See function_representations.pptx slide #1
As long as I have taught algebra this has been an issue! Many students still make the mistake that squaring a negative number results in a negative answer. Due to the fact that one function in this lesson is quadratic, I wanted to make sure we addressed this issue at the very beginning of the lesson to try to correct the mistake.
In order to conceptually approach this for students who are having difficulty have them write out what it means to square any number. For example 3^2 is 3 x 3, 5^2 is 5 x 5, so -2^2 is -2 x -2 and most students know that a negative multiplied by a negative is a positive. Note: To make matters worse if students put -2^2 into their calculator without parenthesis many calculators will return an answer of -4.
For a quick "do now" like this I give students some time to work and then give a non verbal cue of having students hold the fist in front of them and give me a thumbs up if they agree with Danielle or a thumbs down if they disagree. I will then have a student explain their thinking and see if they can convince the others (who thought differently) if they changed their thinking. I try not to insert my thoughts and let the students critique each other's thinking until the correct answer surfaces.
Slide #3: This is a quick way to get at both MP1 and MP7. Students will first need to take some time to undertand the situation and then see some structure in the numerical values. It is always interesting to see how students approach this from multiple perspectives. Some will make a list of values, others will write a recursive formula (subtract 2 each day) while others will come up with the explicit formula of f(x)=10-2x. I have students work on this brainstorming individually first and then pair up with another student to share their thoughts and troubleshoot other ideas. As an extension, encourage students to come up with multiple ways to write the function (see examples above) if they have time before other students finish.
Slide #4: This question allows students to grapple with the concept of domain and range in the context of the problem. While the function f(x)=10-2x has its domain and range defined for all real numbers, in this situation the domain would be from [0,5] (when Adam runs out of money) and the range would be from [0,10] (all of the money he starts with until he runs out of money). Have students try to verbalize this in their partnerships and then have multiple students work together (each one building off the response of the previous student) to verbalize it in a coherent way for the class.
Slide #5: This slide emphasizes conceptual understanding of the function in context. I am listenting for students to explain f(2) = 6 as "after 2 days, Adam has $6 left". I put in the last question, f(6)=-2, to get students thinking about the domain once again. While I would not change the answer from slide 4, I want students to be thinking about what this would mean in context (Maybe Adam had to borrow $2 from a friend on the 6th day). Students may begin thinking about how Adam could continue to go in debt each day which would extend the domain in the positive direction and the range in the negative direction. If time allows, you could also discuss if it would be possible to extend the domain in the negative direction? (This would not be possible because you cannot have negative days)
The difficulty with this investigation will be to not go overboard with all the different observations you will want to make as a teacher and ultimately help your students to see. In most cases, students will be completely unfamiliar with quadratic and exponential functions.
Important: it is not the goal of this lesson to discuss attributes of these different families of functions. To do so at this point would be, in my opinion, overwhelming for most students. As an extension, if students take the conversation there you can follow and facilitate as you see fit.
The purpose of this portion of the lesson is simply: input and output. If students understand the function as a machine where x goes in and f(x) comes out then it should not matter what the function's equation is. In this investigation, I want students to put most of their focus on the table with the inputs and the outputs. The graph is meant to help deepen their understanding of the relationship between these inputs and outputs and how values in a table can be reflected and represented graphically. Lastly, students will use the graph to determine the domain and range of the functions.
Teaching point: For the exponential function it may be a good idea to give students access to a calculator so that those who lack the background with negative exponents do not get too bogged down with calculations. The noticing for this function (at this point in time) is that if the x values get bigger in a negative direction (farther from 0) then the corresponding g(x) values will get smaller and smaller. Some students may want to plug in values beyond -3 to see that the output will never be negative.
The last question on the investigation about "the most interesting shape" is to call student's attention back to the fact that not all functions are linear (as was the case in middle school). Depending on the expression that a function is defined by, the graph can have any number of appearances.
See function_representations.pptx slide #6
This final question is going back to MP1 by challenging students with an unfamiliar question that they may have to think about and grapple with. I am really not looking for anything specific here except to determine students understanding of the square root symbol and why the input for that symbol cannot be negative. As a scaffold it is helpful for some students when I draw in points on the function such as (0,0), (1,1) and (4,2) and show that by taking the square root of the input you get the output (students have seen square roots in grade 8 both explicitly and when working with the Pythagorean theorem).