Wednesday, February 10, 2016

Equations of Lines (Day 100)

According to the blog calendar, today is the 100th day of school -- often celebrated with a big party in first grade classrooms. This is mostly because 100 is traditionally the number to which students are expected to learn how to count.

But we see a change with the advent of the Common Core Standards:

Count to 100 by ones and by tens.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

So now counting to 100 is a kindergarten, rather than first grade standard. For some strange reason, the counting target for first graders is now 120. Considering that the counting target for second graders is 1000, both before and under Common Core, I wonder why 200 wasn't chosen as a target between 100 and 1000, rather than 120.

Last year, on the 100th day of school, I linked to an old thread which discusses whether making kindergartners count to 100 is reasonable. This year, I link to the traditionalist Dr. Ze'ev Wurman, who comments on this and several other Common Core standards. (So yes, this has turned into yet another post that is labeled "traditionalists".)

Regarding the kindergarten counting standard, Wurman writes:

The counting to 100 is unwisely aggressive. As a consequence, in grade 1 it is only extended to 120. A more reasonable sequence would be to count to 20 in Kindergarten and to 100 in grade 1.

In the above link, Wurman compares many of the Common Core Standards to those of Singapore. I used to mention Singapore Math often on this blog, but I stopped as I felt that its secondary standards were too difficult for American students. On this page though, he focuses on the island nation's elementary standards. Notice how he lists many of the Common Core Standards for kindergarten as being first or second grade standards in Singapore -- but meanwhile, many of the seventh grade standards in Common Core are fifth or sixth grade standards in Singapore. In short, it takes eight or nine years (Grades K-7 or K-8) to cover what Singapore does in six (Grades 1-6).

Of course, Wurman also comments on the geometry standards (under eighth grade). We already know that Wurman is not a fan of transformational geometry. But as usual, I do agree with Wurman and the other traditionalists for the early grades, so counting to 100 should be a first grade topic.

To maintain consistency with the Common Core, kindergarten teachers would now celebrate the 100th day of school, while first grade teachers should wait until the 120th day of school. The only problem with this is that the 120th day of school marks the end of the second trimester -- so it's often the day of a (district-mandated) test, not a party.

Today on this 100th day of school, I subbed at a high school, where there was a slight mix-up. I was originally scheduled to sub in a foreign language class. But when I arrived, I ended up covering a science teacher who was out only fifth and sixth period. So until then, I helped out in a special education math class -- the teacher was present, but the aide was out.

Officially, all of this teacher's classes are Integrated Math I. But since these students were low-level, they were nowhere near the unit on systems of equations that the mainstream Math I students were working on last week (let alone the unit on simplifying radicals, as in Honors Math I).

One of the classes, second period, was extremely low-level. The students were reviewing solving one-step (and some two-step) equations in preparation for solving inequalities. The teacher really works hard to make sure that the students aren't mixed up. Many Pre-Algebra, Algebra I, and Math I teachers have the students draw a line through the equal sign. But this teacher also has the students draw a box around the entire term that includes the variable. She tells me that this especially helps the students with two-step equations, where they might not know which step to undo first.

But for this blog post, I want to focus on the first and fourth period classes. These students are slightly beyond the second period class, as they are working on graphing functions, including identifying the domain and range.

The students were given a packet, which I believe is based on MathLinks (Grade 8, of course). The students had various problems to work on, such as:

Express each relation as a table, as a graph, and as a mapping diagram.
1. The relation represents ages of students and the number of words they can write per minute.
{(5, 10), (6, 20), (6, 23), (7, 35)}

State the domain and range of each relation.
3.  x    y
    2    5
    7    8
    8   15
   11  12
   15  19

I mentioned the mnemonic DIX-ROY, for "Domain/Input/x-value," "Range/Output/y-value." I first discussed DIX-ROY here on the blog back in October. Yes, I know that several math teachers have used that similar mnemonics in their own classes, but I always think of DIX-ROY, HOY-VUX, and Slope Dude as Haganisms, associated with the famous math blogger Sarah Hagan.

In fact, this teacher has two mnemonics of her own posted in the room for integer operations. One of them is called "Party or War." This rule states that if we are adding two numbers with the same sign, it's a "Party" and we must add their absolute values -- the sign of the sum is the same as the numbers that are partying together. If the two numbers have opposite signs, then it's a "War" and we must subtract their absolute values -- the sign of the sum is the same as the winner of the war.

The other mnemonic is for multiplication and division. It is called "Church." A building is drawn with two negative signs -- one on each side -- and a large positive sign (the "cross") on top. To multiply or divide two integers, cover up the signs of the two numbers in the question. Then the uncovered sign is the sign of the answer. (This breaks down if both factors are positive, but it's trivial that the product or quotient of two positive integers is positive.)

In fourth period, the teacher told me that one student had been absent and so she told me that I could help the girl out. Here's one way I always like to help students out with functions -- for example, we consider the relation numbered #3 above:

When I say 2, you say ... 5.
When I say 7, you say ... 8.
When I say 8, you say ... 15.
When I say 11, you say ... 12.
When I say 15, you say ... 19.
Therefore yes, this relation is a function.

But let's look at the relation numbered #1 above:

When I say 5, you say ... 10.
When I say 6, you say ... oops! You don't know whether to say 20 or 23.
Therefore no, this relation is not a function.

The girl did very well, getting most of the answers correctly. In fact, she figured out how to find the domain and range given a graph (of isolated points) faster than the students who hadn't been absent!

Today's Geometry lesson is closely related, as it's about linear functions and equations. This is what I wrote last year about today's lesson:

Equations of lines don't appear in the U of Chicago geometry text at all. Instead, we go back to U of Chicago's Algebra (that is, Algebra I) text. There, equations appear in the following sections in Chapter 8, "Slopes and Lines":

Lesson 8-4: Slope-Intercept Equations for Lines
Lesson 8-5: Equations for Lines with a Given Point and Slope
Lesson 8-6: Equations for Lines Through Two Points
Lesson 8-7: Fitting a Line to Data
Lesson 8-8: Equations for All Lines

Obviously, in Lesson 8-4, the equation y = mx + b appears. But as it turns out, the point-slope form doesn't appear in the U of Chicago at all. Instead, the slope-intercept form is used to find equations in both 8-5 and 8-6. Then again, I notice that most students have trouble remembering point-slope, so we might as well teach only the slope-intercept form. In Lesson 8-8, the standard form Ax + By = C of a linear equation appears.

A question that is often asked during this lesson is, why so we use m to denote slope? One urban legend is that it refers to the French word monter, meaning to climb. An English cognate of this word is "mountain," and of course mountains have slopes. The problem with this theory is that there is no evidence that the mathematician Rene Descartes ever used the letter m. One would think that if m had a French origin, Descartes -- you know, the French creator of the Cartesian plane on which slope is usually measured -- would have been the first to use it. A discussion appears at this thread:

Notice that John Conway -- the mathematician I previously mentioned as an advocate of the inclusive definition of trapezoid -- is a participant in this 21-year old thread. (That's right -- when Conway wrote in this thread, I was myself a young geometry student!) Conway suggests that m may stand for "modulus of slope." One teacher tells his students that stands for "move" and b stands for "begin," since this is how students learn to graph lines in slope-intercept form.

The origin of b is little more well-known. It refers to the idea that a is the x-intercept of a graph, and b is the y-intercept. The equation of an ellipse centered at the origin uses a andb in this way -- but if the ellipse is translated so that its center isn't at the origin, a and b no longer stand for the intercepts.

Finally, back when I was student teaching, my students came up with their own mnemonic for the slope-intercept formula. The letters y = mx + b stand for "your mom's ex-boyfriend."

Slope-Intercept Property:
The line with equation y = mx + b has slope m and y-intercept b.

Once we have the slope-intercept formula, we can algebraically prove that its slope is mand that its y-intercept is b. The y-intercept is the point that lies on the y-axis -- and as we saw yesterday, this is the point whose x-intercept is 0. Setting x = 0 in the slope-intercept formula gives y = b -- that is, the y-intercept is b. To find the slope, we let (x_1, y_1) and (x_2, y_2) be two arbitrary points on the line:

(y_2 - y_1) / (x_2 - x_1)
= (mx_2 + b - (mx_1 + b)) / (x_2 - x_1)
= (mx_2 + b - mx_1 - b) / (x_2 - x_1)
= (mx_2 - mx_1) / (x_2 - x_1)
m(x_2 - x_1) / (x_2 - x_1)
m. QED

Finally, we can show how the standard form Ax + By = C is also the equation of a line. We notice that if B = 0, we have the horizontal line x = A. Otherwise, we may divide by B:

Ax + By = C
        By = -Ax + C
          y = (-AB)x + C / B

and so we have a line with slope -A / B and y-intercept C / B. QED

I told myself that if I taught any class that working on linear functions during this week, I'd post it here on the blog. And so this is exactly what I'm doing. I decided to post only an assignment based on the last page of the packet, the Lesson Performance Task, since this page is most closely related to our Geometry lesson with a linear function -- and it asks for an equation. Performance Task questions are great to ask because they appear on the Common Core tests -- indeed, both PARCC and SBAC have separate testing days for Performance Tasks.

You might wonder whether it's wise to post a lesson about domain and range in Geometry. Notice that the transformations (reflections, rotations, etc.) that we discuss in class are actually functions, where the domain is the preimage and the range is the image. Indeed, we often use function notation to denote a transformation as T(x, y). The awkward thing is that in algebra, the input is the x-value and the output is the y-value, but with geometrical transformations, the input has both an x- and y-value, and so does the output.

Here are the worksheets:

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