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 non-linear 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, $y = 5-x^2$ 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 non-linear 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 non-linearity 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 $$\frac{y_2-y_1}{x_2-x_1} = \frac{{mx_2 + b} – (mx_1 +b)}{x_2-x_1} = \frac{m(x_2-x_1)}{x_2-x_1} = m.$$

OK, next week I’ll write a blog post about the comma.

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, input-output 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.

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.

 

I can’t help writing this off-cycle 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 not-for-profit 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.

 

 

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.