I’ve been meaning to let you know about the Illustrative Mathematics blog, which launched a few weeks ago. It has blog posts by members of the IM community about our grades 6–8 curriculum and about teaching practice, including a whole series on the 5 practices framework of Smith and Stein. Also, we will be cross posting any IM related posts I write here over there as well. I hope you find our new blog useful!
Author: Bill McCallum
Why is the graph of a linear function a straight line?
In my last post I wrote about the following standard, and mentioned that I could write a whole blog post about the first comma.
8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on a straight line.
The comma indicates that the clause “whose graph is a straight line” is nonessential for identifying the noun phrase “linear function.” It turns the clause into an extra piece of information: “and by the way, did you know that the graph of a linear function is a straight line?” This fact is often presented as obvious; after all, if you draw the graph or produce it using a graphing utility, it certainly looks like a straight line.
When I’ve asked prospective teachers why this is so, I’ve gotten answers that look something like this:
We know that a linear function has a constant rate of change, $m$. If you go across by 1 on the graph you always go up by $m$, like this:
So the graph is like a staircase. It always goes up in steps of the same size, so it’s a straight line.
This is fine as far as it goes. It identifies the defining property of a linear function—that it has a constant rate of change—and relates that property to a geometric feature of the graph. But it’s a “Here, Look!” proof. In the end it is showing that something is true rather than showing why it is true. Which is to say that it’s not a proof.
Still, the move to a geometric property of linear functions is a move in the right direction, because it focuses our minds on the essential concept. We all know that any two points lie on a line, but three points might not. What is it about three points on the graph of a linear function that implies they must lie on a straight line?
Because a linear function has a constant rate of change, the slope between any two of the three points $A$, $B$, and $C$ is the same. So $BP/AP = CQ/AQ$, which means there is a scale factor $k =AQ/AP = CQ/BP$ so that a dilation with center $A$ and scale factor $k$ takes $P$ to $Q$, and take the vertical line segment $BP$ to a vertical line segment based at $Q$ with the same length as $CQ$. Which means it must take $B$ to $C$.
But (drumroll) this means that there is a dilation with center $A$ that takes $B$ to $C$. Dilations always take points on a ray from the center to other points on the same ray. So $A$, $B$, and $C$ lie on the same line.
I don’t really expect students to get all of this, at least not right away. I’d be happy if they understood that there is a geometric fact at play here; that seeing is not always believing.
How do you tell if a function is linear?
Over at the IM Virtual Math Coach we got a question about the following grade 8 standard:
8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on a straight line.
I could write a whole blog post about that comma in the first sentence, but for now I want to focus on the question of what exactly are the student expectations entailed by this standard. Certainly students should be able to recognize that $y = mx + b$ defines a linear function; and they should be able to show a function is not linear by finding points on the graph with different slopes between them. (By definition, a linear function is one with a constant rate of change, that is, a function where the slope between any two points on its graph is always the same.) However, the following PARCC released item suggests the possible expectation that students be able to tell if a function is linear or not purely from looking at its defining equation.
My guess is that the writer of this item was thinking that students would detect nonlinear functions by noticing the functions with squared and cubed terms (they could also use the point method but that doesn’t seem likely). That’s not a good path to lead them down. Sure, $y = 5x^2$ is not linear. What about $y = 5 – x^2 + (x+1)^2$? By the look of it it is even more not linear! But of course it is in fact linear, because the expression on the right is equivalent to $2x + 6$. The grade 8 standards don’t expect students to be making such simplifications as expanding squares of binomials.
This is another example of the confusion between expressions and functions. The expression $(x+1)^2 – x^2$ has nonlinear terms in it, but the function it defines is linear because it is equivalent to $2x + 1$. Equivalent expressions define the same function.
There’s another confusion revealed in this item, the confusion between equations and functions. Look at option C. Is it intended to be a distractor? Will a student who chooses it be marked wrong? Such a student would have a case for protest on the grounds that $3x + 2y = 4$ is not a linear function because it is not a function, it is an equation. Certainly if you choose to think of $x$ as the input and solve for $y$ to get the output you can think of it as a function, which would indeed be linear. You could also go the other way around and choose $y$ as the input and get a different linear function. It is conventional when $x$s and $y$s are floating around to think of $x$ as the input and $y$ as the output, but you can flout convention without being mathematically incorrect.
The moral of this story is, I suppose, that it is easier to tell when a function is not linear than to tell when it is linear. Testing for nonlinearity involves just picking a few points on the graph; testing for linearity involves picking every possible pair of points on the graph and verifying that the slope between them is always the same. It’s instructive to do this with $y = mx +b$: if $(x_1,y_1)$ and $(x_2,y_2)$ are two different ordered pairs satisfying this equation, then $$\frac{y_2y_1}{x_2x_1} = \frac{{mx_2 + b} – (mx_1 +b)}{x_2x_1} = \frac{m(x_2x_1)}{x_2x_1} = m.$$
OK, next week I’ll write a blog post about the comma.
The image of function in school mathematics
Well, oops, took a little break from blogging there. But I’m back now.
In the course of working on an article with the same title as this blog post for a publication about Felix Klein, I did a Google Image search on the word “function,” with the following results.
I find this fascinating for a number of reason. First, notice the proliferation of representations: graphs, tables, formulas, inputoutput machines, and arrow diagrams (those blobs in the second row). A corresponding search in a French, German, or Japanese gives a very different result. Here is the Japanese one (where I searched google.co.jp for “関数”).
This has a much greater focus on graphs, and no arrow diagrams. The French and German ones are even more focused on graphs; a search in Spanish, however, gives results similar to English.
What does all this mean? Well, I’ll be discussing this more in the article I’m writing, but for now I have just a couple of observations.
First, the English language search reveals a preoccupation with examples of relations that are not functions, with examples of graphs, arrow diagrams, and tables. I have always found this preoccupation perverse: the examples are artificial, and there is very little to be gained from them except an ability to answer questions about them on tests. This preoccupation is not evident in the searches in other languages.
Second, take a look at these two representations, from the English and Japanese language searches:
.
Together they represent a case study in designing representations. Can you see what’s wrong with the one on the top? A function is supposed to have only one output for every input. Does an apple slicing machine have that property?* The representation on the bottom, on the other hand, clearly represents how to think of addition as a function with two inputs.
I’m interested to hear readers’ thoughts on the representations that come up in these and other searches. Maybe someone who can navigate Baidu can tell us what the Chinese results are.
*To be fair, the authors of the web page with the apple slicing machine are clearly aware of the problem. But their contortions to get around it only emphasize the fundamental flaws in the representation.
Math is not linear, but time is
A short post today with a question for our readers.
A number of years ago there was a popular piece by Alison Blank titled Math is not linear, which gave a number of ideas about the order in which we teach mathematics. A curriculum writer has to grapple with the fact that, although math is not linear, time is. Hermione Granger’s time turner does not actually exist. Tuesday comes after Monday, and Tuesday’s lesson comes after Monday’s lesson and, in the end, a teacher has to decide what to teach on each day; that is, they have to decide on a linear order in which to teach mathematics. The gist of “Math is not linear” is that that order need not be a dry march through a logical hierarchy of topics. You can, as Blank says, go on tangents, foreshadow topics to come, connect back to previous topics, and give students problems that create a need for a new topic. These are all great ideas.
Our question is: what other ideas do people have to make sure that the sequence of lessons in a course makes sense to students and makes sense mathematically? Do you recommend any books or articles that might help answer these questions? We have some ideas and will be writing some posts about them, but want to hear from the community as well. Please feel free to share your thoughts in the comments, or on Twitter with @IllustrateMath, #timeislinear.
Illustrative Mathematics 6–8 Math
I can’t help writing this offcycle blog post to celebrate the release of Illustrative Mathematics 6–8 Math last Friday, a proud achievement of the extraordinary team of teachers, mathematicians and educators at Illustrative Mathematics (IM), one that I didn’t dream of when I started IM almost 7 years ago with a vision of building a world where all learners know, use, and enjoy mathematics.
Conceived initially as a project at the University of Arizona to illustrate the standards with carefully vetted tasks, IM has grown into a notforprofit company with 25 brilliant and creative employees and a registered user base some 40,000+ strong. Our partnership with Open Up Resources (OUR) to develop curriculum started a little over 2 years ago when we submitted a pilot grade 7 unit on proportional relationships to the K–12 OER Collaborative, as OUR was then known. In the fall of 2015, not understanding that it couldn’t be done, we agreed to write complete grades 6–8 curriculum ready for pilot in the 2016–17 school year.
One of the things I love about the curriculum is the careful attention to coherent sequencing of tasks, lesson plans, and units. The unit on dividing fractions is an example, appropriate to mention in the middle of this series of blog posts with Kristin Umland on the same topic. It moves carefully through the meanings of division, to the diagrams that help understand that meaning, to the formula that ultimately enables students to dispense with the diagrams. It illustrative perfectly our balanced approach to concepts and fluency. Kristin and I will be talking about that more in the next few blog posts.
A world without order (of operations)
What would such a world look like? Like this:
$$
(((3\times(x\times x)) – (7\times x)) + 2).
$$What a world it would be! A world without ambiguity! A world where PEMDAS would just be P! A world where they would have to relocate the parenthesis keys to a more convenient location on the keyboard!
Parentheses, and order of operations, tell us how to read the meaning of an expression, how to parse it, not what to do with it. In the expression above, every matched pair of parentheses contains something of the form $$
(\mbox{blob}) * (\mbox{another blob}), \qquad (\mbox{where * stands for $+$, $$, or $\times$}),
$$ unless the blobs are just numbers or letters, in which case we don’t surround them with parentheses. We always know exactly what things we are adding, subtracting, or multiplying. Starting with the outermost parentheses, we see it contains the sum of 2 and a blob. Looking inside that blob we see that it contains a blob minus another blob. And so on. The structure of the expression can be represented in a diagram:
So what is order of operations about, and why do we need it? Well, that’s a lot of parentheses up there, so it is useful to have some conventions about when things are understood to be a blob, without actually putting in the grouping symbols (blobbing symbols?). First, any sequence of multiplications and divisions is understood to be a blob (that’s the precedence of multiplication and division over addition and subtraction). Second, in a sequence of additions and subtractions, or of multiplications and divisions, you read from left to right. (Actually, there is disagreement about this last one in the case of multiplication and division, but never mind.) The first rule allows us to write the expression above as
$$
((3\times x\times x 7\times x) + 2).
$$The second rule allows us to leave out all the remaining parentheses. And, of course, we have other conventions about representing multiplication by juxtaposition, and about exponent notation, which allow us to write
$$
3x^2 – 7x + 2.
$$
Calling it order of operations is problematic because it can be misconstrued as suggesting that there is a specific order in which you must perform operations. There isn’t, except insofar as you sometimes have to wait to perform an operation until you have calculated all the blobs in it. But, for example, there is no law that says you have to do the multiplications first in $101\times5699\times56$ and, in fact, it is more efficient to factor out the $56$ and do a subtraction first. Order of operations tells us how to read this expression: it’s a difference of two products, not a product of three factors the middle one of which is a subtraction. But it doesn’t tell us how to compute it. The word “order” in “order of operations” is best understood as referring to order in the sense of hierarchy, as in the diagram above.
Outside of textbook school mathematics the order of operations is a matter of common law, not constitutional law, and it’s a bad idea to make a federal case out of it on assessments. For example, dinging a student for interpreting $x/2y$ as $x/(2y)$ rather than $(x/2)y$ would be unreasonable; many scientists would do the same thing. If there is any danger of ambiguity we should put the clarifying parentheses in.
A few final thoughts:

 thanks to Brian Bickley for suggesting the topic for this post
 there’s a nice discussion of the history of order of operations over at the Math Forum
 and bonus question: do we have to give multiplication precedence over addition? Could we do it the other way around?
New look, new title
I have updated to a new WordPress theme, partly because I thought it was time for a makeover, and partly to see if it would cure some of the problems people have had commenting. In the process I decided to change the title of the blog. My recent writings have been about school mathematics generally, and I hope they are of use to teachers everywhere, whatever their standards. I will still write occasionally about the Common Core, and I will still answer questions over in the forums. I have changed the settings in the forums to allow anonymous posting for people who have trouble logging in. I may have to change that back again if it causes security problems. And, speaking of security, the site has been protected by SiteLock since last summer’s hacking, which means that you will occasionally encounter a captcha screen.
And, by the way, the url mathematicalmusings.org also points to this blog.
Truth and consequences: talking about solving equations
The language we use when we talk about solving equations can be a bit of a minefield. It seems obvious to talk about an equation such as $3x + 2 = x + 5$ as saying that $3x+2$ is equal to $x + 5$, and that’s probably a good place to start. But there is a hidden assumption in there that the equation is true. In the Illustrative Mathematics middle school curriculum coming out this month we start students out with hanger diagrams to represent such equations:
The fact that the hanger is balanced embodies the hidden assumption that the equation is true. It is helpful for explaining why you have to perform the same operation on each side when solving equations; if you take two triangles from the left side you have to take two triangles from the right side as well in order to preserve the balance. This leads to a discussion of how performing the same operation on each side of an equation preserves the truth of the equal sign.
But what happens with an equation like $3x + 2 = 3x + 5$? In this case, the hanger diagram is a physical impossibility: the right hand side will always be heavier than the left hand side. I can imagine that students who have an idea of an equation as “the left hand side is equal to the right hand side” might be confused by this situation, and think this is not a proper equation. Especially when they reduce it to $2 = 5$. Students learn to say that this means there are no solutions, but it’s hard to make sense of that response rule without understanding what’s really going on with equations.
The fact is, an equation with a variable in it is neither true nor false, because it is merely a phrase in a longer sentence, such as “If $3x + 2 = x + 5$ then $x = \frac32$.” This sentence is true, but the phrases within it are not sentences and have no inherent truth or falsity. When we perform the same operation on each side of an equation, we are not only preserving the truth of the equal sign but also preserving the consequences of the equal sign. If we use ifthen language when talking about equations, then we can make sense of equations with no solutions. A sentence like “If $x$ is a number satisfying $3x + 2 = 3x + 5$ then $2 = 5$” makes perfect sense. It’s the mathematical equivalent of “If the moon is green cheese, then I’m a monkey’s uncle.” It’s a way of saying the moon is not green cheese . . . or that there is no solution to the equation.
The middle schooler’s version of ifthen language might not always use the words “if” and “then.” You might say “Imagine there is a number $x$ such that $3x + 2 = x + 5$. What can you say about $x$?” Just as you say “Imagine this hanger is balanced and the green triangles weigh one gram. How much do the blue squares weigh?” I think it’s a useful approach with students to remember that equations are a matter not just of truth, but of truth and consequences.
Why is a negative times a negative a positive?
OK, I can hear the groans already. There are many contexts for answering this question and they are dubious in varying degrees because the real answer is “because I said so.” That is to say, the rule for multiplying negatives is a convention; adopted for good reasons, but a convention nonetheless. Those good reasons are mathematical: we want to make sure that when we extend multiplication and addition to negative numbers the properties of operations still apply. In particular, we want the distributive property to apply. Meditate on this:
$$
3\cdot(5 + (5)) = 3\cdot5 + 3 \cdot (5).
$$
The left side is really $3 \cdot 0$, so it had better be zero. So the right side had better be zero as well. The first term on the right side is 15, so the other term had better be $15$. So $3 \cdot (5) = 15$. We want the commutative law to hold, so we had better say $(5)\cdot 3 = 15$ as well. Now meditate on
$$
(5)\cdot(3 + (3)) = (5)\cdot 3 + (5)\cdot(3).
$$
The same reasoning tells us that $(5)\cdot(3) = 15$.
Trouble is, all this is really hard to explain to middle schoolers, so people invent contexts. One context I’ve seen has something to do with sending out bills. If you receive 5 bills for 3 dollars then you have $5 \cdot (3) = 15$ dollars. Sending out is the opposite of receiving, so if you send out 5 bills for 3 dollars, you have $(5)(3)$ dollars. But once you receive payment, you have \$15. So $(5)(3) = 15$.
One problem with this is that you have to buy more conventions to believe it: the convention about negative amounts of money representing debt, the convention about negative receiving being kinda sorta like sending out. That’s a lot of conventions to prove something that is, as I said, a convention itself. Another problem is that all this context really shows is that $(3) = 3$, five times. The multiplication in this context is really just repeated addition; it doesn’t work for numbers that are not integers. You can’t send out 5.6 bills.
There is one context that I think does a better job here, and that is $\mbox{distance} = \mbox{speed} \times \mbox{time}$. This does work with nonintegers, and you can make sense of all of the quantities involved as negative numbers. Let’s assume that an object is moving along the number line, and that you measure its position at different times, setting your stop watch to 0 when it passes through the origin. Negative distance is distance to the left; negative speed is speed from right to left; and negative time is time before you started measuring. (Later we use the terms displacement and velocity, but there’s no need to introduce them right away.)
So if the object is moving at $5$ m/sec, where is it at time $3$ seconds? Well, it’s moving from right to left and it has 3 seconds before it hits the origin, so it is 15 m to the right of the origin. So $(5)(3) = 15$.
Was I cheating there? Is this context subject to the same objections I made about the money context? Didn’t I just make up a whole bunch of conventions about negative distance, time, and speed? I think these conventions pass the cognitive sniff test better. They don’t seem as artificial to me. You can really make quantitative sense of negative distance, speed, and time. It feels more like the real world and less like an accountant’s convention. (No offense to accountants intended.) In a way, we have replaced the mathematician’s desire to have the properties of operations continue to hold with the physicist’s desire to have the laws of physics continue to hold.
So where is the distributive property in all of this? I think it is built into our physical intuition about this context. If I travel for 3 hours, and then for another 2 hours, I can figure out how far I have gone by just adding the times and multiplying by my speed, or I can add the distances traveled in each time period. That’s the distributive property. If you dig into the reasoning I gave for the object moving at $5$ m/sec in the light of this common sense, questioning each claim, you end up with something not too far from the mathematical reasoning I gave earlier.
By the way, this is the approach we take in the Illustrative Mathematics middle school curriculum. Finding contexts for mathematical ideas that are faithful to the mathematics is difficult and requires real sensitivity to both the mathematics and the way students think. Our brilliant curriculum writing team is up to that challenge.