Lesson 0-4: Op Art (Day 4)

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

Lesson 0.4 of Michael Serra's Discovering Geometry is called "Op Art." Serra explains what this is:

"Op art (optical art) is a form of abstract art that uses straight lines or geometric patterns to create a special visual effect."

Optical art is closely related to the concept of optical illusions. I don't even want to attempt to draw some of the more complex optical illusions by hand, so I just use a Google search instead. On the other hand, the impossible objects page comes directly from this site:

http://brainden.com/impossible-objects.htm

Unfortunately, only the Penrose triangle printed properly. It is named for Sir Roger Penrose, whom Serra describes as a British mathematician and avid puzzle enthusiast. He is in fact still alive -- he just turned 88 this month. I also mentioned him on the blog recently during our reading of Hawking  -- very close to his birthday, probably. Penrose is both a physicist and a topologist, which explains why he appears in both Hawking and the Geometry text.

So I actually had to draw in one of the impossible drawings -- "Three prongs from two?" (called "the devil's fork" by Brainden). Well, I suppose if I can draw it, then so can our students.

Here is the Blaugust prompt for today:

• What is your favorite quote?  How can you share/use it in your classroom?

Just before I was hired at the old charter school, my side-along reading book was Morris Kline. He included quotes throughout his book. Let me cut-and-paste from that year, when I described Kline's first -- and most important -- quote:

Mathematics is the gate and key of the sciences.... Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy. -- Roger Bacon (13th century English philosopher)

Kline then proceeds:

"Perhaps the most unfortunate fact about mathematics is that it requires us to reason, whereas most human beings are not convinced that reasoning is worthwhile."

And this quote applies directly to our classes: Perhaps the most unfortunate fact about math class is that it requires us to reason, whereas most students are not convinced that reasoning is worthwhile. So I am reading Kline as my side-along reading book because much of what Kline writes about math applies to the math classes that we teach.

So many students wonder, "When will we ever use math?" Kline provides his answer to this question:

"The primary motivation for the development of mathematics proper and the primary reason for the great importance of this subject is its value in the study of nature" -- that is, science.

This is the underlying theme of the first chapter and ultimately the entire book. Math matters when, and only when, it can be applied to science.

A few months later, I began teaching at the old charter school. It was the first time I participated in Blaugust -- and the only previous time that I responded to Shelli's quote prompt. Let me cut-and-paste the quotes I mentioned that day:

Back in my Father's Day post, I mentioned the quote "If you don't know the answer, at least know where to find it." I consider it to be part of my classroom motto.

I've also quoted the famous MTBoS blogger Sarah Carter, especially with regards to her three function-related mnemonics (Slope Dude, DIX-ROY, and HOY-VUX). She explains these three at the following link:

http://mathequalslove.blogspot.com/2014/02/hoy-vux-redux.html

Actually, according to the following more recent link, DIX-ROY should be DIXI-ROYD, with the extra I and D standing for "independent variable" and "dependent variable":

http://mathequalslove.blogspot.com/2015/04/2014-2015-algebra-1-unit-1-interactive.html

I probably should use Carter's DIXI-ROYD now, especially since "dependent variable" showed up in one of my eighth grade Benchmark Testing Week questions.

Looking back at these old quotes, I'm sad to say that I never really mentioned either quote in class after the day I blogged them.

The first quote about knowing where to find answers is useful in a Google world, where students need to know where to find answers on the Internet. It also helps in a class where the students have interactive notebooks, so that the answer to "Where?" would be in your notebook. My problem, of course, was management -- I was always having to tell the class to be quiet. Thus I was seldom, if ever, in a position where the students are dutifully doing research and I needed to tell them to know where to find information.

Meanwhile, Carter is, of course, one of the two famous Sara(h)s of math. It's rare when a Blaugust day goes by and no one is mentioning either her or Sara VanDerWerf, especially since so many of their ideas are relevant for the first day of school.

Anyway, I would have used Carter's DIXI-ROYD during the Functions (F) strand of the eighth grade Common Core Standards. My problem was that I never really reached the F standards -- at the time, I based the order of standards on the Illinois State STEM text. These standards often jumped around haphazardly, as opposed to the traditional text where the standards went in order.

If I had naively followed the traditional order, I would have reached F near the middle of the school year (think F for February). Instead, I ended up covering only F.1 -- a standard so basic that only "input" and "output" were mentioned among the key function-related terms. Thus DIXI-ROYD never came up.

Even Kline's quote didn't find much use in my classroom. Once again, students were so busy talking that they didn't even bother to ask "Why do we have to learn this?" And of course, I could have made the connections to science better if I had actually taught science as I was supposed to.

Therefore the reason that I didn't use any of the three quotes says more about how disorganized (in both management and curriculum) my classroom was than whether the quotes themselves are good.

There's one more quote I wish to mention in this post. Whom am I quoting here? Let's just say I'm quoting several Facebook and Twitter users:

8 / 2 (2 + 2) = ?

That's right -- I'm going to discuss that viral math expression. Actually, this expression was first posted several years ago -- for some reason, it has suddenly resurfaced.

The way the question is written is very important here. Notice that the division in this problem is written with the division sign we learn in third or fourth grade. As it turns out, this symbol actually has a name -- an obelus. The obelus is difficult to write in ASCII. Instead, I wrote the division above using a slash or fraction bar -- a solidus. This is also significant if we consider entering this expression into a TI calculator. The key we need to press is labeled with an obelus, but on the screen, it appears as a solidus.

OK, then, so let's try entering this expression into the TI calculator to see what we get. I actually have three graphing calculators -- the TI-83 I've owned since I was a young high school student, the TI-82 I found in my classroom at the old charter school, and a TI-81 that I purchased at a yard sale for $1.

TI-81: 8 / 2 (2 + 2) = 1
TI-82: 8 / 2 (2 + 2) = 1
TI-83: 8 / 2 (2 + 2) = 16

And there we have it -- not even the TI can give a definitive answer!

On the surface, this seems to be a straightforward Order of Operations or PEMDAS problem. Of course, the operation in parentheses comes first, so we add 2 + 2 = 4. And so the problem hinges upon whether we complete the division or multiplication first. If we multiply first then the final answer is 1, whereas if we divide first then the final answer is 16.

Notice that whereas the obelus is used for division, there is no multiplication sign at all. Instead, multiplication is indicated by juxtaposition -- that is, by writing the factors next to each other. This is done all the time in Algebra I and above -- so 2x means 2 times x.

When I was a high school senior taking physics, the teacher reminded us that only juxtaposition should used to indicate the multiplication. The cross and dot symbols that we learn in third grade and middle school, respectively, should be reserved for the cross and dot products used with vectors. So the only way to multiply two numbers (scalars) is by juxtaposition.

Also, some people say that it's the parentheses that indicate multiplication. In reality, it's the juxtaposition that indicates multiplication, not the parentheses. Thus in the following expression:

2(2 + 2)
2(4)
8

we must multiply 2 by 4 due to juxtaposition, not the parentheses (which were originally there to indicate that we must add 2 + 2 first). It's just that without the parentheses, 2(4) would become 24 (in other words, twenty-four). It's impossible to use pure juxtaposition when both factors are simple numbers, so parentheses must appear in such problems.

OK, so now we return to the Order of Operations and PEMDAS. I once learned as a young student that the P in PEMDAS stands for all "grouping symbols," not just parentheses. This is why some authors prefer GEMDAS (G=grouping symbols) to PEMDAS. I also learned that there are more implicit grouping symbols along with the explicit parentheses, brackets, and braces.

One of these is the fraction bar, or solidus. It's a grouping symbol in that any operations in the numerator and denominator must be completed before the division indicated by the solidus -- even if it's addition or subtraction (which ordinarily occur after division). This is difficult to show in ASCII, but instead I can use underscore and underline:

__6__
 2 + 1

Here the addition must be completed before division to respect the solidus as a grouping symbol. So the answer must be 6/3 or 2. On no TI calculator is it possible to enter the solidus grouping symbol:

TI-81: 6 / 2 + 1 = 4
TI-82: 6 / 2 + 1 = 4
TI-83: 6 / 2 + 1 = 4

Instead, explicit parentheses must be placed around the numerator and/or denominator:

TI-81: 6 / (2 + 1) = 2
TI-82: 6 / (2 + 1) = 2
TI-83: 6 / (2 + 1) = 2

Our issue here is whether juxtaposition is itself an implicit grouping symbol -- that is, whether the two factors linked by juxtaposition should be multiplied before any other operation, even if PEMDAS would normally dictate otherwise. For example, in solving the equation:

2x + 4 = 10

we see that Algebra I teachers often tell students to subtract 4 before dividing by 2 because 2x is a term that we shouldn't break up -- as if juxtaposition were a grouping symbol. (I admit that I'm guilty of this.) Of course, we know that it's really because the multiplication of 2x comes before adding 4, and so we must undo the addition first. This is evident when we reach equations such as:

2x^2 = 18

(where ^2 is ASCII for "squared"). We must divide by 2 before taking the square root, because the exponentiation of 2 comes before the multiplying by 2, and so we must undo the multiplication first.

So juxtaposition is emphatically not a grouping symbol. But there are sometimes when we really wish that it were:

8/2x = 1

Notice that this is just our original problem with x = 2 + 2. In an actual Algebra II book, such a problem might be written as:

_8_ = 1
 2x

But here the grouping symbol is actually the solidus, not juxtaposition. We must multiply 2 by x first, not because of juxtaposition, but because we must simplify the denominator before division.

In an actual Algebra II class, the expression 8/2x never comes up, since we're likely to simplify this as 4/x first -- and the expression 4/x contains no ambiguity. On the other hand, 1/2x (and even 3/2x) might really come up. In fact, suppose we are asked to graph the rational function:

y = _1_
       2x

If we enter Y1 = 1/2X, we obtain the following graphs:

TI-81: a hyperbola
TI-82: a hyperbola
TI-83: a straight line

This is consistent with the three values of the viral expression above. The TI-81 and 82 calculators give higher precedence to juxtaposition over division, and so these obtain 1 for the expression and a hyperbola for the graph. On the other hand, TI-83 treats juxtaposition as ordinary multiplication. It thus occurs after the division since both have the same precedence (MD) and so are evaluated left to right -- it obtains 16 for the expression and a linear graph.

Most likely, at first the TI makers gave higher precedence to juxtaposition because it was convenient to enter y = 1/2x to indicate the hyperbola. When it was time to make the TI-83, it was decided that giving juxtaposition a higher precedence than division was inconsistent. I don't own a TI-84, but I assume that it would follow the TI-83 rules since 84 is closer to 83 than to 81 or 82.

What this means is that we must be careful when entering rational expressions into the TI-83, since neither the solidus nor juxtaposition is a grouping symbol:

y = _1_     must be entered as Y1 = 1/(2X)
       2x
y = ___1___     must be entered as Y1 = 1/(X(X+1))
       x(x + 1)
y = _____1_____     must be entered as Y1 = 1/((X + 1)(X - 1))
       (x + 1)(x - 1)

That last expression seems to include too many parentheses, but in reality all are necessary. On the other hand, we have:

y = 2x     may be entered as Y1 = 2X/3
       3
y = x(x + 1)     may be entered as Y1 = X(X + 1)/3
            3
y = (x + 1)(x - 1)     may be entered as Y1 = (X + 1)(X - 1)/3
                3

Why aren't parentheses required in these expressions? It's because multiplication is associative -- and while division isn't associative, we can rewrite division as multiplication by the reciprocal and take advantage of the associativity of multiplication in certain cases:

Y1 = 2X/3 and Y1=(2X)/3 both mean "multiply 2, X, and the reciprocal of 3."
Y1 = 3/2X (on TI-83) means "multiply 3, the reciprocal of 2, and X."
Y1 = 3/(2X) means "multiply 3 by the reciprocal of 2X."

In one case we take the reciprocal of only 2, and the other we take the reciprocal of 2X. It's the same reason why:

Y1 = 3 + (2 - X) doesn't need parentheses, but
Y1 = 3 - (2 + X) needs parentheses.

Both 3 + 2 - X and 3 + (2 - X) take the opposite of only X, but 3 - 2 + X takes the opposite of only 2, and then 3 - (2 + X) takes the opposite of 2 + X.

Thus even though 8 / 2 (2 + 2) will never actually appear in an Algebra II class, it's worth discussing just before graphing rational functions, since it explains why so many parentheses are needed -- and we don't have students graphing lines when the intended graph is a hyperbola.

Notice that on all three calculators, 8 / 2 * (2 + 2) = 16 -- that is, we press the times key and a star symbol (*) appears on the screen. In other words, when we use explicit multiplication instead of juxtaposition, all three calculators use PEMDAS strictly.

This raises another question as to why the viral question is so confusing -- in what class do we use both an obelus to indicate division and juxtaposition to indicate multiplication? After all, we associate the obelus with elementary school division and juxtaposition with high school Algebra I/II. The answer is clear -- in middle school.

In sixth grade, we see the following standards:

CCSS.MATH.CONTENT.6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

CCSS.MATH.CONTENT.6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2)..

Notice that in the division of fractions standard, an obelus is used for division. But then in the factoring standard, juxtaposition is used for multiplication.

Of course, in reality, we wouldn't see the obelus and juxtaposition in the same problem. There is indeed a sixth grade standard for Order of Operations:

CCSS.MATH.CONTENT.6.EE.A.2.C
Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

In practice, a PEMDAS problem asked of sixth graders probably wouldn't use both an obelus and juxtaposition for multiplication. And the seventh grade standards use the solidus for division (e.g., in the divide rational numbers standard).

Let's look at today's Blaugust participant, Cheryl Leung:

https://matheasyaspi.wordpress.com/
https://matheasyaspi.wordpress.com/2019/08/20/dabbling-with-desmos/

I wrote about Leung last year for Blaugust. I remember her as the sixth grade teacher whose only two classroom rules are "Be Kind" and "Be Brave" -- and sure enough, one of her posts from earlier this month is about those same two rules.

This post is about the Desmos online graphing calculator. As I explained before, Desmos is the official calculator used by the SBAC.

Leung explains that she uses Desmos for the following three activities:

1.   Battle Boats – This is a lesson on graphing on the coordinate plane.   Usually, I have students play Battleship when I teach this concept, so this is an extension of that idea that uses Desmos.   I think it will be an easy entry point for our first foray into Desmos.
2. Inequalities on the Number Line – This is a great lesson that enables students to construct an understanding of how/why the graph of an inequality comes about as they plot points on the number line that fulfill the inequality and then see how the graphs change when all the points their classmates also plotted are added to the number line.  I really like the way that it builds the idea of a ray as the solution set and the way that it builds the idea of a closed or open circle.
3. Graphing Stories – This is a nice lesson matching graphs to stories.   It is similar in concept to a lesson that I already teach.   After looking at this lesson, I decided to use this as a model of sorts for creating my own first Desmos lesson.     I like the idea that if it crashes and burns, there is a back up plan for second period.   Thus far, I have built the introduction to the lesson and a card sort activity.   I still need to build a closing formative assessment.

Unfortunately, the sixth grade class I taught at the old charter school didn't use Desmos (using other software such as IXL instead). It's possible to perform the above activities without Desmos, but of course the software makes it much easier to do so.

But guess what -- now I'm curious about how Desmos would evaluate 8 / 2 (2 + 2) -- especially considering that Leung is a sixth grade teacher (the year PEMDAS is taught). I suspect that Desmos doesn't use an obelus, and so that issue never arises.

Here are today's worksheets: