## Thursday, March 31, 2011

### Two for Thursday, 2 and 40

This group is from 2002.

Problem 2

Difficulty: Easy and fun.

Problem 40

Difficulty: Fairly difficult for your students.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Wednesday, March 30, 2011

### The daily Two, from 2002: 1 and 41

Starting a new set!  This group is from 2002.  As with the previous ones, I'll post two at a time so you won't be tempted to work through all 41 at once.  You have to eat, you know.  Just looking out for your basic health.

Problem 1

Difficulty: Easy and fun. I really don't get why students freak out over these simple fraction problems. Okay, I do understand ... not enough practice.  So here's another practice problem.

Problem 41

Difficulty: Until you see it, "What?" When you see it, "D'oh!"
Simple geometry and two nifty trigonometric doodads.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Tuesday, March 29, 2011

### Tuesday Child is full of grace ... 20, 21, and 22.

Problem 20

Difficulty: Easy, but fun. A new way to look at averages and the effect of an addition on the average, depending on n.

Problem 21

Difficulty: Linear programming.Any bets this will somehow work out to an integer?

Problem 22

Difficulty: Easy. Once you've got the details right.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Monday, March 28, 2011

### Monday's Child has Two: 19 and 23

Problem 19

Difficulty: easy. Notice we didn't specify which color.

Problem 23

Difficulty: Very clever geometry question.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Sunday, March 27, 2011

### Delicious pair o'Problems: 18 and 24

Problem 2*3²

Difficulty: easy-ish. A little trigonometric - algebraic manipulation and "Surprise!"

Problem 2³*3

Difficulty: Easy. At least once in every test, in some form or another, the creators work the year into a problem.  This test was for 2007.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Saturday, March 26, 2011

### Are you teaching Calculus next year?

I've been doing it for how many years now? 27 years (I think) ... so much time, so many really cool kids. I guess one thing I really like about teaching calculus is that I can't recall a bad student.

Anyway, for those who are setting up a course for next year:

Text (by author):
• It will probably be Larson or Stewart. Stewart is the more common college text, but Larson is the more HS-friendly. I personally use Larson 7th. I've taught from this book since the third edition and I feel very comfortable with it.
• Hughes-Hallet-Gleason did things in an odd organization, IMNSHO. In a tutoring situation, I had this thing and it was oddly laid out, with integration happening before the end of differentials. You might like it, but I'm too used to the Larson / Stewart / Thomas Finney organization.
• Ostebee-Zora is dense for highschool students.
• Thomas and Finney was cool ten years ago but seems to have fallen out of favor. Don't know why.

If you want to stretch your boundaries and go 21st Century on your kids, there are many online texts.  I'd love to try this and I'm planning on putting selected sections on the Moodle. WolframAlpha will be required for some assignments.
• Neat place to start is the California site: http://www.clrn.org/home/
• http://www.whitman.edu/mathematics/calculus/
• MIT OC: http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
• http://www.math.temple.edu/~cow/
• http://archives.math.utk.edu/visual.calculus/
• and of course, Salman Khan on YouTube.
VERY IMPORTANT
AP insists you fulfill their audit process - mostly they need a syllabus, but you have to complete it if your school wants to put the "AP" label on it. Someone at your school may have already fulfilled that requirement -- make sure to check. Soon.  Here's my syllabus if you want to copy it.

The Summer Before
If you can, and can afford it or if your school with cover costs, hit up a summer AP seminar. The Summer AP Institute at St. Johnsbury VT is one of the best.

The environment is as close as you can get to smart-people nirvana. One week surrounded by a bunch of AP teachers in an informal setting in the Vermont summer. Learn, work, relax, eat REALLY good food, play bocce on the lawn and drink free beer with a slew of brilliant people till dinner. Like you can get better than this? It's \$1100 but worth it ... especially if your school will pony up. Skip the dorm and use the savings and a little extra of yours and take your wife -- a hotel isn't much more and it's a wonderful town.

Yeah, what the Hell, TI?
(from Randall at XKCD) And the marginally colored TI-Inspire doesn't get you off the hook, either.

Technology
• Every kid should have a TI-84 or the equivalent.  I don't require them to buy one but I'll warn them that the ones available in the room for borrowing are often changed and reset by the algebra I kids, making it difficult for the occasional user to get started. "Radians or degrees", language set to Deutsch, etc.  There is a new wrinkle, though:  The graphing calculator app for iPhone and Android devices.  It's got a better screen resolution and it only costs \$3 for the phone they already own.
• Excel is useful - Google apps works, too. Just be ready to reserve time occasionally in the computer room.
• Pencil and paper
To be continued.

### Calculus Syllabus for the AP Audit.

This was enough to satisfy the review process wonks.  Feel free to use it verbatim as I can't claim to have made much up on my own. It is mostly a regurgitation of the table of contents.

I certainly don't feel that anyone should spend an iota of psychic energy on the syllabus - spend it making plans and working toward the best you can do for your students.

Case in point: "Definition by limit: epsilon-delta." was de-emphasized in AP years ago, is usually ignored by most college professors for MAT121 but emphasized by the Audit review process. I couldn't get the audit accomplished without lip service to it.

A change from this syllabus for me: I am changing from a school-hosted ePortfolio to a Moodle, blog and Wiki based on my personal server. There are several places to find free hosting for those. Additionally, I will be requiring certain homework to be done on WolframAlpha since no one can afford Mathematica.  I'd love to require an iPad but I'll have to win the lottery first!

Calculus Syllabus

Instructional Materials
• Textbook: Calculus, 7th edition by Larson, Hostetler, Edwards; Houghton – Mifflin Publishing Company.
• Technology: TI-83 / TI 84 graphing Calculator
While materials can be borrowed from the school, it is highly recommended that students purchase both the book and the calculator. The book will be useful for many years and the utility and convenience of writing notes in the margins and highlighting passages cannot be minimized.
• When appropriate, the class will make use of Geometer's Sketch Pad, excel, Google apps.
• a gmail account would be most helpful.
• E-Portfolios will be a part of the Course. See ____ for account and access information. Work will be done in groups, handed in separately (and included here). Typical Contents:
• Your two sets of school-wide showcase materials
• Chapter projects/labs assigned monthly.
• Past years FRQ questions worked out and scored by each other.
• Inter-curricular questions.

Functions Review
• I. Sign Graph solutions
• II. Coordinates, Slope
• III. Linear Equations: [point slope, slope intercept, general]
• A. Formulae [ distance, angle, perpendicular, bisectors]
• IV. Polynomial functions – library of Functions
• A. Domain, range, discontinuities, roots, synthetic division
• B. Odd, even, Symmetry
• C. composite functions g(f(x)), (g*f)(x), (gof)(x)
• D. Slope = derivatives, differentiability,
• E. "How to say it in Math." Introduction of proof techniques and format.
Limits
• I. x =>a, one sided limits at discontinuity, peak, break in curve, inf. disc, etc.
• II. [ graphically, algebraically, numerically]
• III. definition of derivative as lim h =>0, delta, epsilon (refer back to "More or Less")
• IV. Limits : add sub / mult / div
• V. Infinite Limits
• VI. Continuity, max-min theorem, Intermediate Value Theorem, Computer/calculator-aided exploration of situations in which lim f (x) x a D.N.E.
Derivatives of Polynomial Functions
• I. Definition by limit: epsilon-delta.
• II. Computer-aided exploration:
• A. Using the definition of the derivative as a limit of slope
• B. secants to plot f'(x)vs x. The relationship of the graph of f'(x)to f(x).
• III. Power Rule, Constant Rule, Sum Rule, Product Rule, PowerRule2, Quotient Rule
• IV. Implicit Differentiation
• V. Chain Rule: composite functions
• VI. "Explain" "Show that" "Why or Why Not" – how to justify an answer. Includes review of past free-response questions, and scoring methods, practice developing an appropriate answer with explanations. presentations by students
• VII. Trigonometry
• A. one-day review of basic functions, then Derivatives of trig functions
• VIII. Linear
• A. Tangents and Normals, Linear Approximations and Differentials – ZOOM and compare – analytic results vs. calc – relate back to epsilon/delta
• B. Newton’s Method – writing a recursive program
• C. Calculator computation of derivatives and graphs of derivatives.

Derivative Applications
• I. Sign Graph analysis recap
• II. Graphing
• A. First derivative: increase, decrease,
• B. Second derivative: concavity
• III. Symmetry
• IV. Asymptotes, limits, dominant terms, derivative of hyperbola vs parabola: behavior at extremes
• V. Extrema Theory: Relative and Absolute Maxima and Minima;
• A. 1st derivative test and the Extreme Value Theorem; Intervals,
• B. Continuity;
• 1. Why are the notions of continuity and limit crucial to the Calculus?
• 2. Informal discussion of "function niceness."
• 3. Definition of continuity theoretically compared to on the TI.
• C. 2nd derivative test for maxima and minima
• VI. Recap of "Explain" "Show that" "Why or Why Not" – justify an answer. "How does one know f(c) is minimum?" 1999 FRQ BC:4 Student Sample A and others.
• VII. Maxima and minima: Problems
• VIII. Related rates of change, physics, chemistry, Real Problems, dist-vel-acc-jerk, best solutions. Include here the instantaneous numerical calculations.
• IX. Rolle's Theorem, Mean Value Theorem, Intermediate Value Theorem
• X. Indeterminate forms and L'Hopital's Rule
• XI. Fall School-wide Showcase: Presentations and Explanations (similar to typical portfolio except this is a chance for students to present some of their work to parents, faculty and members of the community.)
Indefinite Integrals
• I. Indefinite Integrals - Primitives (antiderivatives) differential equations, Separation of variables, basic integration formulae, constant of integration.
• II. Finding the constant of integration (C)
• III. Substitution Method - U and dU
• IV. Integrals of Trigonometric Functions
• A. Basic Functions
• B. Using Trig identities to solve integrals
• V. The Area Under a Curve - Definite Integrals
• A. Rectangles & Summation (Sigma notation)
• B. Riemann Integral = Lim as number of rectangles incr. to infinity
• VI. Calculating Areas by Summation
• A. Rectangles
• B. Trapezoids
• VII. Fundamental Theorems of Calculus -
• A. First F.T. of C. -- definition of Primitive F(x): derivative F(x) is f(x)
• B. Second F.T. of C: definite integral is equal to F(b) - F(a)
• VIII. Substitution
• IX. Approximations of Definite Integrals
• A. Trapezoidal Rule A=h/2 (1 2 2 2 2 ...... 2 1)
• B. Simpson's Rule A= h/3 ( 1 4 2 4 .... 2 4 1 )
Integral Applications
• I. Distance - velocity - acceleration - jerk
• II. Areas: horizontal rectangles - dy, f(y), g(y); vertical rectangles - dx, f(x), g(x)
• III. Volumes, by revolution
• A. Creation: use potter’s wheel as example; zucchini-carving to show slicing; cardboard discs, wood tiles. Group–think: Given a lathe template for vase, find the volume
• B. Volumes of revolution around x-axis and y-axis using Discs, Washers, Shells
• IV. Known Cross Section
• V. Average Value, average rate of change
• VI. ** Optional extensions of integral theory
• A. Length of Plane curves, Area of Surface of Revolution
• B. Physics: Moment of Inertia, Center of mass, Centroids, Nonuniform mass distribution, Work done by variable forces (concurrent physics preferred)
Transcendental Functions
• I. Inverse Functions and their derivatives
• II. Inverse Trig functions
• III. Derivatives of the Inverse trig functions and associated integrals
• IV. Natural Log and Derivative
• A. INT (1/cabin) d(cabin) = log (cabin) + C = houseboat
• V. Exponential and logarithmic functions
• A. derivatives and integrals
• B. Applications [ rates of growth, decay, compound interest, Richter scale]
• VI. Spring School-wide Showcase
Integration Methodology
• I. Basic Forms
• II. Integration by parts [Berger method]
• III. Trig Functions, odd powers and even powers
• IV. Trig Substitutions
• V. Integrals involving .....
• VI. Partial Fractions [Alfonso’s Method, Matrix solutions, Millitello cover]
• VII. Numerical integration
** Post – Test Optional
• I. Conic Sections – Definitions and uses of calculus – areas, etc.
• II. Hyperbolic Functions –
• A. Definitions, Derivatives and integrals
• B. Cables, Catenaries (Gateway Arch and Power lines), and the suspension bridge.
• C. Inverse Hyperbolics
• III. Polar Coordinates
• A. Graphing and common figures,
• B. Derivatives and integrals in Polar coordinates
• II. Infinite Sequences and series
• A. e, sin, cos, pi expansions
• B. Taylor, MacLaurin etc.

### Just don't call it tracking

It doesn’t take a rocket scientist—or even a cognitive scientist—to know that kids (and adults) learn best when presented with material that is challenging—neither too easy so as to be boring nor too hard as to be overwhelming. Like Goldilocks, we want it just right. Grouping kids so that instruction can be more closely targeted to their current ability levels helps make teaching and learning more efficient.
Apparently, it takes someone a lot more intelligent than a rocket scientist because it's news to all the guidance counselors and schedulers that I've talked with.

### Happy Saturday. Here's your daily double: 17 and 25.

Day Seventeen

Difficulty: easy-schmeasy, if you subtract. More complicated if you try to add.

Problem 25 = 5².
(I wonder how often you can switch the digits and still write an equivalent expression.)

Difficulty: Medium. More damned logarithms, but there's a neat little prime factorization going on, too.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Friday, March 25, 2011

### Gruesome Twosome: 16 and 26

Sweet Sixteen

Difficulty: easy-ish. Another SAT-type reasoning question. Fun to watch the students work out the ramifications.

Problem 26

Difficulty: Medium. Is it wrong of me to get really annoyed at this question?  It's not terribly difficult, but it is so ... artificial.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Thursday, March 24, 2011

### Daily Problematic Pair: 15 and 27

Day Fifteen ... We're almost there.

Difficulty: pretty easy. Kids will get wrapped up in the P(A) and if-then, but it's an example of a problem that is much easier if you imagine the grid of all possibilities. Definitely a candidate for the SAT.

Problem 27

Difficulty: Medium. It's funny how easily the kids get thrown by an infinite series.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Wednesday, March 23, 2011

### Daily Think. Problems 14 and 28

Day Fourteen
Difficulty: easy-ish. It's similar to a lot of SAT questions in that you don't actually solve for a, b or c but use them all in concert to find some other value.

Problem 28

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Tuesday, March 22, 2011

### Why I support Unions.

The Union will block spurious, petty and mean administrators from firing the competent as well as the incompetent. This is good because the administrator usually has no real idea which is which.

The Union will ensure that teachers get a fair shake when they are dealing with someone who has no real need to be fair.  This is not the owner of a company who is watching every dollar out of his own pocket -- this is a transient employee who has too much power and not enough knowledge.

They float through once, twice a year in a choreographed display of feigned interest in what I am doing. Three weeks before, I get a note saying when they'll arrive and which class they'll be observing - get a lesson plan ready and do the pre-observation meeting. Then they'll watch the class, make a few notes. In the case of the more recent attendees at admin training workshops, they'll attempt to record my speech verbatim in the vain hope that recording every word spoken will somehow tell them more than just listening attentively. This is real?

We meet for the post-op and I'll be given a letter that specifies what they observed. Some of the more insightful comments were "Good lesson" and "knows the material." One of the less insightful was "Separates the kids into two groups by gender." I taught algebra in the chorus room -- apparently Mr. HIP didn't notice that the aisle down the middle separated the kids into two groups and that they took the same places as they normally did during chorus -- I don't do seating charts and usually pay no attention to where students sit. No, he thought I was being discriminatory.

I went 6 years before I was observed again. How was that principal supposed to know anything about me or anyone else in the building? How were they (five in six years) supposed to determine that X should go?

When an admin played games, got in the face of someone improperly, tried to ruin a teacher, there was always the Union stepping in to make sure that things were proper. That is its best role, to act as a buffer, to make sure that each i is dotted and t crossed. I was lucky (knock on wood). I had developed a CYA model and kept tons of paperwork and evidence. When parents complained or school boards took a closer interest, I always had a defense. What if I weren't paranoid?

The Union negotiates a contract. The contract is held up as a object of derision by many on the right - "Look at this egregious waste of taxpayer money. Look at this contract. We shouldn't allow teachers to quote the contract when they might get into trouble!" What an incredible thought, that a contract shouldn't be followed by both sides. "Gullible Boards sign outrageous contracts!" Actually, they sign a contract that both sides agreed to.

Show me any industry that breaks contracts at a whim. I'll wait.

Everybody complains about the LIFO, salary schedule, no merit pay clauses in the contract. When the dust settles, these clauses are seen as the most manageable. Since the admins are rarely around long enough to figure out the names of the teachers in the building, how are they supposed to know which one is worth keeping? The one who has been there for ten years or the kid just out of college who has no idea of what she's getting into? If you eliminate the salary schedule, then you'll have people like me getting \$15,000 signing bonuses and thousands more than the English teachers. Really? Same job. Same experience. Think there won't be any bad feelings? Think the school will run just fine anyway? If you don't have a contract, then I'll demand more money and different working conditions because I know the school is desperate. Maybe I'll train for six weeks and be a dilettante TFA who does the school a favor by my presence and leaves when a real job is available.

The Union works for that contract and makes sure that both sides abide by it.

As for merit pay, this is a fantasy dreamed up by a billionaire. If I get a bonus every year because I suck up to the principal, how long do you think there will cooperation in the department? If I don't trust him to know his ass from a hole in the ground when it comes to hiring and firing, I sure don't want a large fraction of my income held over my head - "maybe you will and maybe you won't."

Far from being the leech on the blood veins of education, unions are the force that makes education work as smoothly as it does.

### Daily Think. Problems 13 and 29

Day Thirteen
Difficulty: easy-ish. Mostly, it's an interpretation question.

I love this next problem. So very cool.
Problem 29
Difficulty: Medium.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Monday, March 21, 2011

### Daily TwoProblems, 12 and 30

Day Twelve

Difficulty: easy-ish. Does your student remember his trig identities?

Problem 30

Difficulty: Medium.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Sunday, March 20, 2011

### Daily Two 11 and 31

First, an easy one.
And then something more complicated, though personally, I'm not a fan of probability questions.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Saturday, March 19, 2011

### Daily two puzzles for contemplation. 10 and 32

Day Ten
Difficulty: easy-ish.

Problem 32:
Difficulty: Tricky. It seems easy, but then tries to kneecap you.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Friday, March 18, 2011

### Daily Two. 9 and 33

Day Nine.
Difficulty: easy-ish.

Number 33:

Difficulty: Hard.
(or as the kids said, OMFG)
It's actually more complicated than hard.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Thursday, March 17, 2011

### Daily Two. 8 and 34

Day Eight will make you think a second ... or ten.
Difficulty: easy.

Difficulty: Hard-ish
(if they don't know the 3D Pythagorean theorem and have to work it out.)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Wednesday, March 16, 2011

### Daily Two. 7 and 35

Problem 7
Difficulty: so easy the kids kept second-guessing themselves. ;-)

Problem 35
Difficulty: Hard (last page)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Tuesday, March 15, 2011

### Daily Pair of Problem Puzzles. 6 and 36

Day Six will make you think a bit.

Difficulty: easy - ish.

Difficulty: Hard (last page)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Monday, March 14, 2011

### Daily Problem Pairing. 5 and 37

Day Five, Number 5
Difficulty: easy.
and Problem 37 because 38 was already posted and I didn't want to lose any of the comments.

Difficulty: Hard. (Last page)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Sunday, March 13, 2011

### Daily Puzzle Pairing. 4 and 39

Problem (4)
Difficulty: first page.

Problem 39
Note: Remainder.
Difficulty: Hard. (last page)

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Yeah, I know, it should read "larger" and "smaller" but I didn't have the energy to fix it.

## Saturday, March 12, 2011

### Daily, they'll get harder. 3 and 40

Day Three
Difficulty: easy.

I'm going to save my self some time and just include the difficult one with the easy one every day.

Difficulty: Hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Friday, March 11, 2011

### Daily Puzzle Pairing. 41

Since there are so many truly able mathematicians among my readers, I've decided to give them something to keep busy with! This set will start at the hardest and work backwards.

Difficulty: hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

### Daily, they'll get harder. 2

Day two.
Difficulty: easy.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Thursday, March 10, 2011

### Daily, they'll get harder. 38

Daily, they'll get harder.  How hard? Try this one:

Difficulty: hard. Especially when you aren't using a calculator.
However, this does mean that the problem is solvable that way and stuff will cancel when you realize the method to use. Of course, when you don't see the method, it's frustrating as hell.

Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

### Daily, they'll get harder. 1

Day one. Question 1
Difficulty: easy.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest

## Tuesday, March 1, 2011

### Bill Gates and Teacher Spaghetti Sauce

Bill Gates is at it again.  For such a smart guy, it is amazing how dense he can be when it comes to education reform. To wit:
We can “flip the curve,” raising performance “without spending a lot more,” if we “measure, develop and reward excellent teaching." ... of all the variables under a school’s control, the single most decisive factor in student achievement is excellent teaching. To flip the curve, we have to identify great teachers, find out what makes them so effective and transfer those skills to others so more students can enjoy top teachers and high achievement.
SO, Billy, we have to identify great teachers and find out what makes them so effective. Then we have to transfer that to other teachers. That will flip the curve.

Hey !  "Find out what makes them so effective"???

If you don't know WHY the great teachers are great, how in Bloomberg's Bloomers can you say definitively that
1. You can identify these great teachers and find out what makes them so effective.
2. These unknowns are transferable.
3. This transference will raise performance “without spending a lot more."
Sometimes life is like judging pasta sauce.

There might not be a best, only a best for you. That teacher might not be the BEST because you can't define best for more than one type of student. There are more types of teachers than there are types of spaghetti sauce. Go watch the Malcolm Gladwell talk at TED.