Monday, February 22, 2016

Lesson 14-5: Vectors (Day 107)

Today I subbed in a special education class. Second period was an eighth grade history class, but there were also three periods of math. (After lunch, during the conference period, I was actually called to cover two sections of drama!)

The math classes begin with a Basic Facts Test where students had about 40 simple addition, subtraction, multiplication, or division problems to answer in ten minutes. Then the main part of the lesson involved the multiplication of fractions. This is a fifth grade standard:

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

The homework, meanwhile, was completely different -- dividing decimals by whole numbers. This is another one of these topics that appears twice in the Common Core -- in fifth grade using alternative algorithms and sixth grade using the standard algorithm:

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

As we already know, traditionalists would prefer to move the sixth grade standard algorithm down to fifth grade and just drop the alternative algorithm altogether.

Oops -- here we go again with our third straight post to be labeled "traditionalists"! The problem is that I keep covering these special ed math classes where below grade-level students are working with the elementary math standards, and so the traditionalist debate keeps coming up.

Not only that, but on Facebook, I've seen a few parents commenting on Common Core standards again -- including the mother of a second- (now third-) grader I mentioned in an October 2014 post. I might or might not jump in with a comment -- if I do, I'll report on it here on the blog later this week in yet another traditionalist-labeled post.

Regarding today's topic, the multiplication of fractions, there actually is an issue with algorithm. It's been pointed out by some traditionalists that students rarely want to "cross-cancel" when multiplying two fractions. For example, when multiplying 7/8 by 4/3, I would want to cancel the factors of 4 to obtain 7/2 times 1/3, or 7/6 as the answer. I'm surprised that anyone would even want to multiply 7 by 4 to get 28 and 8 by 3 to get 24, obtaining 28/24 before simplifying to 7/6. Yet I've never had success teaching this cross-canceling -- instead I'm forced to teach 7/8 times 4/3 = 28/24 = 7/6.

Oh, and of course I played my usual game in order to encourage the students to participate!

As we already know, many students find fractions to be difficult. I suspect that if you ask adults to name the first math topic that they found hard, "fractions" may be a common response. Many teachers point out that not only can't many high school students, in classes anywhere from Algebra I to Calculus -- work with fractions, but neither can many college math students. Perhaps not until we get to a class of Ph.D. candidates in math can we finally assert that a majority of the students can add, subtract, multiply, and divide fractions.

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.

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!

No comments:

Post a Comment