Tuesday, May 2, 2017

Lesson 14-5: Vectors (Day 145)

This is what Theoni Pappas writes on page 122 of her Magic of Mathematics:

"By focusing our attention on the honeybee, a wealth of mathematical ideas can be observed."

Pappas begins by explaining that the square, the triangle, and the hexagon are the only three regular polygons that tessellate. (This is mentioned in Lesson 8-2 of the U of Chicago text.) Of the three, the hexagon has the smallest perimeter for a given area.

This is closely related to Lesson 15-8, where we learn that of all plane figures with the same area, the circle has the least perimeter. If we restrict ourselves to regular polygons, the one that's the most nearly circular (that is, with the most sides) has the least perimeter.

Furthermore, we can find these perimeters using Lesson 14-1 from last week. Notice that a question about a beehive even appears on last week's worksheet, although area isn't mentioned. Anyway, the point Pappas is making is that the hexagon is the most efficient possible polygon for honeycombs -- which is why the bees choose that shape for them.

Today is the second episode of Genius on National Geographic. Once again, I won't write about the show, but I will write about science. Today, I had sixth graders working on a science project about heat and energy transfer. This was always a sixth grade topic under both the old California Standards (as part of Earth Science) and the new NGSS.

In groups of four, students could choose conduction, convection, or radiation as the focus of their science projects, which they will present on Friday. Here are three examples of projects:

Conduction: Which melts ice faster, metal or wood?
Convection: Does cold water push hot water up?
Radiation: Can the sun melt crayons?

Most of the students appear to enjoy these science lessons. It's always more enjoyable for them when they get to choose their own projects.

This is, of course, the sort of project that I should have given my sixth graders earlier in the year. Oh well, it's better I teach this late than never.

And even though I won't write about the TV show Genius today, I will tie today's lesson right back to Albert Einstein. In 1917, the famous physicist wrote a paper on the quantum theory of radiation and a statistical theory of heat.

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

Lesson 14-5 of the U of Chicago text is on vectors. Much of physics deals with vectors. Force is a vector quantity [as Einstein knew all too well -- 2017 update dw].

I remember back when I was a high school senior taking AP Physics C, and our teacher wanted us to remember one thing about vectors:

Vectors operating at right angles are independent.

This means that if two vectors are perpendicular to each other -- most notably if one is parallel to the x-axis and the other to the y-axis -- then they are linearly independent. Motion along one vector has nothing to do with motion along the other. We notice this most clearly when one is throwing a ball, and we resolve the velocity into its horizontal and vertical components. The horizontal component of velocity is slowed down slightly by the force of air resistance, while the vertical component is slowed down -- and its direction is ultimately reversed -- by the force of gravity.

In fact, I remember learning about acceleration so many years ago, and I was confused as to why an object that is turning or slowing down is said to be "accelerating." After all, I thought that an object slowing down was decelerating. To find out the answer, we must look at vectors. If the velocity and acceleration vectors are perpendicular, then the object is turning. (They must be perpendicular because of the theorem from Chapter 13 (Lesson 13-5) that the tangent and radius of a circle are perpendicular.) So if the velocity and acceleration vectors are in opposite directions, then the object is slowing down. I was confused by the use of the word "acceleration" by non-technical English speakers, where it usually refers to an increase in speed without considering vectors at all.

As it turns out, we prove in Linear Algebra -- a college course beyond Calculus -- that any two vectors that aren't parallel are linearly independent. But the linear independence that we consider the most in physics involve right angles. We don't discuss this idea in Lesson 14-5, but we will look at both velocity and force vectors on the posted worksheet.

Vectors appear in the Common Core Standards, but not in the geometry section. Indeed, they appear in the "Vector & Matrix Quantities" section:

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

(The U of Chicago text uses the first option, a single boldface roman letter. So v is a vector.) The point is that the various domains of the standards don't line up exactly with courses -- not even for the traditional (as opposed to the integrated) pathway. So there are standards that appear in the geometry section that don't appear in a geometry course (i.e., conic sections), as well as vice versa (vectors).

The various Common Core Standards for vectors are spread out among the last three lessons of the chapter, 14-5 through 14-7. One standard that appears in today's Lesson 14-5 is:

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

This is only partly realized in Lesson 14-5 -- component-wise addition will appear in 14-6. But adding vectors end-to-end and by the parallelogram rule do appear in 14-5, and the U of Chicago does give an example where the magnitude of a sum isn't the sum of the magnitudes -- indeed, there's an example where the sum is shorter than one of the vectors being added! The text explains that this is to be expected since one vector is the velocity of the boat and the other is the velocity of the current (and the boat is moving nearly upstream).

So in a way, we are beginning this standard today as well:

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

The text points out that vectors are closely related to translations. Many texts tend to define the translation in terms of its vector. The U of Chicago does the opposite. Instead, translations are defined as compositions of reflections. The Two Reflection Theorem of Lesson 6-2 tells us how to find the magnitude and direction of a translation in terms of the two mirrors. Of course, some would say that the magnitude and direction of the translation really means the magnitude and direction of the translation vector. But the word "vector" can't appear in Lesson 6-2, since it isn't even defined until Lesson 14-5. Instead, we see the following theorem:

Two vectors are equal if and only if their initial and terminal points are preimages and images under the same translation.

This theorem, therefore, shows how to connect the U of Chicago definition of translation (two reflections) with the definition found in other texts (vectors). Notice that in both cases, "equal vectors" are defined to be vectors with the same magnitude and direction.

Finally, the text defines vector addition:

The sum or resultant of two vectors AB and BC, written AB + BC, is the vector AC.

David Joyce criticizes the use of the word "resultant" to refer to vector sum:

The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?)

But then again, U of Chicago rarely uses the word "resultant" and mainly uses the word "sum" -- the word that we normally use when referring to addition. Joyce would prefer to omit the section on vectors, but they are included in the Common Core Standards -- and as I mentioned, the U of Chicago devotes three sections to vectors, not just one!

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