This is the same school that I subbed at on Tuesday and Wednesday. Thus I'm aware that this class is continuing to explore Track and Field. Today the students have free play on the field where they can either play another sport, or practice one of the five events that they learned earlier this week:
- Hurdles (60 meters)
- High Jump
- Shot Put
- Long Jump
- Sprints (60 meters)
During each of the five periods I cover today -- including zero period -- I practice one of the five events, in the order that I list them above. As I wrote earlier, participating in P.E. is a personal tradition that goes back to the earliest days of this blog.
Today is Sevenday on the Eleven Calendar:
Decade Resolution #7: We sing to help us learn math.
This one is tricky, since P.E. is nothing like a math class. Indeed, I wrote that I often don't sing at all on P.E. days.
But I wanted to follow this resolution today. Therefore I perform a song anyway. I choose one that I haven't sung in a while -- Square One TV's "Count the Ways." I associate this song with this time of year (about two weeks before Valentine's Day). In a way, this is also somewhat related to P.E. and heart rates, since it mentions how my heart beats for you "70 times a minute."
Lecture 19 of Michael Starbird's Change and Motion is called "Mountain Slopes and Tangent Planes," and here is an outline of this lecture:
I. Measuring the altitude at different places is a good example of a quantity (the altitude) that varies according to position, which is given by two coordinates (latitude and longitude).
II. Let's explore the area of a rectangle.
III. The derivative we have discussed before has an analog in this situation.
IV. Suppose we alter both at once.
V. The analog of the tangent line is the tangent plane.
VI. We can describe the analog of the integral in situations with more than one varying quantity.
VII. Real-world situations often involve several variables.
In today's lecture, Starbird introduces the idea of Multivariable Calculus. (This is the first college class taught beyond BC Calculus. I once met a high school student who was taking this course.) He begins by showing us a surface such as mountainous terrain.
The professor reminds us that l * w = area of a rectangle. This can be written as a function of two variables, as in A(l, w) = l * w. He asks some "Calculus questions" -- at (5, 2), a 5 * 2 rectangle, how fast will the area change if we increase the length or the width?
Well, for each unit increase in length, the area increases by 2 square units. Thus dA/dl at (5, 2) = 2 -- and the d's are actually script d's (or deltas) to denote the partial derivative. Meanwhile, for each unit increase in width, the area increases by 5 square units. Thus dA/dw at (5, 2) = 5. (Of course, the partial derivative symbols aren't easily created in ASCII.)
Starbird also draws the picture for increasing both the width and length by some delta-v. The calculation of a derivative here is possible though a bit more complicated.
The professor tells us that just as tangent lines to a curve exist, so do tangent planes to a surface. The ocean appears to be a flat plane (the tangent plane), as does a mountain (locally). At a maximum (peak) or minimum (valley), the tangent plane is flat.
Starbird now asks about integrals in Multivariable Calculus. Just as ordinary integrals give us the area under a curve, integrals of surfaces give us the volume under that surface. For example, a cone:
cone = integral_ integral_ Omega (1 - sqrt(x^2 + y^2))dx dy where "Omega" is the unit circle
(He describes this as "really neat" notation, though it's not so neat in ASCII.)
To show us how to compute the double-integral for a volume of bread, the professor cuts the bread into thin slices. The sum of all the thin slices gives the volume. The moral of the story is that Calculus is the best thing since sliced bread.
It's only fitting that today's the first day of volume in our Geometry course. Fortunately, we won't need Multivariable Calculus to find these volumes.
This one is tricky, since P.E. is nothing like a math class. Indeed, I wrote that I often don't sing at all on P.E. days.
But I wanted to follow this resolution today. Therefore I perform a song anyway. I choose one that I haven't sung in a while -- Square One TV's "Count the Ways." I associate this song with this time of year (about two weeks before Valentine's Day). In a way, this is also somewhat related to P.E. and heart rates, since it mentions how my heart beats for you "70 times a minute."
Lecture 19 of Michael Starbird's Change and Motion is called "Mountain Slopes and Tangent Planes," and here is an outline of this lecture:
I. Measuring the altitude at different places is a good example of a quantity (the altitude) that varies according to position, which is given by two coordinates (latitude and longitude).
II. Let's explore the area of a rectangle.
III. The derivative we have discussed before has an analog in this situation.
IV. Suppose we alter both at once.
V. The analog of the tangent line is the tangent plane.
VI. We can describe the analog of the integral in situations with more than one varying quantity.
VII. Real-world situations often involve several variables.
In today's lecture, Starbird introduces the idea of Multivariable Calculus. (This is the first college class taught beyond BC Calculus. I once met a high school student who was taking this course.) He begins by showing us a surface such as mountainous terrain.
The professor reminds us that l * w = area of a rectangle. This can be written as a function of two variables, as in A(l, w) = l * w. He asks some "Calculus questions" -- at (5, 2), a 5 * 2 rectangle, how fast will the area change if we increase the length or the width?
Well, for each unit increase in length, the area increases by 2 square units. Thus dA/dl at (5, 2) = 2 -- and the d's are actually script d's (or deltas) to denote the partial derivative. Meanwhile, for each unit increase in width, the area increases by 5 square units. Thus dA/dw at (5, 2) = 5. (Of course, the partial derivative symbols aren't easily created in ASCII.)
Starbird also draws the picture for increasing both the width and length by some delta-v. The calculation of a derivative here is possible though a bit more complicated.
The professor tells us that just as tangent lines to a curve exist, so do tangent planes to a surface. The ocean appears to be a flat plane (the tangent plane), as does a mountain (locally). At a maximum (peak) or minimum (valley), the tangent plane is flat.
Starbird now asks about integrals in Multivariable Calculus. Just as ordinary integrals give us the area under a curve, integrals of surfaces give us the volume under that surface. For example, a cone:
cone = integral_ integral_ Omega (1 - sqrt(x^2 + y^2))dx dy where "Omega" is the unit circle
(He describes this as "really neat" notation, though it's not so neat in ASCII.)
To show us how to compute the double-integral for a volume of bread, the professor cuts the bread into thin slices. The sum of all the thin slices gives the volume. The moral of the story is that Calculus is the best thing since sliced bread.
It's only fitting that today's the first day of volume in our Geometry course. Fortunately, we won't need Multivariable Calculus to find these volumes.
Today I will be doing Lesson 10-3 of the U of Chicago text, on the fundamental properties of volume.
(Also, I might add that Lessons 10-1 and 10-3 also flow naturally from last month's 8-8 and 8-9. Both the formulas for a circle appear in the surface area formula of a cylinder -- the circumference of a circle leads to the lateral area of a cylinder and the area of a circle leads to the full surface area including the bases.) But some people might point out that this would confuse the students even more. Instead of doing all of the surface area formulas at once (as the U of Chicago does) and all of the volume formulas at once, we'd keep going back and forth between surface area and volume. But another argument is that it's better to do all of the prism formulas at once, then all of the pyramid formulas, and finally all of the sphere formulas.
(Also, I might add that Lessons 10-1 and 10-3 also flow naturally from last month's 8-8 and 8-9. Both the formulas for a circle appear in the surface area formula of a cylinder -- the circumference of a circle leads to the lateral area of a cylinder and the area of a circle leads to the full surface area including the bases.) But some people might point out that this would confuse the students even more. Instead of doing all of the surface area formulas at once (as the U of Chicago does) and all of the volume formulas at once, we'd keep going back and forth between surface area and volume. But another argument is that it's better to do all of the prism formulas at once, then all of the pyramid formulas, and finally all of the sphere formulas.
The cornerstone of Lesson 10-3 is a Volume Postulate. The text even points out the resemblance of the Volume Postulate of 10-3 to the Area Postulate of 8-3:
Volume Postulate:
a. Uniqueness Property: Given a unit cube, every polyhedral solid has a unique volume.
b. Box Volume Formula: The volume of a box with dimensions l, w, and h is lwh.
c. Congruence Property: Congruent figures have the same volume.
d. Additive Property: The volume of the union of two nonoverlapping solids is the sum of the volumes of the solids.
Just as we derived the area of a square from part b of the Area Postulate, we derive the volume of a cube from part b of the Volume Postulate:
Cube Volume Formula:
The volume of a cube with edge s is s^3.
And just as we can derive the area part of the Fundamental Theorem of Similarity from the Square Area Formula, we derive the volume part of the Fundamental Theorem of Similarity from the Cube Volume Formula:
Fundamental Theorem of Similarity:
If G ~ G' and k is the scale factor, then
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.
Volume Postulate:
a. Uniqueness Property: Given a unit cube, every polyhedral solid has a unique volume.
b. Box Volume Formula: The volume of a box with dimensions l, w, and h is lwh.
c. Congruence Property: Congruent figures have the same volume.
d. Additive Property: The volume of the union of two nonoverlapping solids is the sum of the volumes of the solids.
Just as we derived the area of a square from part b of the Area Postulate, we derive the volume of a cube from part b of the Volume Postulate:
Cube Volume Formula:
The volume of a cube with edge s is s^3.
And just as we can derive the area part of the Fundamental Theorem of Similarity from the Square Area Formula, we derive the volume part of the Fundamental Theorem of Similarity from the Cube Volume Formula:
Fundamental Theorem of Similarity:
If G ~ G' and k is the scale factor, then
(c) Volume(G') = k^3 * Volume(G) or Volume(G') / Volume(G) = k^3.
Last year I stopped posting Euclid's propositions at this point, and so I do likewise this year.
Today is an activity day, the only such day this week. Last year, I connected the worksheet to a special activity I did for Lesson 10-1, which I didn't post with the worksheet on Wednesday. Since my old Lesson 10-3 worksheet refers to the 10-1 activity, it makes sense to restore that project for today's activity day.
This is what I wrote last year about that project:
Today's lesson is on surface areas. But recall that back at the end of the first semester, I mentioned Dan Meyer, the King of the MTBoS (Math Twitter Blogosphere), and his famous 3-act lessons. I pointed out how one of his lessons was based on surface area, and so I would wait until we reached surface area before doing his lesson.
Well, we've reached surface area. And so I present Dan Meyer's 3-act activity, "Dandy Candies," a lesson on surface area.
Meyer includes some additional questions for teachers to ask the students, but I only included what fits onto a single student page. Those who want the extra information can get it directly from Meyer:
http://www.101qs.com/3038
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