When I think back to when I learned about a mathematical function, it was in honors precalculus in 1983. We had to stand at our desks one by one and recite the definition of a function. If we didn’t nail it, our teacher told us to sit down and we’d have to do it the next day.
A function is a relation in which for each element of the domain, there exists one and only one element of the range.
Junior year was a biggie and I also had to memorize the first 18 lines of the General Prologue from Chaucer’s Canterbury Tales… in Middle English. (Click here to listen to what that sounds like.) So the function definition was chump change in comparison; I got it right on the first try. Not so with the Chaucer. I remember the Chaucer, however, to this day and it makes for an excellent party trick.
When I first started teaching, functions and their notation were creeping into the Algebra II curriculum, and by 2000 they had wormed their way into the Algebra I curriculum. But, in Algebra I it was basically, “Whenever you see ‘f(x)’ it is the same as y,” and, “if this graph passes the vertical line test, then it’s a function.” When the Common Core State Standards (CCSS) were approved by multiple states, including Mississippi where I live and teach, and subsequently PARCC began releasing its test items (click here to access a PARCC Algebra I practice test), my initial reaction was:
Disclosure: This is my usual reaction to ludicrousness. I MEAN, WHO ARE THESE PEOPLE AND HAVE THEY EVEN TAUGHT IN A REGULAR CLASSROOM EVER BEFORE IN THEIR LIVES? FOR MORE THAN FIVE YEARS? After the shock wore off, I did what I always do, I pushed up my sleeves and got to work. I realized that if our students have to be fluent with the definition of a mathematical function, adroit in manipulating the notation, and work with functions given multiple representations (graph, table, equation, story), then they need to start learning about functions from day 1.
To that end, I reflected on how we could we could renovate what we were already doing (which was working pretty well) to hit the function early and often. Here’s our current flow with some sample item renovations (we didn’t renovate everything, we still ask the traditional questions too!) and some links to some of the resources we use:
UNIT 1: At first we just define function as an input / output machine. Students learn the notation, do a few exercises. This goes nicely with our first unit which is basically order of operations and solving linear equations. Students do not learn about domain and range and they do not graph. Tables, stories, equations. [We also do some baby stats in this unit for review… mean median mode range, box and whisker plots, etc..]
UNIT 2: Our second unit is inequalities. In this unit we solve linear inequalities and compound inequalities and graph solutions on number lines. Now we also compare function values.
I realized that piecewise functions would go perfectly in the inequalities unit, and since there wasn’t anything written on an algebra 1 level, I wrote some notes and a circuit.
UNIT 3: This is when we historically developed the idea of a function, introduced the bulk of the nomenclature, and really got into multiple representations. relation / function / domain / range / vertical line test / linear / non-linear / quadratic / exponential. Much of what we do is the same as what we did in the past, but we have incorporated the vocabulary “even” and “odd” to describe functions. Understanding domain and range from a more sophisticated perspective is paramount. I wrote a matching activity to cap this unit many years ago and it is still viable today.
UNIT 4: This unit used to be “graphing lines and writing equations of lines” but now it has morphed into The Linear Function. Instead of graphing y = 2x – 4, we graph f(x) = 2x – 4. Instead of writing the equation of the line through the points (-2, 1.5) and (5, -7), we write the equation of the linear function, g(x), through those points. We talk about what it means to be linear (constant rate of change). I could go one and on. And we do. For about three weeks. [We also do linear regression / line of best fit in this unit, because, well, statistics is coming into each math course and must be hit early and often too, not as a stand-alone unit, but that is the subject of another blog post.]
Unit 5: Solving linear systems algebraically and graphically, solving non-linear systems graphically, and solving systems of linear inequalities graphically. Most of this is pretty traditional Algebra I fare, but at this point in the year, I throw questions on their warm ups like this:
This is not a hard question if students are comfortable with the notation and realize it’s a situation where they can set the equations equal to each other and solve graphically with their hand-held technology. After a few lessons on multiplying polynomials (Unit 6), students might not even go to their technology and opt to solve algebraically. Notice how the teacher has to carefully select problems that teeter on the edge of students’ understanding. In the above warm-up example, the x-squared terms subtract out leaving a linear equation since we have yet to solve quadratic equations.
UNITS 6 & 7: Operations with polynomials and factoring. These units are in preparation for intensive work with the quadratic function, however, we do find zeros of functions at the end before we go into the quadratic function in depth. Here is a typical renovation:
UNIT 8: The Quadratic Function in depth. All about the parabola, solving quadratics not just by factoring, word problems, etc. This unit is so full and long that I will keep the description short. The end.
UNIT 9: The Exponential Function in depth (this is a short unit and students appreciate that after all of the work with the quadratic function which spanned essentially three units). This really works well as a “looking back on other functions” unit to discuss similarities / differences with the other functions. [Stats: linear, quadratic, exponential regression ]
UNIT 10: Statistics wrap up. No functions in here. But as we practice for the state test, students have been faced with the function from day 1 and so they aren’t dividing by 2 after they evaluate f(2). If you have taught math, you know exactly to what I am referring right now, and if you haven’t, lucky you.
Shortcomings: We really need to do more work with transformations of functions and I am hoping to work that into the already too long Unit 8. We’ll see.