The Bell Work today gives me a chance assess students knowledge of function notation. It is a quick review of function notation before we start the main activity for the day.
Evaluate f(x)=3x+8 for f(2), f(3x) and 3 * f(x)
After making sure students have the correct answers, Students explain why the answers for f(3x) different than 3f(x)? Understanding this difference is important when working with functions such as the trigonometric functions and radical functions. My goal is to have students see that f(3x) makes the domain value 3 times bigger while 3f(x) is making the range 3 times larger. If necessary, I give students values for x to evaluate f(3x) and 3f(x) to see how they are different. An example is: Evaluate f(3x) if x=2. Explain your process. Now evaluate 3f(x) if x=2.
Discussing this as a class helps students understand the differences.
I have always struggled to help students understand that sin(2x) is not equal to 2sin(x) or sin (x+y) is not equal to sin (x) +sin (y). In the last couple of years I have realized that the issue is not the trigonometric function but more about understanding function notation.
My students have worked with function notation as f(x), g(x), etc. they have not spent time with functions that are written as area(x) or sin(x) By writing functions as area(x) and sin(x) my goal is for students to understand the structure of the notation and not just see it as "y". It is the answer, output or range.
Students work in groups on the function notation exploration. The activity begins with evaluating some functions then comparing results to see if properties of real numbers work with function notation. I picked common errors to analyze such as sin(2x) compared to 2sin(x) and sin(x+2) as sin(x)+sin (2). As the students work I move around the room and help student with evaluating function notation. As I work with the students I see a lot of students get confused when the function is something other than f(x), g(x), etc. I ask students what the question is asking. What is the name of the function? What is the problem telling you to do with the function? Once students see that the name is area or sin, students are able to complete the problems.
It is interesting to see how students deal with problem 7 on the worksheet. Some students do each expression completely while others write the problem and immediately see it as not equal. Some students worry when the answers on problem 7 are all no. They think you always have at least one that is yes. I let them know that I am going to switch things up this year what has been the expected may not be the case in this class. You have to verify and explain what is going on as well as determining if an answer exists.
When most groups have stopped working I bring the class back together. I will ask students to put the answers for problems 1-6 from the task on the board.
We then discuss any questions and correct any errors. The students explain to each other what may be an error after discussion the class comes to a consensus. I guide the students to the correct conclusion if I see the class is not finding the correct answer.
Then, we will discuss Problem 7. Students are asked to put up answers to all the problems they could answer. Many of the groups say this is confusing. I ask a group that completed the first question in problem 7 to show how they did the problem. After the problem is shared I help students learn how to analyze the work. We discuss what the expression is saying. We discuss that area(2x) means that the domain value is 2x. We look at the work on the board and see how the student put 2x in for each x. Now the class analyzes 2 * area(x). I ask what the expression is saying. We look at what the student did to evaluate that expression. I ask why did the student say the 2 expression are not equivalent.
After we do the first pair of expressions we look at the next pair. I give students a minute to see if the expressions are equal. I have a student to answer and explain their response. Students can explain verbally or share work on the board.
The class continues until each pair of expressions have been analyzed. I have students look at the second pair of expressions.
I want students to realize that function notation is not like simplifying an algebra expressions in which the variables are real numbers. As the year progresses I will refer to this activity when we have errors with function notation.
As class ends, I want my students to discuss the following question:
Based on today's activity, do you think that it is possible to multiply two different functions (i.e f(x)*g(x))? Why or why not?
We discuss this for a few minutes. This closure leads to the next section where students see new notation for functions operations and review function operations.