The first session of Math Through Technology took place last Thursday and the participants were up to the task: everybody had their laptops open and were eager to get going learning to solve hard math problems using their computers. We used Geogebra to graph functions and find derivatives, all to help Galileo prove the acceleration of dropped objects is constant.
They were all capable of finding patterns in the algebraic equations for slopes of lines and curves, so they easily found the pattern for all polynomials.
Next week they'll be looking for the pattern for finding areas under lines and curves. This group is so bright I think I'll stand aside and let them do the exploring.
Recently I talked to an engineer who said it's a problem for his firm that the college interns show up not knowing any of the engineering software that's used daily in an office. They know all the theoretical stuff, but they have to learn the practical, technological side on the job. I told him I'm trying to help him!
Sunday, September 9, 2012
Sunday, August 12, 2012
Solving Quadratics the Hard Way
A post to a math Google Group asked about the effect of computerized equation solvers on math education. Don "The Mathman" Cohen replied that during his work with his students he's solved quadratic equations (like x2 - 5x + 6 = 0) no less than 12 ways! This list includes continued fractions and iterating using a calculator. Schools normally only teach 3 ways: factoring, completing the square and (my favorite) the quadratic formula.
Being in my calculus mode recently, I wanted to add Newton's method to the list. In the 1600s Newton used his new invention, calculus, to approximate roots of polynomials like the one below:
A "root" of a function is the x-value where the line or curve crosses the x-axis. In the figure above, the root is the red dot. You guess the root is at x0, then you plug x0, the function of x0 and the derivative of the function at x0 into the formula
to get x1, the next approximation for the root. Do that again and x2 is even closer to the red dot in the picture.
I just wrote a short program in Sage to print 9 approximations and used x = 5 as my first guess:
f(x)= x^2-5*x+6
g(x)= derivative (f,x)
a=5
d=1
while d < 10:
b=a-(f(a)/g(a))
print N(b)
a = b
d = d + 1
Here are the approximations:
Newton and his derivatives show up in the Math Through Technology Program starting September 6!
Being in my calculus mode recently, I wanted to add Newton's method to the list. In the 1600s Newton used his new invention, calculus, to approximate roots of polynomials like the one below:
A "root" of a function is the x-value where the line or curve crosses the x-axis. In the figure above, the root is the red dot. You guess the root is at x0, then you plug x0, the function of x0 and the derivative of the function at x0 into the formula
x1 = x0 - f(x0)/f'(x0)
to get x1, the next approximation for the root. Do that again and x2 is even closer to the red dot in the picture.
I just wrote a short program in Sage to print 9 approximations and used x = 5 as my first guess:
f(x)= x^2-5*x+6
g(x)= derivative (f,x)
a=5
d=1
while d < 10:
b=a-(f(a)/g(a))
print N(b)
a = b
d = d + 1
Here are the approximations:
3.80000000000000 3.24615384615385 3.04060269627280 3.00152476019449 3.00000231782539 3.00000000000537 3.00000000000000 3.00000000000000 3.00000000000000Sure enough, 3 is a solution of the equation: 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0. An initial guess of 0 gives the other root of 2. The method gives the root closer to your initial guess.
Newton and his derivatives show up in the Math Through Technology Program starting September 6!
Tuesday, July 31, 2012
Diffy Q's
My cousin, who's now an engineer, hated school and even Calculus until he finally took a class he could use: Differential Equations (affectionately known as "diffy q's").
Your algebra, geometry and precalculus teachers will tell you the reason to learn all the stuff in their classes is that you'll need it in calculus. But what's calculus for?
In the 1600s Descartes put algebra and geometry together for the first time, allowing others like Fermat to make progress on finding slopes of curved functions and areas under curves. Newton and Leibniz systematized the process and it became Calculus with a capital C.
But it all would have remained abstract, "pure" math if not for the fact that scientists were able to use the new math notation (Leibniz's, anyway) for modeling situations they found in nature. Situations where you didn't know the value for something (heat, for example) but you knew how fast it changed were all of a sudden able to be modeled using differential equations like dy/dt = ky.
Of course Newton made the most notable use of his new invention to solve the differential equations he encountered in his astronomical studies. His solutions proved Kepler's laws to be correct and that his own famous laws applied to objects on earth and in the heavens.
Textbooks are written every day about solving differential equations using algebra and integration. Instead of the change in y you get a formula for y itself. Take the differential equation
dy/dx = y - x
We're looking for a function y whose derivative is the original function y minus x. In school you jump through all kinds of hoops to find the solution for y is
y = ce^x + x + 1
Take the derivative of that function and sure enough, it's the function minus x. Well, those differential equations have all been reduced to algebraic formulas thousands of times, and more with every school year. Furthermore, most of the differential equations that are encountered in the real world have no explicit solution with x's and y's. This situation was already well known in the 1700s, leading Euler to develop "numerical methods" of approximating solutions using "Euler's Method."
If you know one value (an initial value, say) of y but otherwise you only know how it changes, you can start at that value and just take steps using the direction given by the differential equation. If the steps are small enough, you can get a good approximate graph of the function y. Euler's Method has been improved over the years, and now it's even made into highly instructive Java applets. Every so often somebody codes these methods into a new free differential equation solver software to help, like Calcode or Maxima.
So we do Algebra and Geometry to do Calculus, but we do Calculus in order to solve Differential Equations. The faster we get to Diffy Q's the better. I introduce them in the third meeting of the Math Through Technology Program, starting September 6th.
Your algebra, geometry and precalculus teachers will tell you the reason to learn all the stuff in their classes is that you'll need it in calculus. But what's calculus for?
In the 1600s Descartes put algebra and geometry together for the first time, allowing others like Fermat to make progress on finding slopes of curved functions and areas under curves. Newton and Leibniz systematized the process and it became Calculus with a capital C.
But it all would have remained abstract, "pure" math if not for the fact that scientists were able to use the new math notation (Leibniz's, anyway) for modeling situations they found in nature. Situations where you didn't know the value for something (heat, for example) but you knew how fast it changed were all of a sudden able to be modeled using differential equations like dy/dt = ky.
Of course Newton made the most notable use of his new invention to solve the differential equations he encountered in his astronomical studies. His solutions proved Kepler's laws to be correct and that his own famous laws applied to objects on earth and in the heavens.
Textbooks are written every day about solving differential equations using algebra and integration. Instead of the change in y you get a formula for y itself. Take the differential equation
dy/dx = y - x
We're looking for a function y whose derivative is the original function y minus x. In school you jump through all kinds of hoops to find the solution for y is
y = ce^x + x + 1
Take the derivative of that function and sure enough, it's the function minus x. Well, those differential equations have all been reduced to algebraic formulas thousands of times, and more with every school year. Furthermore, most of the differential equations that are encountered in the real world have no explicit solution with x's and y's. This situation was already well known in the 1700s, leading Euler to develop "numerical methods" of approximating solutions using "Euler's Method."
If you know one value (an initial value, say) of y but otherwise you only know how it changes, you can start at that value and just take steps using the direction given by the differential equation. If the steps are small enough, you can get a good approximate graph of the function y. Euler's Method has been improved over the years, and now it's even made into highly instructive Java applets. Every so often somebody codes these methods into a new free differential equation solver software to help, like Calcode or Maxima.
So we do Algebra and Geometry to do Calculus, but we do Calculus in order to solve Differential Equations. The faster we get to Diffy Q's the better. I introduce them in the third meeting of the Math Through Technology Program, starting September 6th.
Monday, March 19, 2012
Math Stuff That Used To Make Sense Part I: Rationalizing Denominators
In the 21st century math teachers are still making students "rationalize the denominator," because a square root in the denominator is a crime against nature. Well, there actually used to be a good reason for not wanting a never-ending, never-repeating decimal in the denominator of a fraction, but we'd have to think back before calculators.
Supposedly, the first mathematician to insist the diagonal in a perfect 1x1 square can't be expressed as a whole number or a fraction was put to death. Irrational numbers have been given a bad name, but they come up very often in geometry and trigonometry, thanks to the Pythagorean Theorem. The "special triangles" so beloved by the SATs contain Ö2 and Ö3.
Why can't you just leave square roots in radical form? Well, in the real world, you never give your friend directions like "go Ö17 miles down the road and turn left." There is an effective but confusing and annoying algorithm to compute square roots which reminds me of long division and not in a good way.
So there used to be tables of square roots in the back of textbooks, or students used to memorize square roots using mnemonic devices, in case you weren't allowed to use your notes or a table (or a slide rule) on the test.
"For the square root of 2
I wish I knew" = The number of letters in each word is 1, 4, 1, 4. The square root of two is 1.414...
"To know the square root of 3
O charmed was he" = 1.732
"To know the root of 5
So we now strive" = 2.236
That was the reason for making students reduce irrational numbers like Ö75 to 5Ö3. They're both 8.660... but before calculators we memorized the square root of 3 but not the square root of 75. We wanted a form we could get a couple of decimal places out of, and multiplying 1.732 by 5 was a cinch compared to having to do the square root algorithm on 75 every time. Of course, now with calculators, there's no longer any need to reduce roots like this.
Similarly, before calculators, expressing the cosine of 45 degrees as 1/Ö2 would make your heart sink. Imagine trying to do long division, dividing 1 by a never ending, never repeating decimal. Echhh. So you "rationalized the denominator" and came up with Ö2/2. Dividing a long number by two is pretty easy in comparison. Both expressions give you 0.7071...
But nowadays? Come on: Ö2/2 and 1/Ö2 require the exact same number of keystrokes on your calculator to get 0.7071.... I submit there is absolutely no reason to continue to teach this outdated arithmetic to students, and to penalize students who write 1/Ö2 instead of Ö2/2.
Saturday, February 18, 2012
Real Math vs. School Math
I made a video presenting some background on how math is taught in schools. My opinion is that learning to really use math tools should take a lot less time than we're led to believe. Hope it starts a dialogue!
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