Get The Solutions Manual

While textbooks required for demanding math and physical science courses generally provide explanations and examples designed to help students review covered material or make better sense of teachers' lectures, the sample exercises included are often too limited or few in number to paint a complete picture of the topics presented. Answers are usually given for odd-numbered homework problems, but oftentimes this just isn't enough.

The issue for many students in these tough classes is how to get those particular answers.

Wouldn't it be nice if there was a companion book that would actually list the step-by-step solutions for each odd-numbered problem in a given math or physical science textbook ... not just the answers?

In most cases, there is such a book. It's called the "solutions manual," and possession of one of these miracle volumes can sometimes make all the difference between ease and stress, progress and frustration, success and failure.

The solutions manual is the perfect compliment to the required text for any important math or science course. When stuck in the mud on a particular homework exercise, students can simple appeal to the solution shown in the manual, allowing them to answer their own questions and continue their forward progress with the assignment. Instead of pulling out their hair over a particularly vexing problem, students can use the solutions manual to quickly figure out what they've been doing wrong, and then apply the techniques illustrated in the manual's solution to other similar exercises, saving valuable time and energy (e.g. solutions to even-numbered exercises can typically be inferred from those given in the manual for corresponding odd-numbered problems). In this way, the solutions manual acts as a virtual tutor, enabling students to make much more efficient and effective use of time spent doing homework or studying for exams.

I always recommend that students about to take a difficult math or science class invest in a copy of the solutions manual for the textbook required for that class. The best way to find a copy is to visit the publisher's web site or do a Google search using the full name of the text followed by the author's name and preceded by the phrase "solution manual" or "solutions manual" (without quotes). Once you have an ISBN number for the appropriate solutions manual, you may be able to find a less expensive copy by searching for it on amazon or eBay. 

(Note: the term "solutions manual" usually refers to the "student solutions manual" containing solutions to odd-numbered problems; the "instructors solutions manual" shows solutions to all problems in the text, but is available only to course instructors).

Don't pound the table, become discouraged, or fall behind ... just get the solutions manual!

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Contact Chris

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

What NOT To Bring To Show And Tell

Most of us have fond memories of grade school "Show and Tell" times, when we as youngsters could bring "cool stuff" to share in front of the class.

"Show and Tell" fosters confidence, introduces little ones to the art of public speaking and presentation, and injects welcome variety (and uncertainty ...) into the daily classroom routine. Thankfully, this old-fashioned pedagogic technique has not gone the way of the Dodo, and is still practiced in many early elementary school classrooms throughout the country.

Nevertheless, some "cool stuff" really ought to stay at home.

An elementary school near Dallas was evacuated recently when a second-grade student innocently brought a deactivated hand grenade to present during his class's "Show and Tell" period.

The principal took the prudent step of emptying out the entire school until police could determine that the neutered explosive device posed no threat. Though it contained a pin, the grenade was empty and harmless, and ultimately no one was hurt.

Can you imagine the lecture this kid got? My ears burn just thinking about it.

An article in Yahoo! News documents the incident.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Pythagorean Legends, Pythagorean Triples

The Pythagoreans were a secret society in ancient Greece, a mystical brotherhood dedicated to mathematical study. Their motto was "All is number." Esoteric groups such as the Freemasons and Rosicrucians claim descent from the Pythagorean Brotherhood.

The central assumption of the Pythagorean brotherhood was that the entire universe could be explained by careful application of whole numbers or their ratios (i.e. fractions, or rational numbers). Legend has it that when a member of the cult of Pythagoras proved that even something as simple as the diagonal of a square could not be measured using whole numbers or their ratios, he was summarily executed to avoid insulting prevailing Pythagorean doctrine.

Nevertheless, Pythagorean theories have been applied over the centuries to such diverse problems such as building the great pyramids of Egypt, establishing musical scales, and mapping the orbits of planets.

The good ol' Pythagorean Theorem is one of the most fundamental formulas in all of mathematics. Falsely attributed to Pythagoras of Samos, this famous rule defines the relationship between the lengths of the sides of all right triangles:

a^2 + b^2 = c^2.

Pythagorean Triples are trios of whole numbers that conform to the requirements of the Pythagorean Theorem, and therefore measure the three sides of a right triangle: the two "legs" forming the right angle, and the "hypotenuse" (not "hippopotamus") opposite the right angle.

Here's a fun and simple way to generate Pythagorean Triples from any given natural number (indeed, all Pythagorean Triples can be generated this way):

1. Pick a natural number larger than two. Call this number n.

2. If n is even: let y be 1 and let x be half of n; otherwise (if n is odd) first subtract 1 from n, then let y be half of this new number and let x be one more than y.

3. The following equations will generate Pythagorean Triples for any such x, y:

a = x^2 – y^2

b = 2xy

c = x^2 + y^2

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Car Talk Puzzlers

NPR's uproariously funny Car Talk could be the single funniest thing on radio. On more than one occasion it was all I could do to maintain control of my car as I listened to the hilarious rantings of Tom and Ray Magliozzi, better known as "Click and Clack ... the Tappet Brothers," during their weekly automotive question and answer show (to get an idea, check out the current list of Car Talk staff members, presumably vetted by the show's ace legal firm: "Dewey, Cheetham, and Howe").

Far from being among the dimmer bulbs in the lamp, these grease monkeys are actually scientists with MIT degrees, no less. A featured part of each broadcast is the weekly "Puzzler," and a fair number of these fabulous brain twisters are mathematical rather than automotive in nature (here's where the guys show their MIT roots).

Click here for an archive of Puzzlers from past shows ... or here to buy a recently published collection of favorite Car Talk Puzzlers at amazon.com for as little as $.01 plus S & H!

To listen to Car Talk (online, NPR affiliate station, Sirius Satellite Radio, podcasting, etc.), click here.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Finite Simple Group (Of Order Two)

Whoever said math isn't musical has never heard The Klein Four.

If you've got a couple of minutes, take a break, give yourself a treat, and check out this hilarious, cleverly written, well performed a cappella composition:

Finite Simple Group of Order Two

Although knowledge of group theory isn't at all required to enjoy the performance, the brilliant plays on words that comprise the lyrics will make real sense only to those with some knowledge of higher mathematics.

Here they are:

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Finite Simple Group of Order Two

The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true

But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two

I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeroes, it's a cruel trap
But we're a finite simple group of order two

I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")

I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.