An SAT Problem or Teaching/Learning Beneath the Surface

Ok, so I don't have another anecdote from my grandson today so I'll have to get back to mathematics -- problem-solving, teaching and learning.

The video below deals with an algebraic equation in 2 variables which should be straightforward for your stronger Algebra II or Precalculus student. But will it be? I invite you to predict how many in your classes will answer it correctly, then try it out. After all, it is multiple choice, so statistically some should get it right by some means or other!

Hopefully, the purpose of the problem and the video will become clear to all of you. If we want our students to demonstrate better reasoning and an understanding of important ideas in math, we need to feel comfortable in teaching for meaning and understanding. This doesn't mean we stop teaching algorithms and procedures, however. Exactly what all this means and how to do it is the reason for this blog. I certainly never claimed to know the answers or any other mystical secrets. I only know that I never gave up trying.  Sometimes my efforts failed miserably, but I hopefully learned from these attempts.

It would mean a lot to me if you share your thoughts here or on my You Tube channel, MathNotationsVids, where you will find my other videos.




Note: Another subtle point I should have made in the video---
y(x-4) = 0 → y = 0 OR x = 4
It is important for us to stress this point and distinguish it from "AND" logic. If the equation were in the form:  y² + (x-4)² = 0, we would have (y = 0) AND (x = 4), whose graph would be the single point (4,0). Another instance where an exercise on the board can lead to a rich, fruiful and profound discussion. If all of this is seen as taking too much time away from content, remember this is precisely the kind of change in curriculum and instruction that Prof. Schmidt has been trying to tell us about for over 15 years! Well, I'm preaching to the converted, aren't I...
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"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific

Stuck on a Calculus Problem? Ask St. Patrick!!

With existing technologies and all the help available on the web I am incredulous that students still feel lost at sea the night before a major exam or just getting through an assignment. I've asked so many students what they do when they're frustrated by some math problem at 10 PM: "So who do you call? CalcBusters!"Seriously, they often just look at me as if I have two heads. Well, what are their options?

(a) Call their teacher/professor? Uh, not likely...

(b) Email their teacher/professor? Assuming one has this email address, what are the odds you will receive an immediate reply which will illuminate everything...

(c) Go into a Calc help chat room enabled by your teacher or one set up by some student. A good option if anyone is actually online at that moment. Your teacher or another student can establish guidelines for this so that students know a help session will always be available from 9 PM to 11 PM for example. See (e) below for a similar idea.

(d) Call or email a friend (remember you will then have only 2 lifelines left!). Of course this presumes you have a friend who understands it better than you and can communicate a solution over the phone or via email. Remember what time it is...

(e) Go to a Calculus forum/discussion group in which you can post your question and someone with the knowledge will reply in short order. This is a viable option as there are now many such help groups out there and I will review these and provide links in another post.

OR...

Go to YouTube and find a free video tutorial demonstrating a similar problem in detail.
For example, suppose you're floundering with "integrating by partial fractions." You can just Google "YouTube partial fractions calc video" or something like that and, presto, you are transported here. Ok, the video covers a more sophisticated problem involving a rationalizing substitution as well as partial fractions, but you immediately see dozens of related videos in the sidebar. There are many excellent free videos online from many talented teachers/professors but on this post I will feature one of the best.

If you're interested in seeing an exceptionally clear presentation of calculus or other math topics you cannot do better than Patrick's videos which he offers at no cost. Patrick does not know that I am writing this review so rest assured I am not getting any commission here! The link above is one of his lessons.

He tries to limit his lessons to 10 minutes to make the file size manageable. His writing on the whiteboard is crystal clear, his organizational skills are exemplary and his speaking voice is soooo calming. Further, his explanations are mathematically precise and include just enough rigor to make the purists out there happy without sacrificing clarity. Compare his videos to the amateurish attempts I have posted on this blog - uh, there is no comparison.

Patrick also has a website where you can find all of his free videos. Look here first on YouTube or go directly to his site.
Patrick's background (from his site)
About me: I have been teaching mathematics for over 8 years at the college/university level and tutoring for over 15 years. Currently I teach part time at Austin Community College, but have also taught at Vanderbilt University (a top 20 ranked university) and at the University of Louisville.

He runs a tutoring service in Austin and I'm quite sure he is doing well considering the quality of what he is offering for free. And his videos are not exclusively calculus. Enjoy!

Boring Hole in Sphere Calc Video - Finally!!

Remember when I originally posted this problem back in January? Look here.

Here is the original problem:

A hole is drilled (bored) completely through a solid sphere, symmetrically through its center. If the resulting "hole" is 6 inches in height (or depth), show that the remaining volume must be 36π inches cubed.

OBJECTIVE: Motivation, explanation and application of method of cylindrical shells for finding volume of solid of revolution

TOTAL LENGTH: about 45 min

Please Note: These videos are not intended for students who want quick simple explanations for standard homework or typical exam items. This problem is above that level and the explanations are lengthy and very detailed!


Well, this 'video' is fragmented into 7 parts, the transitions are amateurish, it was composed over a few days (therefore different outfits!), cheap props and the quality is well, you know...

In spite of all the negatives, I'm hoping someone will find this helpful. Remember I'm doing this to cover a broad audience -- the Calc I/II student who wants understanding and clarity (not skipping steps!) to the AP student/Math-Sci-Engineering major who wants some theory and rigor. I'm also demonstrating some aspects of pedagogy here for the new calculus instructor who may have to prepare a similar lesson.

As mentioned in the video, there are many wonderful websites and videos which will provide better graphics, animation and quality. A couple of links are provided below. However, my purpose here to provide a highly detailed development of a classic calculus problem which reviews the method of cylindrical shells for volumes of solids of revolution.

Finally, my original intent was to find the volume that was removed by at least two methods and to generalize to a hole of depth h, but this is way too long as it is! Of course, I don't expect many views or comments but it will be out there for anyone who might have use of this for as long as this blog exists! I'm really hoping comments will look past the low-tech aspect and address the content and pedagogy.

Instructors
Please feel free to share this with your students or for whatever purpose you may have.
As stated above, the total length of all parts is about 45 minutes, the length of a typical hs class period, so it wouldn't make sense for the classroom. You might want to recommend students view this after learning the basic idea of the 'shell' method as reinforcement or after assigning this problem for hw or extra credit. These days students are savvy enough to locate, on the web, solutions and videos to most any problem we assign, so be careful! (You already knew that!)

Some Recommended Links
Volumes of Revolution - Cylindrical Shells
As mentioned in the video, patrickJMT is as good as it gets for clear, simple and mathematically accurate explanations.

Volumes - Cylindrical Shell Method
Wonderful explanations and excellent graphics and animation of the shell method (in Flash) from one of the best calculus sites on the web - utk (U Tenn Knoxville)

There are many other outstanding sites - I apologize in advance for omissions here. Just keep searching until you find the one that works for you!

As always, I am responsible for any errors - don't hesitate to point them out! At least we made it before XMAS 2008 ended!

The videos below are connected, so you might want to watch them in sequence.
However, the actual solution to the problem starts in the 5th segment below.
Read the descriptions of the segments to guide you in deciding where to begin. If you do not want a lengthy introduction, and already know the shell method, skip down to the 5th clip.


These first two video clips provide an overview for what I intend to cover.
Also the key relationship R² - r² = 9 is developed.







These next two segments motivate and derive the method of cylindrical shells.








The actual solution to the problem starts below!











Yes folks I know how drawn out this all was. I will try to improve on these but I will take an hiatus from my busy movie production schedule for awhile!

Happy New Year!

Calculus Video: Optimization (Max-Min) - NEW Improved Version!

Well, I felt badly about that error I made in the original version from a few days ago (which will now be deleted). I also decided to change some portions, electing to solve for the critical values using algebra rather than by the graphing calculator. Finally, I took a tighter view of the whiteboard so that the writing will appear larger. There will be some glare on the board which I hope will not be too distracting. I hope you will find this more helpful and again I apologize for any confusion caused. If you stored the original video, I would ask you to delete that.

HAPPY HOLIDAYS!

The problem in the video below demonstrates important concepts as well as the standard procedure for solving optimization problems. There is also a brief discussion of a heuristic I have found very useful when teaching these kinds of applications. As always I depend on you to share your thoughts. I keep saying this knowing there might not be too many comments!

Calculus Video Pt2 - 11-14-08: Related Rates

As promised, here is a video of a related rate application. This one is fairly straightforward but it does demonstrate the procedure to follow when doing the more sophisticated types.

These videos build on the foundation videos I posted which developed the Leibnitz form of the derivative and the Chain Rule as well as the basics of implicit differentiation. I strongly encourage the student to view these first. If you're more familiar with the topic and want a quick review, then start anywhere!



As always, I depend on my readers to make suggestions to improve the quality of these videos, both mathematically and technologically. Please indicate any errors I may have made so that I can correct them or at least indicate where they are in the video.

Finally, there are DVD sets of Calculus lessons out there as well as other videos you can find on YouTube or by searching. I am interested in knowing if you feel there is a reason for my producing more of these. I hope you enjoy it and, if it helps just one student feel a bit more comfortable with these topics, then it has been worth the effort!

Calc Videos : Chain Rule, Implicit Differentiation Pt. 1

I 've been promising this for awhile now so I thought I would at least upload the first 3 video segments. The more complicated implicit differentiation example will follow...

Note: There are usually some minor glitches in these productions, no exception here!
(1) In the first video below I inadvertently referred to one of the variables as 'dependent' rather than independent.
(2) On a few occasions the writing near the bottom of the board gets cut off but it should still be possible to follow along.

The three video segments below are linked. I had to split the original video to make it manageable.

Video Mini-Lesson: Cone in the Sphere Problem

As a result of the numerous views of a calculus problem I published in November, I decided to present the following video mini-lesson. As before, I had to break it up into parts to control the file size for uploading. I hope this has some value for those who were looking for a more detailed discussion of this question. Much of this is highly appropriate for precalculus students.

Note: Before playing the videos below, a correction and comments:
(1) In error, I referred to the cross-section of the cone as an isosceles right triangle. Make that isosceles only!
(2) The video and audio quality is far from perfect. Bear with me on this!
(3) I didn't discuss the case where the height of the cone is less than or equal to the radius. This will not produce maximum volume but should have been noted. I will have more to say about this later.
(4) There is so much more to discuss about this question, in particular, the result that the cone of maximum volume has height equal to (4/3)R or that the center of the sphere divides the altitude into a 3:1 ratio. These may be discussed in upcoming videos. In particular, as suggested in the videos below, there will be a treatment of the 2-dimensional analogue of this problem, namely, the isosceles triangle in the circle problem.
(5) These video 'mini' lessons are designed for the university or secondary calculus student (probably comes too late for the college final exam) or for anyone wanting a refresher. Beyond my personal style of presentation, there are pedagogical issues (instructional tips) that arise in the videos that might be of interest to someone teaching calculus for the first time.

If you're getting bored of watching the same chalkboard and my same drab outfit, well, it is a low-budget video! I hope you will let me know if this proves helpful and if you'd like me to continue these. As mentioned previously, I will also be employing other technologies for demonstration purposes.

Happy Holidays!

The Dissection of the Square Problem - Part III - A Video Surprise

Ok, we've been discussing the square dissection problem for a few days now. The original problem was here and Solution Method I was here.

Now don't start laughing at these crude, low-tech videos! The 1st video is 2-3 minutes long. It provides a quick overview of the squares problem and a demonstration of why the product of the areas is the same for either pair of non-adjacent rectangles. It ends abruptly with part of the derivation I gave in the previous post.

The remaining presentation is split into 3 segments in order to control the file size (Blogger has a problem with larger files). In these segments, I derive a formula for the maximum product using the Arithmetic-Geometric Mean Inequality (AM-GM) and apply it to the square dissection problem. The first of these clips reviews this important and useful inequality, presenting a standard algebraic derivation. The 2nd of these 3 clips presents a geometric derivation and the final clip shows how we can derive a formula, m⁴/16, for the maximum possible product of the areas in the general case of a square of side length m.

As I say on the video, there's no SmartBoard technology, no Mimio, no whiteboard with vivid color dry erase markers! Just an old chalkboard my kids use in the basement, some inexpensive chalk and sheets of paper towel to erase. I wore my favorite 'Pop-Pop' sweatshirt - sorry, no formal wear.

Let me know if you find this helpful (once you stop chuckling!). At least you'll be able to match a name to a face and it isn't Archimedes! If I get few if any comments, I'll know that my 15 minutes of fame have expired.