The students I work with often miss extended periods of class time. Because the idea of functions is so central to algebra and students often have misconceptions, I like to spiral back to this lesson and the idea of a function machine. I am hoping to solidify the main points about functions that I want students to understand.
We start class with a big image of a function machine on the board. I tell students I have this machine, and something is happening inside of it. I begin to put numbers into the machine. For example, I tell students that I am putting the number 2 into the machine. Then, make some of kind of noise and tell them that something is happening to the 2 inside the machine. I ask them if they can guess what will come out of the machine after this something happens. I might choose to add 3 to each Input. I let students guess until they have a 5 come out of the machine and then write the number 5 somewhere near where it says Output. I continue on with more numerical examples for the same rule. I make sure to emphasize that the same thing is happening to each Input once inside the machine. It shouldn't take long for students to realize that each number is increasing by 3 inside of the machine.
Next, I tell students the machine will now do something different inside. I like to move on to a non-numerical example next so students get the idea of a what a function is without being overly concerned with actual numbers. I might put in a word, and have the number of letters in the word come out. I want students to focus on this idea of what is happening inside of the machine, not so much the math of numbers.
Next, I show students that I have made a big mess of the board trying to keep track of what this machine is doing. I ask them if they can think of a way to organize this information that won't be so messy. I try to elicit the idea for a table, or more specifically, an In/Out table. Next, I have students write a rule in words for one table as an example (perhaps under the table). I encourage students to use In/Out language. I might ask them, "In this table, what do you do to the In to get the Out?" I make sure to emphasize again with students that the Out value depends on what goes In to the function machine. This is a key idea about functions and will also help students later with graphing dependent variables.
In the next section of the lesson, I want to relate the idea of a function to a problem that has a context that students will understand. I give students a problem like the first one in Connecting the Dots. We will return to this task in full later in the unit, but for today, I want to focus on making multiple representations for the first problem only. We read through the problem together, and I ask students to represent the problem in as many ways as they can. This should be review for them, but again, for students who struggle, it may seem new. I circulate around the room and prompt students to make tables, graphs, and equations to represent the piggy bank problem.
When we discuss the problem together, I want to make sure to circle back to the idea of the function machine. I'll ask students what would be happening in this particular situation inside of the machine?
Next, I want to make (or remake) the point that we often identify four ways of a looking at a function: as a table of values, as a graph, as an equation, and as a relationship between quantities. I have different students show simple examples of each of these that relate to the piggy bank problem.
Next, I want to clear up some misconceptions about functions and equations with students and make sure they feel comfortable using function notation. First we talk about the domain and range of a function. We can use the piggy bank example to think about what are possible inputs for this problem. I want students to understand that we could not put in a negative day for example, because that would not make sense in terms of how Savannah saves her pennies. I have students write the following definition for domain in their notebooks:
Then I want to elicit a similar idea about the range. I'll ask students what are the possible values for the out column of the table? Can Savannah have a negative amount of pennies in her piggy bank? What about 0 pennies? I have students write down the definition of range:
Next, I'll ask students if they can summarize today's work and we can come up with a definition for a function. I might ask them if there's any way I could put the number of days Savannah has been saving as the input and get out two different answers. This is a key idea about functions that I want students to remember: there are no two ordered pairs with the same input and different outputs. We might talk about how the machine would have to be broken in order for this to happen. I let them know that a working definition for a function is:
Lastly, I want to clear up any misconceptions or confusion students may have about using function notation. We talk about f(x) as being the most common notation and how x is the most common way to represent an input. f(x) then represents the output. I let students know that sometimes other letters are used to denote what is happening with a relationship. For example, they might see a function written as h(t) where h stands for height and t stands for time. Here, height is dependent on the amount of time that has gone by, so the function notation is useful for us to remember what is happening in our situation. I try not to just say that f(x) is the same as y although sometimes students equate them to be the same. I think this creates further confusion about the difference between functions and equations.
We covered a lot of information about functions today. I like to give students time to reflect on what they've learned and absorb some new content. I close class with an Exit Ticket activity about functions. I might ask students: What will you remember about how a function machine works?
Connecting the Dots: Piggies and Pools is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.