into a normal distribution -- this assumes your kids are normally distributed and in an AP class that may not be the case. Besides with this approach you will always have to fails some one even if they are all pretty good students.

Throw the test up the stairs ...

So what to do? After all you do have to assign some grade. My suggestion is to read "Assessing True Academic Success" by Dan Kennedy. A perfect approach, probably not. But it will give you some things to think about and suggest a curving scheme that you might like and that has some thinking behind it.

The take-away from this article is "You control the grading algorithm."

The Shell Game

2009-11-23T09:15:00.001-06:00

It's been a while since I've written anything here. The usual excuse: too busy with work. Here is a little idea you may like.

Every year about this time as AP Calculus teachers get ready to teach applications of the definite integral the question comes up about whether to teach the method of cylindrical shells. There are several things to consider.

First, the method is not tested on either AP calculus exam; so you will not put your student at a disadvantage if you omit the topic. Any volume of rotation question on the exam can be done with the disk/washer method. But then it is technique that all the texts cover and most college teachers expect students to know. So if there is time it is worth including. It is probably best to include it when you do other volume problems. Due to time considerations some teachers save the topic until after the exam, during the final weeks of school.

However, what I'd like to discuss here is the fact that you never actually need the method of cylindrical shells. You can always avoid it. That said, let me add that shells is usually faster and more efficient than the idea that follows. The example shows how to work around the shell method (pun).

This technique will work for a function that is one-to-one on the interval of integration. If your function is not one-to-one, the interval must be broken into sub-intervals on which the function is one-to-one.

Here is an example: Find the region enclosed by the graph of the lines x = 1, y = 1, the y-axis and the graph of

is revolved around the y-axis to form a solid figure. Find the volume of this figure.

Proceding by the disk/washer method the volume is given by

Note the dy and the limits of integration that correspond to the dy.

The usual next step is to write the integrand in terms of y so that the antiderivative can be computed. Cubic equations can be solved, but who remembers how? And even so the next example may involve a sixth degree polynomial, so let's assume you cannot solve for x in terms of y. At this point textbooks give up and recommend the method of cylindrical shells.

But it's not necessary that the integrand be in terms of y, it is only necessary that the integrand contain only one variable. So consider the substitution:

Making this substitution we get

The resulting integrand is a polynomial that can be easily expanded and antidifferentiated. Note the change of the limits of integration to correspond with the dx.

Do AP students need to know this? No. It is more of a curiosity or a little "extra" for them. But it is a nice idea and also illustrates a slightly different substitution.

Using CAS for testing

2009-10-06T06:50:00.000-05:00

The ETS recently posed a survey concerning the use of a symbolic manipulation calculator on their exams. Here are my thoughts on the survey.

First, the survey seemed to be aimed at determining whether a symbolic manipulation calculator (SMC) would give someone with a SMC an advantage on the standard questions that appear on the ETS exams (AP, SAT etc.). The questions did not seem at all the kind were designed for SMC use. Too bad.

The argument is always along the line of if using a graphing calculator (GC) or a SMC helps people learn more or better or deeper. That's different than what should be expected or required on an exam of one's own or of the ETS. So I'd like to discuss two questions to see the difference between good testing practice and using SMCs to do and learn math.

2003 AB 76 gave a velocity function v(t) = 3+ 4.1cos(0.9t) and asked for the acceleration at t = 4. The point of the questions was to see if the student know the relationship between velocity and acceleration. The problem could be done be hand; this requires the use of the derivative of a constant rule, derivative of a sum rule, the constant multiple rule, the derivative of the cosine and the chain rule. To do this by hand would necessitate the use of "easier" numbers and one of the special trig values. A student who makes a mistake on any of these gets the wrong answer and leaves the examiner unclear as to whether the student knows that the derivative of velocity is acceleration. Therefore without the use of a GC the question is pointless.

But you say all those other things are important; a calculus student should know them. I agree -- but they were tested elsewhere on the same test. The velocity - acceleration thing was the point here.

2003 AB 87 gave f ' (x) = sqrt(x)/(1 + x + x^2) and asked for the x-coordinate of the point of inflection. The fastest and, I am sure, the intended method of solution is the use the GC to graph the given derivative and identify its extreme value which corresponds to a point of inflection. This concept is often tested and can be found on several free-response tests.

The questions could be done by graphing f '(x) by hand, but even with a simpler function with easier arithmetic this is time consuming and not practical. You could also find the second derivative and find its zeros by hand. Again not really practical with the given function. It is correct approach but one that does not tell us that the student knows the graphical relationship between the graph of derivative and the POI.

Using a SMC you can more easily find the second derivative and its zeros. So you can get the answer, but even that takes more time and thought than the intended method. So here the SMC will work but is a disadvantage time wise. And, as before, does not tell the examiner if the student knows the relationship between the extreme values of the first derivative and the POI.

The point is that technology can make testing more focused on what you want to test. Using more complicated functions and less "easy" numbers is a way of pointing the student to his GC or SMC hoping he will show that he understands the concept being tested.

Whether you choose to let students use a GC or SMC on your tests is up to you; you can certainly test these and other concepts without them. Certainly you can write questions for which a SMC will not be of any help. Or you can use the SMC to focus on the concept you are interested in. I think that's the point in testing with a SMC.

But the other reason to use a SMC or other technology in your class and requiring your students to know how to use it is to help them learn math better. I could live with no technology on test, but not with no technology in class. The reason to require technology use on tests is to be sure it is used in class.

Technology Changes Curriculum - Or does it?

2009-09-13T10:36:00.001-05:00

Technology changes things. That's sort of an understatement.

The first big curriculum changes resulting from technology occurred some 40 years ago. Four function calculators became available. I remember my first one cost $70. At that time something between 2 weeks and a month was spent in Algebra 2 classes teaching computing with logarithms.

For you young folk: back in "the old days" calculations involving multiplying, dividing, power, and roots we routinely done using logarithms. The logarithms were listed in tables. A logarithm table for 3-digit numbers is two pages long, 4-digit numbers 10 pages and 5-digit numbers 100 pages. While in college I worked summers for a civil engineer; he had an even larger table that took up an entire oversize book -- and he used it. There were also tables of the logarithms of the trigonometric functions.

The were no calculators. We had in the office a machine to do multiplication and division. Huge thing; rows and rows of buttons. You entered one number and then to multiply you turned a crank for the other number one digit at a time. For the next digit you turned another crank which move the whole carrage over one place so you could crank in the next number. To divide you turned the crank backwards. Hence the term "crank out the answer." Someone good with logarithms could do the job just about as fast.

The other thing close to a calculator was the slide rule. But the slide rule is just an analog representation of logarithms.

Students in Algebra 2 were taught all the tricks of calculating with logarithms: how to deal with the decimal point, how to interpolate between the values in the table to get more accurate answers, how to use co-logs to avoid subtracting and of course how to use the table in the other direction to change the final answer back to a regular number (called finding the antilog).

Multiplication and division were done by adding and subtracting the logarithms; powers and roots by multiplying or dividing by a single-digit number. I remember after teaching how to set up the problem and find the logarithms that one year the students did all the set up work, looked up the logarithms and then got out their calculator to do the adding and subtracting! That's when I know logarithms were on their way out.

There are lots of topics in mathematics that were invented to make calculations with numbers easier and more efficient. Logarithms was certainly one of them, but there are others. The advantages of four-function calculators and soon thereafter scientific calculators were obvious.

But, here's the thing, computer algebra systems, CAS, do not seem to have had the same effect on the curriculum. Why? There are many basic computations that we still require students, not just to know, but to be proficient at doing by hand. I'll list a few below, but so that I don't sound too far out, let me first say this:

Just as logarithms are and always will be important and students need to understand them, so too the things in the list are important and students need to understand them. But they don't have to be good at doing them! Just as very, very people who graduated high school in the last 30 odd years could come close to doing a computation using logarithms, students in the future don't need a by-hand proficiency with these thing. They need to understand what's going on; they do not need to be good at doing it.

Okay, here's my short list:

Simplify radicals
Rationalizing denominators (or numerators for that matter)
Factorig quadratic polynomials (those few that actually factor)
The Quadratic Formula and the "discriminant" (although completing the square is a keeper, but not for solving equations)
Cotangent, secant and cosecant (these have not been taught in Europe for decades, if ever. Just noticed that the spell check on this Blog doesn't recognize "cosecant" as a word; maybe there is hope.)
Newton's Method for approximating roots of any equation
Antidifferentiating (there are tables of over 1,000 antiderivatives in print, these too can go)
Synthetic division
Descartes Rule of Signs
The Law of Sines and the Law of Cosines
Cramer's rule and the discriminant
Matrix arithmetic and find the inverse of a matrix

So that's my short list. Anyone care to add anything?

They were invented for a good reason; but that reason no longer applies.

Why hasn't technology changed these things? I think the major reason is that they appear on the state graduation exams -- now there's a good reason to teach them!

CAS question

2009-09-07T18:18:00.001-05:00

As I have made no secret of the fact that I think symbolic manipulation software should have a major role in mathematics education starting no later than Algebra 1, I tend to get into a lot of discussions with folks who do not quite agree with me.

Not too many years ago graphing calculators made their appearance in high school math classes. They were pushed down from the top, so to speak. By requiring students to have them for the 1995 AB and BC calculus Advance Placement exams, teachers were forced into using them. What happened? Almost overnight those teachers and their colleagues saw how much more students could learn and understand using graphing calculators. I doubt very many would give them back today.

The very first models graphed and did numerical computations. Very quickly the ability to make tables of values of functions and software to do sophisticated statistical calculations were added.

(Side track: That first year, 1995, I used the TI-85 which did not have a table feature and half my class showed up with HP-48s (don't ask) which did not have a table feature either. What to do? I wrote a program for the TI-85 to make tables and one of my students wrote a table program for the HP-48 (in assembly language no less). We shared them and then everyone had a table program)

Today, computer algebra systems, CAS, include those features and also include the ability to do symbolic manipulations. (And more actually: dynamic geometry software and computer like document handling are features of the newest models.) While CAS do all the graphing and numerical work that graphing calculators do, the acronym CAS has come to refer to the symbolic manipulation features alone.

So here is my question: Since the graphing, table, statistical, and geometry features of the CAS have been embraced, why not also symbolic manipulation features?

Any ideas?

Making the Grade - 3

2009-08-29T10:10:00.002-05:00

So if percentage grading doesn't make it and the various scaling schemes have little to recommend them, what do I recommend?

Before I answer that I suggest you read ASSESSING TRUE ACADEMIC SUCCESS: THE NEXT FRONTIER OF REFORM by Dan Kennedy of the Baylor School in Chattanooga, Tennessee. This article deals with a lot more than scaling test and is well worth reading. Dr. Kennedy is the co-author of a popular calculus textbook and has been involved with the AP Calculus program for many years. His wide ranging ideas on assessment make a lot of sense (to me at least).

As I discussed previously, scaling is often used to raise students' grades when the teacher has given a test that is too hard, when the mean score was too low. Dr. Kennedy writes, "If you want your students to think on the test, then you will have to give them a question for which they have not been fully prepared." Certainly posing questions for which the class is not fully prepared will make the test "too hard." His answer is to stretch the students by asking challenging questions on the tests and the protecting then students' grades by scaling the results. He points out that "[you] control... the grading algorithm." His conclusion is that "we ought to present students with challenging, relevant, useful, and varied assessments all of the time, and then scale the grades to conform to our expectations." (Emphasis in the original)

Here is Dr. Kennedy's suggestion on a method for scaling test grades. Take two order pairs. The first is (class mean on this test, desired class mean) and the second (the highest test score, 99). Write the equation of the line containing these two points and use it to scale the grades. (The "desired class mean" can be set in advance and adjusted slightly and can be different for different level classes. It can be the historic mean for each course.)

He finds that,

Freed from the shackles of unreasonable numbers, I can now challenge my students to do just about anything, then see how far they can go. They, in turn, have been freed from the burden of getting a certain percentage right, so they can concentrate on doing as much as they can as well as they can.

What more could you ask for?

Again, I suggest you read the entire article. It suggests other methods of assessment that the author has used in mathematics classes.

While no scaling method is perfect and will work for everyone, I think that this one has a lot to recommend it. First and foremost you can and will ask challenging questions. You also will be able to test that do not total 100 points. If your test is too long for the time allowed, well it's too long for everyone and, because of the scale, no one will be hurt.

Making the Grade - 2

2009-08-22T19:06:00.004-05:00

In a previous post I discussed percentage grading. The big problem with percentage grading is that it does not allow for differences in the difficulty from test to test, class to class or year to year.

So if you do not use straight percentages, what do you do? You "curve" or "scale" the grades.

But curving (I'll use that term from here on) can be and is being, done lots of ways. And some of these may be worse that percentages. Let's look at a few common ways and see how they work.

(Disclaimer: I've used some of these myself and with the benefit of hindsight wish I had not.)

One common approach is to list the grades in order from highest to lowest and look for natural breaks to divide them 4 or 5 groups. The high group gets As (or numbers above 90), the next group gets Bs and so on. This is often used when the grades were not as good as the teacher would like; it is never used when the percents come out okay as it may lower grades. It's just a way to recover from a hard test.

Then there is the "Square Root Curve." Here you multiply the square root of the percentage correct by 10 to calculate the grade. An 81 becomes a 90, a 49 becomes a 70, a 36 becomes a 60, etc. This has the effect of raising all the grades and raising the lower grades the most. (Max/min question: What grades is raised the most? Answer 25.) Now even if you use this all the time and give very easy tests you will not lower anyone's grade. But what is the assumption behind this? Darn if I know. You just raise everyone's grade using some pseudo scientific formula. Maybe it appears "mathematical" since the formula has the strange radical sign in it. I've even seen a state exam that I am pretty sure was curved this way.

While we are on state exam, I also saw one where I am pretty sure they used 4 data points: 0 was scaled to 0, 87 (the highest possible score) to 100, the minimum passing grade to 55 and a high score (distinguished or something) to 85. Then they used a cubic regression through these 4 points and used that to curve the other scores. Obviously, a lot of mathematical thinking behind that idea.

One could normalize the scores and then curve using the bell curve. 3 standard deviations above the mean = A, two a B, one above to one below a C, 2 below a D and 3 below an F. (Of course with a small class you may not get any As or Fs.) This makes the assumption that the grades should be normally distributed. I don't think that's true especially in an AP class. Also most of your grades will be C: that won't do.

Then there is the tried and true: throw all the paper up the stairs and those that go the highest get an A. Good as any.

The AP exams are curved. The goal is that the same amount of knowledge will get the same grade from year to year. This is done to allow for the difference in difficulty of the exam from year to year. By reusing questions from previous exams the Educational Testing Service can judge the difficulty of the current test compared to previous years and set the cut point accordingly. The scores are not normalized and there is no predetermined percentage of students who are given a 5, 4 ,3 2, or 1. This is great for the ETS, but not practical for a single teacher.

So what to do? My next post will discuss the best system I've hear of. Perfect? No, but pretty good.

Making the Grade - 1

2009-08-16T10:24:00.006-05:00

It is sort of obvious that when a teacher puts 87% on a quiz, a test or a report card that it means something. But what does it mean?

I have no idea.

Of course, I used to do it, but after many years I have come to the conclusion that percentage grading make no sense. There is no acceptable assumption behind using a percentage to indicate how much or how little a student has learned.

A teacher writes a test. The students takes the test. The teacher grades the test and the grade is percentage of the total possible points on the test that the student answered correctly. Sounds very reasonable, efficient and clear.

But it is not reasonable and cannot be. How does the minimum "passing" grade, say 65%, establish that a student who only earns 64% not know enough to pass, while a student with 65% does? How does a score of 65% on one teacher's test compare to a 65% on the test that the teacher in the next room wrote on the same material, or the teacher in the next school district, or the next state?

Then there is the problem of all the students scoring below our expectations. We've all given a test that was too difficult - by which I mean that all the scores were lower than we would like. What to do?

There are lots of things: give a retest, give students the opportunity to earn back some or all of the points they lost by correcting their mistakes, or make sure the next test is so easy that everyone can bring their average up to where it was.

A big favorite is to "scale" the test. Students particularly favor scaling because they get some extra points and don't have to do anything for them.

After teaching a few years teacher can become very good at knowing how to write a test that is not too difficult and the class average will be close to what you (and your principal) would like it to be.

But any and all of these deny the validity of percentage grading in the first place. It doesn't make the grade.

While no system is perfect, the alternative I favor is a type of scaling. In my next post I will discuss some scaling schemes that are commonly used, but also do not have good assumptions behind them. A third post will explain the system I like.

Meanwhile, think about what, if anything, a percent grade really indicates.

Home Again

2009-08-13T21:28:00.007-05:00

For the last three weeks I've been leading AP Calculus Suimmer Institutes in Virginia, Alabama and Massachusetts. I'm finally home and have been in the office this week. I hope that now I'll have a little more time to write.

The AP Calculus EDG has been going on and on about two things: the em-dash and the square root of 9. Both discussion have been long (as in lots of E-mails), somewhat repetitive, but nonetheless informative.

The em-dash, —, to use its typographical name, is the symbol used to represent (1) subtraction, (2) the opposite of a number or expression, and (3), when used with a numbers as opposed to an expression, to indicate that the number is negative as in —3. It is having three uses that makes it confusing to beginning and not-so-beginning students. There was a time when it was suggested that the em-dash be used only for subtraction and a raised en-dash be used for the other two uses. Makes a little sense since the opposite of 3 is negative three; but then the opposite of x might really be positive. And then calculators do have a "subtraction" key and an "opposite" key.

This can be confusing especially since the three uses are closely related to each other. The best suggestion was not more symbols, but simply to honestly face the problem and teach the kids the three meanings, how to tell the difference, and how to use the proper one.

I'm afraid I may have started the other thread when I related that a teacher asked me "What is the square root of 9?" I had to ask him if he meant or a solution of the equation . This to led to many replies with all sorts of opinions many as to whether was the same as the square root of 9. Then the function came under discussion: is this the "square root function" or the "radical function."

Again the moral is that we have to be careful how we explain things to students and be very precise in how we (or our textbook) defines things.

One additional thing that came from this discussion was the suggestion that if the square root "function" is the inverse of the squaring function then its range must be restricted to make it a function. This is probably the first time that students come across this situation and it is a great place to stop and explain why this restriction is necessary and why it is allowed. Because the next time this comes up is when you are defining the inverse trigonometric functions and restricting their ranges.

So maybe we all have too much time on our hands this summer - no that can't be right. But the discussions were interesting and point to places where teachers need to be very clear about what's going on.

Oh Hell, Another Hour of Algebra

2009-08-01T15:03:00.006-05:00

Do mnemonics have a place in learning math?

The yearly discussion of mnemonics is (unfortunately) in full swing on the AP Calculus discussion group this week. I may be to blame for staring it with a mention of SOHCAHTOA. I was trying to make a point about learning the concepts verses memorizing something that will get you the answer with no understanding.

Some things do have to be memorized, no two ways about it. And I can live with SOHCAHTOA or the title of this post as a way of remembering the right triangle definitions of the trig functions. The thing is I can't spell SOHCAHTOA without saying to myself "Sine: opposite over hypotenuse, cosine: adjacent ..." so for me there's no point to that one. But I can see where it may help an Algebra 1 student.

Then there are useless mnemonics like LAND and GOR mentioned in my last post. These really do more harm than good.

Other mnemonics I'd rather avoid. For example "Please prepare my dinner Aunt Sally" or PPMDAS or PEMDAS for short. Here an understanding of the underlying reasons, the concepts if you will, are essential to anyone doing arithmetic (A rat in Tommy's house, etc) or algebra. The parentheses are there to show you what to do first; and you need to know that. Powers indicate repeated multiplication so just think of as which you have to know anyway, as with multiplication and division before addition and subtraction which should become automatic. (The mnemonic leaves out the "from left to right" part anyway.)

FOIL is a particularity bad one. It falls apart as soon as you try to multiply a binomial by a trinomial.

Formulas are better learned in words that tell you what to do. To this day when I differentiate a quotient I'm saying to myself, "denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared." That's the way I've always told my classes to memorize the formula.

This one which showed up today on the AP Calculus EDG and is one I could live with: DAMMIT (Dispise all mathematical mnemonics, instead THINK).

Just my opinion. Do what works for you. So for now Hi-De-Low..

LAND and GOR or What to Value Absolutely

2009-07-25T10:01:00.029-05:00

This week I led an AP Calculus Summer Institute in Virginia. An instructive differential equation problem came up that took us back to Algebra 1 and the definition of absolute value.

You often see in Algebra 1 the solution of the equation |x | = 3 given in one step as

Now, this is correct, but solving inequalities this way presents major problems as soon as you get to things like

If you go with the plus and minus now you have to deal with

So students are told things like for Less then use and or (LAND)

and for Greater then use or or GOR

The problem is there is no attempt to understand the concept. The are three situations, =, <, >, and three different ways to approach them which need to be memorized.

The right way, which works for any, every, and all absolute value problems is to use the definition of absolute value:

If an expression is positive or zero, then its absolute value is the same expression;
if the expression is negative, the absolute value is its opposite.

Then every absolute value is broken into two cases:

Students should learn the definition in words and how to apply it. (And they should say "its opposite" rather than "its negative.") Then every situation is handled the same way: replace the absolute value with two expressions, or if you know which case the expression falls into, the nuse that part of the definition.

Now for the differential equation. The problem was to solve

with the initial condition that y(2)=0.

After separating the variables and antidifferentiating we arrive at

The question was whether to evaluate the constant immediately or to first continue on to

Substituting at this point gives the correct solution.

While substituting the initial condition into

Apparently gives

This differs from the earlier attempt by the "+" sign. There is something wrong and there is no obvious way of realizing it.

This mistake is this: Near the initial condition point (2,0),

and so

Equation solved. The problem came from a poor approach in Algebra 1 (or earlier).

CAS 1

2009-07-18T12:16:00.003-05:00

COMPUTER ALGEBRA SYSTEMS (CAS)

Computer Algebra Systems have be available since the early 1990's. A CAS usually has the ability to draw graphs, produce tables of values, be programmed and run programs. All of this sounds like a graphing calculator. However, a CAS also has the ability to do algebra in symbolic form.

Some CAS are built in handheld form and look like graphing calculators; Texas Instruments, Casio and HP all make such machines. They are also available as computer software; Mathematica is probably the top of the line, but many simpler systems are available some of which are free and/or open software.

All of this make a package for DOING mathematics. Specifically, they can do all the symbol manipulation for the user. From x + x, to factoring, to finding complicated derivatives and antiderivatives, to matrix algebra and statistics computations, the CAS makes the drudgery of algebraic and arithmetical manipulation a breeze.

Nothing this wonderful is without controversy. There seems to be the opinion that the purpose of school mathematics is to teach children how to do these computations. I submit that this is not the purpose. Rather, the purpose is to know and do mathematics. It is more important that students know when they need to ___ and what the result means than to be able to ___ . (You fill in the blank with whatever you want say, factor, differentiate, antidifferentiate, solve an equation, etc.)

Computations are often the longest and most difficult part of doing a problem and the least useful and informative. You have to get past them for sure. This is where a CAS is used.

Like any tool, students need to be taught how to use a CAS. It changes the way mathematics is done. This is, to my knowledge, is not being done. Why?

For further reading see Algemetic.

Good Read

2009-07-12T11:16:00.005-05:00

Is God a Mathematician?
by Mario Livio
Simon & Schuster, 2009

"How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" A. Einstein.

The is an excellent book that discusses two philosophical points about mathematics: Was mathematics invented or discovered? and why is mathematics so unreasonably effective?

Starting with the ancients, Pythagoras, Plato and Archimedes and continuing through Descartes, Newton, Leibniz up to the present time, Dr. Livio interweaves the work and thoughts of mathematicians and philosophers about where mathematics comes from and why it works so well. The many examples of mathematics discovered (or invented) long before there was any practical use for it add a facinating depth to the discussion (e.g. hyperbolic geometry there ready when Einstein needed it for General Relativity).

Many of the "popular" books on math and science available today are really just so much fluff; this one is not. It is a very good and intelligent discussion understandable to anyone. It will help your students (and you too) to get a much broader view of mathematics in all its bredth and depth.

Dr. Livio is an astrophysicist. He has written several other books that I have read and recommend: The Equation That Couldn't Be Solved (Galois, symmetry and groups) and The Golden Ratio. He also wrote The Accelerating Universe that I plan to read soon.

Ban Slope-intercept

2009-07-08T15:48:00.008-05:00

Ban y = mx + b

Really I mean it!

Given a point (x1, y1) and the slope, m, you first write the equation y = mx + b with the slope substituted in for m. Then you substitute the coordinates of the point for x and y and solve for b. Now that you know all the parameters you can write the equation of the line. Sure it works. Sure it's correct.

Someone a long time ago thought this would be a great way to have kids write the equation of a line. Long ago: I learned it in high school and probably most of you did to. It's there in the textbooks to this day. But it is not the best way to write the equation of a line.

Why not go right to y = y1 + m(x - x1), the point-slope equation? Substitute in the 3 numbers and you're done? Fast, simple, no chance of an algebra or arithmetic mistake.

In addition to the efficiency, it makes more sense. Here's why. Let's start with a "real" example.

Five towns, Alpha, Beta, Gamma, Delta and Epsilon, in that order, are located on a straight road that starts at Alpha and goes uphill rising at the rate of 26.4 feet per mile. Beta is 3 miles (horizontally, not along the road) from Alpha. From Alpha is is 7 miles to Gamma, 10.5 miles to Delta and 14.75 miles to Epsilon. Beta is 827 feet above sea level.

How many feet above sea level are each of the other towns?

How far from Alpha should the "1000 Feet Above Sea Level" sign be placed?

I suggest you let your algebra 1 students work on this for a while. I'm pretty sure that together thy will figure it out without know about the equation of a line; without modeling the road with an equation.

Once they are familiar with the situation, you can lead them to writing the equation that gives the distance above sea level, y, in terms of the horizontal distance from Alpha, x, and use that to answer the questions.

Thy reasoning like this: Since we know the most about Beta we'll start there. We are 827 feet above sea level and 3 miles from Alpha.

To this we add the number of feet the road rises per mile multiplied by the number of miles from Beta, which is 3 less than the number of miles from Alpha: 26.4(x - 3)

(This may be the hard part -- using (x - 3) = the distance from Alpha. so spend some time making sure everyone gets this part.)

The height above sea level is y = 827 + 26.4(x - 3).

And there it is: our elevation is Beta's 827 feet plus 26.4 feet for each mile from Beta. And since Alpha is downhill from Beta, when x = 0, the (26.4)(0 - 3) will give a lower elevation for Alpha.

Striping the "real" parts away we have written the equation of a line through (3, 927) with a slope of 26.4.

When you have the equation of a line you want the y-coordinates in terms of the x-coordinates. So y starts somewhere say , y1, and then changes by an amount m for each unit change in x from x's starting value x1. The point-slope form gives you all that.

The slope-intercept form is really a special case of the point-slope form with x1 = 0. Did you ever consider that it may be easier to teach the general form first and the special case later?

You may want to somehow "prove" this (or maybe not). Here is a way to do that. Given the point (x1, y1) and the slope m, pick a general point on the line and call it (x, y). What makes a line a line is that the slope between any two points is always the same. So write:

SLOPE = SLOPE

Then multiply by (x - x1) and solve for y:

There are reasons beside ease of writing the equation to use this form.

1. The form makes sense: start at y1, increase by the slope times the "change in x."

2. The equation can be "simplified" into any other form if necessary.

3. The equation can be entered in a graphing calculator or other graphing program as is, without simplifying.

4. When you get to calculus the equation of the tangent line to a function, f, at (a, f(a)) has a slope equal to the derivative of f at a. The equation is the same.

and the reasoning is the same (start at f(a) and change this value by f ' (a) multiplied by the change from a to x).

5. Later, when you get to power series, the equation above is always the first two terms of a function's Taylor series centered at a.

The real reason is that it is so much faster and simpler.

Aloha

2009-07-01T20:45:00.013-05:00

For reasons that will probably never be known I started this blog in the 4 days between the end of the AP Calculus reading and the start of my vacation. So I'm on vacation right now and not doing much math (unless you count calculating the tips on all the restaurant bills).

Meanwhile some pictures from Hawai'i. They have lots of different flowers here. Here's one called Queen Emma Lily. There are so many flowers here you can actually smell them in the air as you walk around.

Then there are the beaches. Many, many great beaches with clear cool water. This one is is Po'ipu on Kaua'i.

And new things to see. The final shot is from the rim of the caldera of the Kilauea Volcano on Hawai'i. The volcano is active. To give you an idea of the scale, the rim of the caldera runs across the center of the picture. The black floor is about 300 feet down from the rim. The depression in the background is the Halema'uma'u crater; it about 1/2-mile wide and 1.75 miles away. The far wall (horizon) of the caldera is 2.5 miles away. The gas being vented from the volcano and causing the cloud is Sulfur Dioxide, SO2, in the amount of 2,000 to 4,000 tons per day. It causes Vog. Vog is smog caused by a volcano. Unfortunately, like smog, vog is unpleasant.

So Aloha for now. I'll be back to math next week, but right now I Have to deal with this Mai Tai that my wife of 30 years and one day made me.

Absolutely Extreme

2009-06-19T08:57:00.010-05:00

The mantra of this year's AP calculus exams was "communication."

Many of the free-response questions required students to communicate to the reader not just how they found the answer, but also how they knew their answer was correct. "Bald" answers (a correct answer with nothing else) were not accepted. The directions "justify your answer", "show the work", "explain the meaning of" and like appear eleven times on the AB exam alone.

Students were asked to justify a maximum value of a function tree times on the AB exam. I'd like to discuss how to do that here. Students who found the correct maximum value or its location often, far too often, wrote poor justifications. And sometimes they missed earning the point by a single word.

The "maximum (minimum) value of a function" means the absolute maximum (minimum) value; not the local extreme value.

To justify an absolute extreme value it is necessary to consider the entire domain of the function, not just the local region near the critical point.

There are three main ways to justify an extreme value:

The Candidates' Test: Compute the value of the function at the critical point(s) and the endpoints of the domain and choose the maximum or minimum.
Discuss the sign of the first derivative on the entire domain. Say that the derivative is positive for all values on one side of the critical point and negative for all value on the other.
Discuss the sign of the second derivative on the entire domain.

The examples from the 2009 AB Exam.

2009 AB 2 / BC 2 (b) The Candidates' test works easily here. Setting the derivative equal to zero and finding the interior critical point earned the first 2 points in this question. The function had 2 critical points one of which was also an endpoint. So evaluating the function at 3 points and comparing the values easily justified the maximum and earned the last point.

Derivatives could also have been used.

2009 AB 3 (d) The domain in this questions was k ≥ 0 so, with no endpoint, the Candidates' Test cannot be used. So after earning 3 points (derivative = 0, solution k = 400, and value at the solution = $16,000) the justification point can be earned by writing that

this is the only critical point and that the derivative changes sign from positive to negative here.
or the student could claim that for 0 < k < 400 the derivative is negative and for k > 400 the derivative is positive.

This is where the mistakes were made. Omitting the word "only" or otherwise not saying there was only one critical point and the final point was not earned. Saying that the derivative was positive to the left of 400 and negative to the right was not enough. Saying the derivative changed sign here (without saying "only here") didn't do it either -- the sign could have changed elsewhere.

Finding the value of the derivative at particular points to the left and right is even worse. Now, of course, checking single points is how you determine what the sign of the derivative is, so the temptation is to report this as the justification. That doesn't do it. The justification must consider all the values on each side.

The second derivative was easy enough to compute in both these examples (although few used this approach). A justification here would need to cover the entire domain as well: "k = 400 is the only critical point and P''(k) = -4/8000 so k = 400 is the location of the maximum."

2009 AB 6 (c) gave the graph of the derivative of a function on a closed interval [-4, 4] and showed two critical points at x = -2 and x = 3ln(5/3).

The Candidates' Test could be used here, but the relative sizes of the 4 values of the function

were not clear, and this was a no calculator question. (See how this could be attempted without calculating the values at the end of the previous post in this blog.)

So the first derivative was the way to go here. Again students missed the justification point for not considering the entire domain. Having two critical points required some care (x = -2 was a critical point, but not an extreme value). Students made the same mistakes as in the previous example: not saying this was the only place the derivative changed from positive to negative, discussing the sign of the derivative at particular points to the left and right, not saying that f '(x) ≥0 for all x in the interval (-4, 3ln(5/3)) and so on.

So the lessons here are teach your students the meaning and use of "only." Practice writing justifications of absolute extremes that consider the entire domain and not just the local region near the critical point.

The Fundamental Difference

2009-06-16T21:10:00.019-05:00

I returned today from the AP Calculus reading. Seeing all those exams (304,490 more or less) is great and seeing all the different solutions is quite an experience. I would like to discuss several parts of the problems that all relate to the Fundamental Theorem of Calculus (FTC)

Let's start with 2009 AB2/BC2 part c. Students were given that

They were asked to find

You could almost hear them thinking, "First I have to find w(t), then evaluate it at the two points and subtract. I can do that." And in fact this works. After far too many lines of computation by hand (omitted here) students (who were good at this sort of thing) came up with the correct answer:

But wait. Isn't

and can't you calculate that on your calculator? Do you really need to know w(t)?

Then let's look at 2009 AB3 part (b). Students asked to explain the meaning of

using correct units in the context of the problem. The integrand was given as the cost to produce a portion of cable that is x meters from the beginning of the cable in dollars per meter.

Explain the meaning of a definite integral has been asked often on the AP Calculus exams. Three things are required in the explanation: (1) what it represents, (2) its units and (3) an accounting for the limits of integration.

There were many different approaches, with many convoluted sentences. The most common mistake may have been forgetting the units (dollars) or giving the wrong units (dollars per meter). Answers like, "The cost in dollars of producing the last 5 meters of a 30 meter cable," and "the cost in dollars of producing the part of a cable between 25 and 30 meters from the end" earned the point.

Students have trouble with all that of course. So my suggestion is to think of the FTC:

Now it should be quite clear that "The definite integral represents the difference in dollars in the cost of producing a cable of length 30 meters and a cable of length 25 meters." is the simplest, most straightforward answer. And it works even if we don't know or can't find f(x).

In 2009 AB1/BC1 part (b) students were given a velocity expression v(t) in miles per minute (a graph actually) and asked to explain the meaning of

The absolute value of velocity is speed and the integral of speed is the distance traveled. So letting p(t) = the distance traveled.

So the meaning is easily seen to be the "distanced traveled in miles from t = 0 to t = 12."

Finally, in 2009 AB6 a few students tried to justify an absolute maximum of a function f given the graph and equation of f ' on the closed interval [-4, 4] as explained next. Alas, few if any, were successful with this approach, but I liked the idea.

The absolute maximum occurred at a point where x = M (the actual value of M was given). Students could have used the Candidates' Test and reasoned this way:

The inequalities in the first and third lines above are true since from the graph

Perhaps the reason I liked this is that I never quite thought of using the FTC in any of these ways until I realized what students were doing.

Even if you don't know or don't what to bother computing the antiderivatives thinking of them this way may help your students better understand the FTC as something more than a way to evaluate definite integrals.

First Post

2009-06-16T16:04:00.006-05:00

Just getting this started today, so this is not much of a post. We will start with a picture which has nothing to do with high school mathematics. This is to see if I can get it posted. Hope you like it. It was taken just west of Las Vegas, Nevada on a day last December - a day when it snowed in downtown Las Vegas!

If you like this there are some others here