Lesson 10-4: Multiplication, Area, and Volume (Day 104)

Lecture 20 of Michael Starbird's Change and Motion is called "Getting off the Line -- Motion in Space," and here is an outline of this lecture:

I. How can we describe motion in space?
II. Let's consider the example of a bug moving around the plane.
III. Let's revisit the example of the trajectory of a baseball.
IV. The Pythagorean Theorem is perhaps the most famous theorem in all of mathematics.
V. We found the speed and direction of travel of a car; however, we have not yet figured out how far it travels over time.
VI. Analyzing the path of a mosquito flying around a room entails exactly the same analysis to find its speed, direction, and length of the trip.
VII. Applying the ideas of calculus to different settings is what makes calculus so powerful and what makes calculus books so thick.

Today's Starbird lecture is all about vector calculus. He explains that this is the opposite of the last lecture -- instead of functions with many dimensions in the domain and one in the range, today's functions have one dimension in the domain and many dimensions in the range.

The professor begins with a VW bug circling a track as opposed to a straight road. Thus its position in the 2D plane is a function of time. We can ask about not just the speed of its car but its direction -- in other words, the velocity of the car.

Starbird also returns to the baseball example. Once again, the independent variable is time, but this time the value of the function includes both height and distance from home plate. The velocity vector is 48 ft./sec. upward, 100 ft./sec. forward. So he uses the Pythagorean Theorem to find the magnitude of this vector (the speed of the ball), approximately 111 ft./sec. -- we've seen this technique used in Lesson 14-7 of the U of Chicago text.

Now the professor proceeds to prove the Pythagorean Theorem. He attributes to the mathematician Bhaskara of India. According to Cut the Knot, Bhaskara's proof combines Proofs #3-4:

https://www.cut-the-knot.org/pythagoras/

Starbird now shows some formal examples. He starts with the vector-valued function:

p(t) = (1/3 t^3 - t, t^2)
p(0) = (0, 0)
p(1): x(1) = 1/3 - 1 = -2/3
         y(1) = 1
p(1) = (-2/3, 1)

And then he asks, how fast is the car moving along this path? To answer this, we take the derivative of each component:

x'(t) = d/dt (1/3 t^3 - t) = t^2 - 1
t = 2 min. 2^2 - 1 = 3 mi./min.

y'(t) = d/dt (t^2) = 2t

And then we use the Pythagorean Theorem to find the speed -- at t = 2, we obtain the well-known right triangle 3-4-5, so the speed, 5 mi./min., is easily found.

The professor now asks, how far has the car traveled? He points out that the length of a curved path is tricky, but if it's sufficiently smooth, then it looks like a straight line up close. Thus we approximate the curve by secant segments whose sum is nearly the length of the curve. In the limit, this sum becomes an integral:

v(t) delta-t = sqrt((x'(t) delta-t)^2 + y'(t) delta-t)^2)
Find distance traveled from t = 0 to t = 3
int _0 ^3 sqrt((x'(t))^2 + (y'(t))^2) dt

We now find the speed for the example p(t) = (1/3 t^3 - t, t^2) given earlier. To do this, we must use what we learned from previous lectures:

x(t) = 1/3 t^3 - t -> x'(t) = t^2 - 1
y(t) = t^2            -> y'(t) = 2t (Lectures 2 & 7)

sqrt((x'(t))^2 + (y'(t))^2) = sqrt((t^2 - 1)^2 + (2t)^2)
distance = int _0 ^3 sqrt((t^2 - 1)^2 + (2t)^2) dt (Lec 20 & 8)
              = int _0 ^3 sqrt(t^4 - 2t^2 + 1 + 4t^2) dt (High School Algebra)
              = int _0 ^3 sqrt(t^4 + 2t^2 + 1) dt (High School Algebra)
              = int _0 ^3 sqrt((t^2 + 1)^2) dt (High School Algebra)
              = int _0 ^3 (t^2 + 1)dt (High School Algebra)
              = t^3/3 + t from 0 to 3 (Fund. Theorem of Calculus, Lec 4, 7, and 14)
              = 27/3 + 3 - 0 = 9 + 3 = 12 miles

Meanwhile, last night the Kansas City Chiefs won Super Bowl LIV -- their first championship in half a century.

And now I'm going to post the same thing I posted after the Philadelphia Eagles, Washington Nationals, Toronto Raptors, Chicago Cubs, etc., ended their respective championship droughts. A typical eighth grader who's failed math for seven years decides just to give up and not even try to pass math in his eighth grade year. Yet the Chiefs have failed for 49 years -- seven times as long as our eighth grader -- yet they strove to win in the fiftieth year, and they won.

As I reflect back to last week's P.E. class, that was before the Super Bowl. But it was after the passing of Kobe Bryant, and so I could have used him as an example to motivate the students. Or if I prefer an example from Track and Field (as that was the sport they were practicing that day), I could have mentioned the hardworking Usain Bolt. (No other Track athlete is famous enough to mention as an example here.) Being a hard worker in school isn't just for sports and math -- it's for all endeavors.

Lesson 10-4 of the U of Chicago text is called "Multiplication, Area, and Volume." In the modern Third Edition of the text, multiplication, area, and volume appear in Lesson 10-2.

This is what I wrote last year about today's lesson:

In many ways Lesson 10-4 is more algebraic than geometric. Indeed, this lesson introduces geometric models to justify the FOIL method of multiplying polynomials.

In today's Lesson 10-4, we ask questions such as, how is the volume of a box affected by tripling its length, width, and height? The answer is that it increases by a factor of 27.

One of the review questions asks for the volume of an Uno card. (I wonder whether there are any Wild Draw 4 cards in this deck!) The bonus question asks students to cube a binomial.