Number Line Movement as a Function Intro... Continued

I've been using a new approach to introduce functions to my Algebra 2 students (who presumably aren't completely new to functions). As much as I can, I'm trying to let need dictate the math. At this point our definition is somewhat incomplete, but working well so far. Today we see a need for more explicit notation, because it's not always clear to us what exactly the function rule is:

I was amazed how quickly they got comfortable with function notation, given this set-up, but I am willing to accept that it may be partly due to previous teachers and previous foundations.

From here we jumped right into the idea of function compositions (but not composition notation, because we don't need it yet):

It was entirely logical to follow these with a discussion of inverses that are a bit more complex:

Here are two samples of their student work at this point. There still seems to be some ambiguity around the word opposite (a great entry point for an upcoming lesson). Beyond that, however, their understanding appears to be rock solid. Everybody did a nice job of explaining the function. Only one got mixed up on defining the inverse. A teacher's dream.

Number Line Movement as a Function Intro

I'm bragging here, and these ideas are not all mine, but I did an introductory lesson on functions for an Algebra 2 class today and it rocked. My students are average juniors in high school and we've just finished our first unit on Sequences and Series. We're about to dive into an intense study of polynomial, exponential, logarithmic, rational, and radical functions. I wanted to set the stage with a solid foundation of how functions behave - including transformations, compositions, and inverses. Here's what today looked like.

The students walked into class and there was a large number line taped to the floor of the classroom. I asked for volunteers.

discovering additive identity and translations through number line movement

Now they know what to do. So they go back 'home' and then they subtract 3, rest, and then add three. Ah. The inverse is born:

introducing function inverses through number line movement

What comes next turns out to be mind blowing, and even I didn't expect it. We return 'home' and then multiply by 2:

It seemed so odd to everyone that a simple operation like 'multiply by 2' would result in such seemingly UNuniform motion... which made it all the more amazing to 'discover' the constant rate of change in the distances between neighbors.

Now seemed like a good place to introduce some traditional algebra:

We followed these ideas with a wonderful exploration of our tables and graphs generated from our movement activities. Thank you Desmos, for a wonderful tool. We were able to easily see both our data points and our generalized lines, and the symmetry of inverses was so obvious and yet so cool nevertheless. You can check out our exploration here: https://www.desmos.com/calculator/ecfpw2h4tp

The last item of the day was to define some function terminology:

In the end, we had a good start to to a definition of function:

A function is a relationship between two variable quantities that follows a rule to map inputs to outputs.

It needs work, but they don't know that yet. Tomorrow we'll get into function notation and simple compositions and inverses that use only the traditional operations as they were presented today. Then, we'll return to the number line and explore some other function rules: opposites, reciprocals, absolute value, exponents, roots, etc. When we run into trouble (I PLAN on it!) we'll adjust accordingly.


(Thank you to Max Ray and Michael Pershan for the inspiration from their Teaching Complex Numbers workshop at NCTM 2015. I'll be going back to these ideas when I get to complex numbers, for sure.)

Dear Class,

It seems to me that you are suffering from a mid-year slump – maybe having to do with some disappointment about the recent midterm exam, or maybe something more. Recently I have heard: “You never taught us that.” “Nobody cares about understanding. Only the grade matters.” “I haven’t learned anything in here.” “You don’t like me.” And a few more. Ouch.

You may fault me for not being a teacher who leads you through every step of the way. It’s true. I ask you to practice problems on the homework that are harder than what we did in class. I ask you to be ready for a graded assessment at any time, and without warning. I ask you to be assertive and self-aware - requesting clarification when you need it, both inside and outside of class. I also put problems on tests that I have not previously solved for you in class. I know this makes it more challenging, but to me it’s the best way. I see the world of mathematics as a vast universe of possibility. I get to be your guide for only this tiny little region, but ultimately, I want you to trust your own instincts and skills and venture out without me.

You have a choice to make. One option is to be bold; take a leap out into the vastness and use the tools you already have to find your way through this wonderful world. I promise: you are not alone; you will not perish; and when you find that you need new tools, they will be there, ready for you to learn how to use them. Or, you can sit still and wait for a personal guide. It’s ok; sometimes this is the better way. I know that fear or insecurity or any number of things can freeze any of us in our tracks. Nevertheless, I cannot stop encouraging you to be bold and venture out independently, because I know that a richer and more beneficial world is there for those who do.  There will always be problems on ‘tests’ that have not been previously modeled for you. This much, I can guarantee. I wish we lived in a world where all the problems were previously solved, but I assure you, you will be confronted with problems that I cannot even imagine yet. The question is, what will you do with those?

In the end, it’s not important to me whether you attribute your growth to me or not. I just hope you notice what I do: that you HAVE grown. Don’t let the vastness of the universe lead you to a narrow view of your own successes. You have come a long way. I have evidence to prove it. And while we both know that there’s an even wider view in front of you, that doesn't diminish the excellent work you have done to get this far. It is one of life's great ironies that success and failure travel so well together:

“Failure is simply the opportunity to begin again, this time more intelligently.” Henry Ford

“I know that I am intelligent, because I know that I know nothing.” Socrates

“It’s not that I’m so smart, it’s just that I stay with problems longer.” Albert Einstein 

“I've missed more than 9000 shots in my career. I've lost almost 300 games. 26 times, I've been trusted to take the game winning shot and missed. I've failed over and over and over again in my life. And that is why I succeed.” Michael Jordan

6 Essential Questions in Algebra

A year ago, I began this blog with the goal of uncovering some satisfying essential questions in algebra. These were to be questions that addressed the fundamental essence of algebra, while also being able to extend beyond a single discipline... and of course, they needed to be intriguing to both my students and to myself. A few months later I wrote that one of the qualities of a REALLY good math teacher is having a 'second set of objectives that go beyond the mastery of today's content.' A reader challenged me to identify these objectives, which I slyly avoided. But this week, in honor of my one year blogoversary, I present six essential questions or 'higher objectives' for my algebra classes. It's a start. In the spirit of UbD, expositions are voiced in the language of enduring understandings.

How is algebraic thinking different from arithmetic thinking?

It is my hope that my students will understand that algebra is a language of abstraction, where patterns are generalized and symbols are used to represent unknown or variable quantities. Arithmetic involves counting and manipulation of quantities where algebra relies more heavily on reasoning and generalizing the patterns that are observed from arithmetic procedures. It is my ultimate hope that they come to appreciate the power and utility of generalization.

What makes one solution better than another?

I would like my students to understand that numerical accuracy is only one piece of a good solution. The measure of a comprehensive and satisfying solution involves a subtle balance of precision, clarity, thoroughness, efficiency, reproducibility, and elegance (yes, elegance). I want my students to be masters of the well-crafted solution.

How do I know when a result is reasonable?

I want my students to understand that in math, as in life, context is supreme. There is no 'reasonable' or 'unreasonable' without an understanding of context. I hope that they can refine the skills to analyze and dissect problems that are both concrete and abstract, applied and generalized. I want them to develop habits of inquiry, estimation, and refinement. Ultimately, I hope that they will improve their sense of wisdom.

Do I really have to memorize all these rules and definitions?

Students will understand that mathematics is a language of precision. Without explicit foundations (axioms and properties) and precise definitions, reason gives way to chaos. On the other hand, they should understand that many perceived 'rules' in mathematics are simply shorthand ways to recall a train of logical reasoning (like formulas and theorems). It is my hope that they will appreciate precision but also understand the value of reason over recall.

Isn't there an easier way?

Without destroying their fragile spirits, I want my students to appreciate the benefits of struggle. I want them to realize that insight and higher knowledge are gained by approaching a problem from different angles and with multiple methods and representations. I want them to understand that knowledge about how mathematics works is on a higher echelon than the solution to a particular problem. In my ideal classroom, the students will understand how to spark their inner intrigue in order to move themselves beyond answers to seek connections, generalizations, and justifications.

Do I really need to know this stuff?

By sheer repetition and example, my students will know that the practical applications of algebraic thinking are numerous, especially in the rapidly changing fields of science, engineering, and technology. Beyond these undeniably important applications, they will know that confirmed correlations have been made between success in algebra and improved socioeconomic status. But ultimately, I hope that they will understand that the beauty and intrigue of mathematics is vast, and the limit of its power to improve the quality of their lives is unknown. I want them to glimpse infinity.

What's the Big Idea with Algebra 2?

Lately, I've been following some of the conversation around the big ideas in an advanced algebra/pre-calculus course. The Global Math Department* hosted an interesting panel discussion around this topic a couple of weeks ago. I appreciated the thoughtfulness and complementary ideas of the presenters (John Burk, Dan Goldner, Michael Pershan, and Paul Salomon), and especially the thoughts behind proof and 'the well-crafted solution.' Without entirely reaching a consensus, the focus of the discussions tended to lean towards prediction as the overarching theme for algebra ii. The reasoning was thoughtful and grounded, but this theme did not satisfy me. While I can certainly see it, I also think that prediction is the theme for statistics. Can Algebra 2 and Statistics have the same theme? They can, I suppose, but it is not satisfying enough.

Some of the new bloggers from the New Blogger Initiative also tackled this topic last week.
gooberspeaks got me thinking about the focus on families of functions and David Price included ideas about varying ways of representing functions and modifying their behavior. Kyle Eck has a strong bent towards applications which resonates with the GMD theme of predictions. And all these ideas muddled around in my brain for a long time before emerging as a single construct that currently satiates my desire for deeper inspection.

Algebra 2 is all about: generalizing patterns of behavior in bivariate relationships.

But that's my academic's definition. In the UbD-influenced language of a high school classroom, I'd say that Algebra 2 asks these questions:

• How can we communicate the behavior of a relationship between two ideas?
• Are there rules of behavior that apply to all relationships?
• Why is it important to be able to generalize patterns of behavior?

Functions certainly play a large role here, because it's easier to generalize patterns when there are overt rules of behavior to follow. But just as importantly, we also look at conic sections and the elusive inverses of even polynomials and periodic functions, because these ideas give us essential insight about the comforting nature of functions that are both one-to-one and onto, and about the obstacles presented by relationships that are not.

Graphing also plays a large role, because it is a most excellent tool for alternate representations of bivariate relationships. Seeing patterns emerge in the shape of coordinate graphs can be enlightening long before symbolic manipulation clears a path through the brain... and I thank the math gods for that! I am wary though of too much graphical emphasis, for our well-loved coordinate system has obvious limits as our brains allow us to consider relationships with more variables.

And applications clearly play an important role too, especially in the attempt to answer that third question. But I hesitate to put applications at the forefront of an advanced algebra theme. I think that is perhaps better handled by a physics class. In algebra we are attempting to represent scenarios with a generalized pattern of behavior, and manipulate this generalization to highlight useful information. I think I agree with Paul Salomon in that proof and 'well-crafted solutions' may trump (but certainly not replace) applications in the hierarchy of an overarching theme in algebra.

To end, I'll just say that my desire to ask (and attempt to answer) the big questions is never entirely satiated, but I do so enjoy the conversations that emerge from them. I welcome your thoughts, criticisms, and further insights. The discourse is what makes being a mathematician so much fun.

*Megan Hayes-Golding, where have you been all my life? What a terrific thing the GMD is, and one of these Tuesday nights, I will not have bedtime routines or NBI deadlines to worry about and will be able to attend a session while it is actually happening! Thanks to you and all others who are making this happen.

Parenthetically Speaking

I imagine that you, like me, have taught, or retaught, or referred to parentheses in the traditional manner:

Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules). In an expression like , the part of the expression within the parentheses, , is evaluated first, and then this result is used in the rest of the expression. Nested parentheses work similarly, since parts of expressions within parentheses are also considered expressions. Parentheses are also used in this manner to clarify order of operations in confusing or abnormally large expressions. (from Wolfram Math World)

Wolfram goes on to define seven other mathematical uses for parentheses, including interval notation, ordered pairs (0, 5), binomial coefficients , set definitions , function notation , etc. With so many uses, it's perhaps no minor miracle when students are able to emerge with any working facility of parentheses at all!

Honestly, I feel for my students. Even to me, mathematical definitions can sometimes seem inconsistent and confusing. Like the difference between terms (things that are added) and factors (things that are multiplied).
 I can hardly keep my own head on straight to describe the number of terms (2) and factors (0) in the following expression:

(To be fair, the first term consists of two factors, each containing two terms each, and the second term has four factors.)


And then recently, I reviewed a prealgebra curriculum that described parentheses as symbols that tell us to "treat part of the expression as one quantity." (from onRamp to Algebra) The book goes on to further implore the teacher to forgo the order of operations description in lieu of the 'one quantity' idea.

I knew that... 


So WHY have I NEVER thought to describe it that way??? 

Parentheses are grouping symbols that tell us to treat the group as a single entity. Period. No confusion. 

A function input is a single argument.
An ordered pair is a single location.
An interval is a single, uninterrupted region.
A matrix is a single array.
A set is a single collection.
A binomial coefficient is a single combination.
An expression in parentheses is a single quantity.

For some reason, this simple statement of an idea that seems so obvious is completely enlightening to me. The Common Core lists "Look for and make use of structure," as one of its Standards for Mathematical Practice. To me, this 'single entity' idea is paramount to the mastery of this standard: the key to seeing structure in long complicated algebraic expressions.

I find a subtle beauty in tiny moments of enlightenment, even if it is only my own. It's got to rub off on someone!