Ok, the secret's out. Start ordering your

Archimedes and Eureka -- Nature Abhors a Vacuum T-Shirts.

It was a difficult decision, but the results are in (although we still haven''t heard from Palm Beach County). The top 3 submissions based on the log_{b2}(b^{6}) entries in the first ever MathNotations: Name that Mathematician Challenge are -- in no particular order --

Mathmom

Eric Jablow

tc

I came in a distant 4th (I am going to appeal this). Of course, it was only fitting that I chose the 'Arc-Man' to lead off, since one of my personal favorite posts of all time came earlier this year:

The Genius of Archimedes: Parabolas, Tangents, ....

Anyone recall I attempted my first complicated diagram, suggesting how Archimedes proved that light emanating from the focus of a parabolic surface are reflected in parallel rays (and conversely)?

By the way, one of the better explanations for kids of his discovery of the displacement principle can be found here.

Ok, here are the details:

Mathmom found the following fascinating fact here:

Archimedes invented a puzzle called the Loculus (or the Stomachion,

the Ostomachion, the Syntemachion, or Archimedes' Box). It's like a

huge, complex, tangram. In November of 2003, Bill Cutler used a computer program to enumerate all solutions. Barring rotations and reflections, there are 536

distinct solutions.

Eric actually used something called a book (for youngsters out there, here is a link to explain the meaning of this obsolete term) to find the following information about Archimedes' perspective on pure math vs. science:

ARCHIMEDES OF SYRACUSE (c. 287-212 B.C.), son of an astronomer, was

Greece's star mathematician. By avocation he desired the pursuit of

mathematics proper, and he was wholly and passionately committed to

mathematics at its "purist." But by world reputation he was an

engineer, especially in the field of military engines, even if he

protested that he derived no satisfaction from this kind of work. And

when confronted with the problem of determining whether a golden crown

was made of pure gold or was alloyed with silver, he initiated the

method of Hydrostatics for the purpose.

Scientist-professors will always be the same. Archimedes, when

heading the Weapons Research Group for the Syracuse Department of

Defense, would write letters to friends that he was yearning to return

to the Campus and do nothing but pure research for its own sake. But

he was apparently doing classified work to his last breath, literally

so. And when he found his theorem on Hydrostatics he was so excited

that he insisted on talking about it to the man in the street.

I won't actually publish tc's joke regarding one of Archimedes' engineering feats (young children might find their way to this blog), so I will leave it up to my readers to invent their own...

Finally, I promise, from now on, not to the use the name of the mathematician in labeling the image file! Duh...

## Friday, November 30, 2007

### Eureka! The Results Are In! The Winners Are...

Posted by Dave Marain at 5:47 AM 4 comments Links to this post

Labels: contest, Name that Mathematician

## Thursday, November 29, 2007

### Just Another Square Problem? A Means to an End...

Before announcing the thousands (or less!) winners of the Name That Mathematician Challenge, I came across a problem about dissecting a square ABCD with lines PQ and RS which are parallel to the sides of the square. (see diagram).

Naturally, I decided to make it into a deeper investigation. Students and/or readers will be asked to find the maximum value of the product of the areas of either pair of non-adjacent rectangles formed. There are many approaches here, one of which uses the famous Arithmetic Mean-Geometric Inequality. As usual you will work from the particular to the general, beginning with a specific value for the sides of ABCD.

STUDENT/READER INVESTIGATION - PART I

The given conditions about the diagram are given above.

For Part I, we will assume each side of the square has length 4.

(1) (Particular) If AP = 3 and RC = 2, determine the product of the areas of APTS and RTQC. Do the same for the other pair of non-adjacent rectangles formed. Do you believe this product is the maximum possible as we vary the positions of segments PQ and RS?

(2) (General) Show that the product of the areas of either pair of non-adjacent rectangles formed is less than or equal to 16. For example the product of the areas of APTS and RTQC is ≤ 16.

Notes:

(1) Do you think many students would guess what the configuration would be for the maximum product to occur? Is proving the conjecture much more difficult?

(2) The challenge here is to find an effective use of variables to denote the segments. There are many possibilities, some much more efficient than others.

(3) I will add additional parts to this challenge after receiving comments on Part I. How would you generalize this result further? More interestingly, there is a way to prove Part I using the AM-GM Inequality?

Posted by Dave Marain at 5:58 AM 2 comments Links to this post

Labels: AM-GM Inequality, geometry, investigations, optimization, squares

## Tuesday, November 27, 2007

### A New Feature: Name That Mathematician (and more...)

Update^{2}: Only one submission thus far (and a good one!) so I will give our readers another day to email me some fascinating fact about 'A'! Ironically, Isabel, over at God Plays Dice, has a humorous post on attaching names to faces of current mathematicians. What are the odds! As with all experiments, we'll see how this challenge is responded to before making it a regular feature!

Update: Mathmom, reminded me that one can right-click on a PC and see the name of the image file. Working on a MAC, I forgot that. Well, this first picture might be a freebie, but you still need to wow me with some esoteric fact about him! Next time, I'll rename that file!! I also removed the personal info requirement (again, thanks to mathmom's astuteness!).

Starting today, we are introducing a different kind of challenge. Look at the image near the top of the sidebar. Here are the rules (read these carefully before submitting):

DO NOT NAME THE FAMOUS MATHEMATICIAN IN A COMMENT! (If you did already, I will delete it!).

Instead, send me an email at "dmarain at gmail dot com" with the following information:

(a) The name of the person

(b) One fascinating fact about her/him; it doesn't have to be what made that person famous -- I'm looking for the unusual or curious here, be it mathematical or something personal...

(c) Please cite your source for this fact (include the link) - I need to verify its authenticity

I will choose the top responses received within 24-48 hours of the time the picture first appears. Earlier submissions may receive higher ranking than later submissions. I will announce the winners in a couple of days and, of course, include the fascinating facts as well.

Note: I realize that students today have extraordinary online research skills (although one cannot enter an image in Google!), but, remember, the winners are not only determined by giving a name. Feel free to share this with students. They can enter too!

Posted by Dave Marain at 7:48 AM 2 comments Links to this post

Labels: contest, famous mathematicians, history of mathematics

## Sunday, November 25, 2007

### The Right Combination - A Metaphor for Teaching and Learning Mathematics?

If you were looking for a challenge here in higher math using combinations and permutations, sorry to disappoint you! I felt compelled to write this essay after watching my wife patiently attempting to teach one of my children how to open a combination lock. She doesn't think of herself as a teacher, but, she is, and, in many ways, far more skilled than I ever was.

One of the rites of passage for many middle schoolers is mastering the intricacies of the combination lock for their lockers, somewhat akin to elementary schoolers learning how to tie their shoes. Do you remember the frustration you felt the first few times you tried to solve the puzzle of these locks? Do you recall your euphoria when it magically opened? Consider all of the 'skills' involved and think of the parallels to mastering the algorithms of mathematics:

(1) Fine motor skills required to precisely turn the dial and stop at the correct number

(2) Memorizing the 3 numbers in sequence

(3) Understanding the difference between Right and Left when rotating the dial and retaining the R-L-R sequence

(4) The absolute discipline and precision required - close is not good enough

(5) The dreaded second step of the process needed in going 'past zero'

(6) The extreme feelings of frustration from failing repeatedly and the inclination to give up, yet driven to continue

(7) The elation felt in getting it the first time all by yourself, only to be followed by despair when you can't seem to duplicate the feat!

(8) The feeling of accomplishment when you can do it almost every time without anyone helping you

(9) Is there any substitute for independent practice in achieving mastery here?

(10) How important is motivation here in driving the child to continue in the face of adversity?

What about the challenges faced by the 'instructor' here? If you were the one who helped someone succeed, did you find it frustrating or did you have 'unlimited' patience? Did you have to practice it yourself first and think about breaking this 'automatic' process into simple discrete steps? Did you have to try different verbal instructions (for example, using 'down' and 'up' vs. 'left' and 'right') or different techniques of one approach failed? Did repeated demonstrations in front of the child suffice? Did the child say, "Let me do it by myself?" If you've helped several children learn to 'unlock' the combination, did you use the same approach successfully with each child? Are some youngsters simply unable to 'solve the problem' at that time and need to be given a key lock instead as an accommodation? Is making this concession detrimental to their self-esteem and eventual development or is it reasonable at that time? Will some of these youngsters be able to succeed later if given the opportunity to try again (when developmentally ready)?

Is there a metaphor here for teaching children mathematical algorithms? By the way, can you think of others skills or concepts involved in opening the lock that I overlooked? Pls share!

Now, parents, extrapolate this 'teaching' process to dozens of unique math students every day with a myriad of different algorithms over the course of a school year? Anyone can teach, right?

I realize some of you will see the flaws in this metaphor and will point out all the differences between opening the lock and solving a mathematical problem? I know the parallel is far from perfect but this is something that just struck me and I had to put my thoughts down. You know, like a journal, a diary, a blog... Your thoughts?

Posted by Dave Marain at 6:11 AM 0 comments Links to this post

Labels: algorithms, learning, teaching

## Thursday, November 22, 2007

### The Prime Rate: A Post-Thanksgiving Class Opener for Grades 6-12 that Stirs the Brain

Many math educators use warmup problems to review, challenge or set the tone as students walk in the room. Routines like this are effective in having students 'hit the ground running'. These mini-problems can be on the board, while an overhead transparency of selected answers are displayed. The instructor then has time to circulate, check homework, engage students, and get a feel for the difficulties they had with the assignment. By the way, do most of your students take these warmups seriously?

Here is a warmup that requires more active participation on the student's part. I may suggest a different one in a later post, but I'd really like readers to share some of their favorites!

Math Bee

All students stand at their seats. They are told they will be have to give the next number in sequence, according to some rule that will be explained. They will have 3 seconds to respond (can be adjusted but no more than 5). If incorrect or time runs out, they will be instructed to sit down and the next person will have to give the correct answer and so on. You may want to start them off by giving them the first number (judgment call here). I suggest you allow a maximum of 5 minutes for this activity.

Here's the problem I gave to a group of high schoolers but it is highly suitable for middle schoolers as well:

Using positive integers, think of primes ending in 1 or 3 (you may want to use the technically precise phrase 'whose units digit is 1 or 3'). For example, 21 'ends in' 1, but it is not prime. You must go in order and you are not allowed to ask the number the previous person gave! You will have 3 seconds to respond. If incorrect or time runs out, I will ask you to sit down and the next person will need to give the correct number. We will continue until there is only one star shining or time runs out and we have co-champions!

Learning Objectives:

(a) Reviews primes (How many of your students do you think would be eliminated early by starting with 1? You can always start them off with 3 if you feel this will help. Some students will begin with 11, assuming that you meant 2-digit numbers. By the way, 51 and 91, in particular, typically knock out many, if they get that far! Finally, are you thinking this question is not particularly relevant for high schoolers? Count how many questions relate to primes on the SATs!)

(b) Improves listening skills and concentration (How many of your students do you think will forget the last number given either because their minds are wandering or from trying to think ahead to their turn?)

(c) Learning how to think under pressure. (Although we know some students will 'freeze up' or be embarrassed if they are eliminated, they will not be alone! Typically, about half of the students in an above-average class will be sitting down on the first pass through the class! With a high-achieving class of very strong students, you may need to make several passes to reach a winner. If there are a couple left after 3-4 minutes, proclaim co-champions.

Let me know how this goes if you try it after the Thanksgiving break. What variations would you use to make this more effective for your students? What other kinds of problems are suitable for this 'Bee?' Were my predicted statistics way off? Did you predict that students would get past 100?

Again, please share some of your favorite warmups!

Posted by Dave Marain at 6:50 AM 2 comments Links to this post

Labels: math challenge, primes, warmup

## Wednesday, November 21, 2007

### A Thanksgiving Thought...

[Reprinted from Jan 3rd, 2007]

I'd like to share a sentiment that was emailed to me by a close friend. I decided to re-publish this on the eve of Thanksgiving. You may have already seen this, but it does send a message about giving thanks for what we have. (Thank you, Elisa, for sharing this):

One day a father of a very wealthy family took his son on a trip to a farm with the firm purpose of showing his son how poor people live. They spent a couple of days and nights on the farm of what would be considered a very poor family.

On their return from their trip, the father asked his son, "How was the

trip?" "It was great, Dad." "Did you see how poor people live?" the father asked. "Oh yeah," said the son. "So, tell me, what did you learn from the trip?" asked the father.

The son answered: "I saw that we have one dog and they had four.

We have a pool that reaches to the middle of our garden and they have a creek that has no end.

We have imported lanterns in our garden and they have the stars at night.

Our patio reaches to the front yard and they have the whole horizon.

We have a small piece of land to live on and they have fields that go beyond our sight.

We have servants who serve us, but they serve others.

We buy our food, but they grow theirs.

We have walls around our property to protect us, they have friends to

protect them.

The boy's father was speechless. Then his son added, "Thanks, Dad, for showing me how poor we are."

Isn't perspective a wonderful thing? Makes you wonder what would happen if we all gave thanks for everything we have, instead of worrying about what we don't have.

Appreciate every single thing you have, especially your friends!

Life is too short and friends are too few.

HAPPY THANKSGIVING TO ALL MY READERS!

Posted by Dave Marain at 12:12 PM 0 comments Links to this post

Labels: Thanksgiving message

## Sunday, November 18, 2007

### Circles, Chords, Tangents, Similar Triangles and that Ubiquitous 3-4-5 Triangle

[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]

The cone in the sphere problem led me to an interesting relationship in the corresponding 2-dimensional case with a surprise ending. (Only a math person would compare a math problem to a mystery novel!). The following investigation allows the student to explore a myriad of possibilities: from similar triangles to the altitude on hypotenuse theorems to Pythagorean, to chord-chord or secant-tangent power theorems, coordinate methods, draw the radius technique, etc. Sounds like this one problem might review over 50% of a geometry course? You decide for yourself! Just remember -- one person is not likely to think of every method. Open this up to student discovery and watch miracles unfold...

STUDENT ACTIVITY OR READER CHALLENGE

In the diagram above, segment AF is a diameter of the circle whose center is O, BC is a tangent segment (F is the point of tangency), BC = AF and BF = FC. Segments AB and AC intersect the circle at D and E, respectively. Lots of given there! Perhaps some unnecessary information?

(a) If AF = 40, show that DE = 32.

Notes: To encourage depth of reasoning, consider requiring teams of students to find at least two methods.

(b) Let's generalize (of course!). This time no numerical values are given. Everything else is the same. Prove, in general, that DE/BC = 4/5.

(c) So where's the 3-4-5 triangle (one similar to it, that is)? Find it and prove that it is indeed similar to a 3-4-5.

Posted by Dave Marain at 6:43 AM 3 comments Links to this post

Labels: 3-4-5 triangles, circles, geometry, investigations, similar triangles, tangents

## Saturday, November 17, 2007

### The Classic Cone Inscribed in the Sphere Problem: Developing Relationships Before Calculus

Update: View the series of videos here explaining the procedure for solving the cone in the sphere problem below as well as related questions.

Many Algebra 2 and Precalculus textbooks have begun to include those challenging 3-dimensional geometry questions involving 2 or more variables and/or constants. However, we know from the difficulty that calculus students continue to have with these, that we need to do more before students do their first optimization problems in calculus. You know the kind: Determine the radius of the __________ of maximum volume that can be inscribed in a _________ of radius R. These problems have fallen out of favor somewhat with the AP Development Committee, perhaps because they lack that real-world flavor or perhaps because they had become predictable or perhaps too hard. I would argue they have been part of the rites of passage for calc students for many generations for a reason - they blended the spatial reasoning of geometry with the need to identify variable relationships and reduce the number of conditions down to one function of one variable if possible. In other words, they help to develop mathematical sophistication. I 'cut my teeth' on these -- did you? Any calculus teachers reaching this topic yet in AP Calc?

Anyway here's an activity for you Algebra 2 or Precalculus students to prepare them for these challenges. As usual we proceed from the concrete (i.e., given numerical dimensions) to the abstract. Rather than attempt to draw the diagram, which is fairly challenging for me given the tools I have, I will describe the problem verbally. Good luck!

STUDENT ACTIVITY

(1) A right circular cone of height 32 is inscribed in a sphere of diameter 40.

Note: Students need to learn how to make a diagram of this problem situation.

(a) Determine the radius of the cone.

(b) Determine the volume of the cone. [Imagine asking students to memorize the formula!]

(c) Keep the diameter of the sphere at 40. This time, determine both the radius and volume of the inscribed cone whose height is 80/3. The numbers are messy but try to work in exact form (fractions, radicals) before rushing to the calculator to convert everything to decimals. Oh well, we all know what will happen here!

(d) Try another value for the height of the cone, keeping the diameter of the sphere at 40. See if you can produce a volume greater than in (c). Any conjectures?

(2) We could throw in an intermediate step by using a parameter R to denote the radius of the sphere, and use numerical values for different possible heights of the cone, but I'll leave that to the instructor. Instead, we'll jump to the abstract generalization:

A right circular cone of height h is inscribed in a sphere of radius R.

(a) Express the radius, r, of the cone in terms of R and h.

(b) Express the volume, V, of the cone as function of h alone (R is a constant here).

(c) Use your expression for r and your function for V to verify your results in (1).

(d) Calculus Students: You know what the question will be! Oh, alright:

Determine the dimensions and volume of the right circular cone of maximum volume that can be inscribed in a sphere of radius R. Anything strike you as interesting in this result?

Posted by Dave Marain at 7:36 AM 0 comments Links to this post

Labels: calculus, cone, functional relationships, geometry, optimization, precalculus, sphere

## Thursday, November 15, 2007

### Percent Increase vs. Decrease Redux: A Rich Investigation for Middle Schoolers and Algebra Students

[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]

While we are contemplating tc's rectangles inscribed in circles problem (and we will post some solutions in a couple of days as needed), here's a change of pace. Awhile back, there was considerable interest in a percent problem posted on MathNotations involving an artificial scenario in which there were 20% more girls than boys in a group. Remember the heated discussion about the semantics of that problem?

Well, here's another scenario for you to challenge your middle school students or yourself...

STUDENT ACTIVITY or CHALLENGE FOR OUR READERS

Final score in the basketball game: Central 90, Eastside 60.

Jay, who played on Eastside, thought to himself after the game: "If we had scored 15% more points and they had scored 15% fewer points, we would have tied."

Now, how's that for a real-world application of percents. I'm sure you know hundreds of students who would think like that after the big game. Well, it's my blog and Jay is my invention and that's how he was thinking, so there! Of course we know that Jay was confusing increase and decrease of points with % increase and % decrease. But, solve the following:

(a) Increase 60 by 15% and decrease 90 by 15% to show numerically that Jay's reasoning was incorrect.

(b) Increase 60 by 20% and decrease 90 by 20% to show that 20% is the percent Jay had intended.

(c) Determine algebraically that the correct answer is 20%, again starting with scores of 60 and 90.

Note: While parts (a) and (b) can be handled by most middle school students in prealgebra, this question should prove more difficult, even for Algebra I students. However, some will get it and, with guidance, the rest can too!

(d) You surely didn't think we would let you off that easily, did you? Of course, we will now ask you to generalize the result:

Suppose A and B are positive numbers and A is less than B.

If A is increased by X% and B is decreased by X%, the results are the same. Determine an expression for X in terms of A and B.

Note: If you come up with the formula, think about why it makes sense. Any thoughts?

Posted by Dave Marain at 5:51 AM 13 comments Links to this post

Labels: algebra, middle school, percent increase problem, percent word problem

## Tuesday, November 13, 2007

### Drum Roll Please: The Debut of TC's Total Challenge

As you may have read in an earlier comment, I've invited one of MathNotations' most dedicated and talented contributors to go beyond commenting and share some of his creative ideas and insights by being an occasional guest blogger - he has graciously accepted.

For his inaugural offering, tc is challenging you and/or your students to solve a classic calculus problem using non-calculus methods. I have made a few minor edits, but the activity is essentially what tc sent to me.

---------------------------------------------------------------------------------------------------------------

I give you tc's Total Challenge I:

One of my math professors in college used to say there were three ways

of tackling any problem: the right way, the wrong way and the Navy way

(correct, but extremely roundabout).

In this exercise, we will look at three ways (not necessarily the ones

named above) of doing the following problem:

Determine the rectangle of maximum area that can be inscribed

in a circle of given radius r.

Let the inscribed rectangle have sides a and b. The diagonal of the rectangle passes through the center of the circle (this can be shown, but you can assume it is true).

(1) Express r in terms of a and b.

(2) Express the area in terms of a and r.

(3) Instead of maximizing the area, we can maximize the square of the area.

(a) Express the square of the area as a quadratic in a^{2} (you may want to substitute c for a^{2}).

(b) By completing the square, determine the value of a for which the area is a maximum.

(c) Determine the value of b and the maximum area.

(d) What conclusion can you draw about the rectangle of maximum area?

(This is the first way, which I call the Algebra way)

(4) Divide the rectangle into 2 congruent triangles, using a diagonal. Draw a half

diagonal that intersects this diagonal.

(a) Write an inequality for the area of one of these triangles in terms of r alone. The inequality should be of the form Area ≤ _______.

(b) If you can achieve equality, then you have maximized the area of the rectangle! Find out when this occurs, and if it does, find the lengths of a and b. (The Geometry way).

(5) Method 3 - the Calculus way of course.

---------------------------------------------------------------------------------------------------------------

Additional comments from DM:

(i) thanks, tc!

(ii) tc's geometric approach in (4) also suggests a connection to the famous AM-GM Inequality. Visit this link and see if you can make the connection. This is not obvious.

Hint: Apply the AM-GM to a^{2} and b^{2}.

Posted by Dave Marain at 10:18 PM 2 comments Links to this post

Labels: calculus, inscribed rectangle, maxima/minima problem, tc'sTotal Challenges

## Sunday, November 11, 2007

### The Matrix Reloaded: Deriving sin(A+B), cos(A+B) by Rotation Matrices

In our previous post we asked students to verify the sin(A+B) identity for an angle of 75°. Although one might try to generalize the result, there are many other derivations for the sum and difference identities that teachers have seen or used. Those who teach this topic whether it be in an Advanced Algebra/Trig, Advanced Math, Precalculus, or some similarly named course have the choice of deriving the formula, outlining a proof and having students attempt to provide the details or simply motivating the formula. My guess is that, unless you have a highly motivated and strong group of students, the full derivation is not done in class. By the way, an excellent applet for helping students visualize these formulas is located here. One could use this for classroom demonstration or in tutorial mode.

The forms of these trig identities often engage students and some may wonder if there's a pattern one can recognize that occurs elsewhere in mathematics. Using nonsense names for functions, these rules seem to have the form: the squiggle of the first times the squeegee of the second ± the squiggle of the second times the squeegee of the first. Students will hopefully recognize this pattern again when they see the product rule in calculus. Is there an overarching concept that encompasses forms like this? Well, I'd like to suggest one in this post. You can decide for yourself if it's worth developing the required machinery for secondary students (or it can be an enrichment topic or project).

The trig identities mentioned above have the form of a sum (or difference) of products. Where might one encounter this in mathematics? For myself, it's when I studied the products of matrices or the dot product of 2 vectors. v_{1}⋅v_{2} = a_{1}b_{1}+a_{2}b_{2}. Matrix products and vectors are required in some states' curricula (CA for example), so there is a basis for this approach (pun intended!!).

To simplify , we will focus on position vectors with a length of 1 so that their terminal points are all on the unit circle. Students learn early on in trig that each point on the unit circle can be represented using the parameter t or Î¸ as in (cosÎ¸, sinÎ¸) and that Î¸ represents an angle or rotation, counterclockwise for Î¸>0, etc. Another way of viewing this is that the unit vectors (1,0) and (0,1), (which are labeled i and j by most physics teachers), are transformed by this rotation as follows:

(1,0) ---> (cos Î¸, sin Î¸) (*)

(0,1) ---> (-sin Î¸, cos Î¸).

These results are straightforward and students should be able to demonstrate them graphically for values of Î¸ in Quadrant I. For those who recall the terminology from linear algebra, (1,0) and (0,1) are referred to as an orthornormal basis.

[We will use the notation R_{Î¸} to denote the rotation transformation by an angle Î¸ about the origin]:

STUDENT (or reader) ACTIVITY

1. Have students verify or find the following (either graphically or using (*) above) :

R_{90°} (1,0) = (0,1)

R_{90°} (0,1) = (-1,0)

R_{60°} (1,0) = (1/2,√3/2)

R_{60°} (0,1) = ______

R_{45°} (1,0) = ______

R_{45°} (0,1) = ______

--------------------------------------------------------------------------------------------------------------

We will now show how the following 2x2 rotation matrix can accomplish this transformation:

(**)

Note that the first column (vector) is the rotation image of the vector (1,0) and similarly for the 2nd column. This idea of using (orthonormal) basis vectors to construct a matrix representing a transformation is crucial in linear algebra. This requires more development and that is not the purpose of this post. We are asking students to accept this for now, although we will have them verify that this matrix approach produces correct results in specific instances. We will now represent the rotation image of any point on the unit circle by multiplying this matrix by the column vector which represents that point. In rectangular (x,y) notation:

----------------------------------------------------------------------------------------------------------------

STUDENT ACTIVITY (cont.)

2. _{Î¸ } for Î¸ =

(a) 90°

(b) 60°

(c) 45°

(d) -90°

3. Re-do problem #1 above using the matrix multiplication formula (***). Make sure your results match!

---------------------------------------------------------------------------------------------------------------

So how does all of this relate to the sum/difference trig identities? We're almost there! Instead of using (x,y) to represent an arbitrary point on the unit circle, we will use the trigonometric form (cosÎ±,sinÎ±) in column vector form. For now we will assume Î± is between 0 and Ï€/2:

STUDENT ACTIVITY (cont.)

4. Perform the following matrix multiplication:

Does the resulting column vector look familiar? Answer the next two questions and maybe you'll figure out why!

5. The result of #4 is equivalent to a single rotation of what angle?

6. What can we conclude?

----------------------------------------------------------------------------------------------------------------

There's much more to say here but this is enough for now. As usual, I take full responsibility for errors or ambiguities. I await your thoughts. If this is a derivation you've seen before, let me know. One could also derive this using complex numbers or polar coordinates or ...

Posted by Dave Marain at 6:25 AM 8 comments Links to this post

Labels: linear algebra, matrices, rotations, sum-difference identities, trigonometry

## Saturday, November 3, 2007

### Another View of sin(A-B) for a Special Case: An Investigation

[As always, don't forget to give proper attribution when using this in the classroom or elsewhere as indicated in the sidebar]

In a standard trig unit, students learn those wonderful formulas for the sin and cos of the sum and difference of angles. Many creative methods have been developed to derive these formulas and, depending on the ability of the group and teacher preference, these are demonstrated or not. Students are typically shown various mnemonics for recalling them on the big test, but, in this investigation, students will derive sin 15° using only 30-60-90 triangle ratios and the Pythagorean Theorem. We will then compare the result to that obtained by the traditional formulas for sin(45°-30°) or sin(60°-45°) and show equivalence by algebraic methods using radicals. This is not an attempt to develop a general approach to deriving sum/difference formulas, although readers are invited to try a generalization. You may recall other posts on this blog of a similar nature.

THE PROBLEM/INVESTIGATION

Refer to the triangle above. If the print is too small, click on the image to magnify.

∠A = 75° and ∠B = 15°

(a) In the triangle above, locate point D on side BC such that ∠CAD = 60° . Express the lengths of the sides of triangle CAD in terms of a.

[Note: We could avoid the variable a altogether and assign a value of 1 since this is a ratio problem.]

(b) Show that CB = a √3 +2a.

(c) Use the Pythagorean Theorem to show that AB = a √(8+4 √ 3)

(d) Verify the identity (√ 6 + √ 2)^{2} = 8+4 √ 3. Use this to rewrite AB.

(e) Use above results to obtain an expression for sin 15°.

(f) Use the standard trig formula for sin(45°-30°) to obtain an expression for sin 15°.

(g) Show your results in (e) and (f) are equivalent.

An Instructional Aside

When introducing the formula for sin(A+B), for example, teachers sometimes motivate it effectively using numerical values or considering the special case A=B. Here's an alternative:

Consider the special case A+B = 90°

Ask students to verify the formula for sin(A+B) in this special case. Simple, but at least it's something slightly different to pique their curiosity.

Standard Disclaimer:

This investigation is not copied from some other source. As it is original and has not been edited by others, there's always the possibility of error. Please feel free to suggest corrections/edits/extensions...

You know I welcome your comments!

Posted by Dave Marain at 6:32 AM 12 comments Links to this post

Labels: 30-60-90, algebra 2, precalculus, radicals, sum-difference identities, trigonometry