Quadratic Function Think Aloud

The problem in the following blog post can be found in the April 2014 issuse of Mathematics Teacher.

The first think I see when I start this problem is the the function f(x) has three unknowns, x, b, and c. So right away I know before I can find f(4) I first need to determine what the values of b and c are. I also see that I am given two other pieces of information that might help in figuring out the values of b and c in this situation. The first thing I am going to do is use the fact that f(1)=9 and I am going to substitute 1 in place of x in my function f(x). 

So from this information I have found out that when x=1 then f(x)=1+b+c. I also know that f(1)=9 because I was given that information at the beginning of the problem. 

 Now I see that I still have two unknowns b and c, so I still need to use the other information I have to determine the values of these unknowns. 

I was given at the beginning of the problem that f(3)-f(2)=8. Even though I do not know exactly what f(3) or f(2) equal I do know that their difference is equal to 8. So now in the same way that I found f(1) earlier I am going to find f(2) and f(3). 

So now I know what f(3) and f(2) are equal to in terms of b and c, and therefore I can find f(3)-f(2). 

So now I have found that f(3)-f(2)=b+5. I remember that I was also told that f(3)-f(2)=8. 

I have a value for b, this is progress! Now that I know that b=3, I can use this information and substitute b into my earlier equation, b+c=8. (Remember that I found b+c=8 when I found out that f(1)=9) 

So now I know that b=3 and c=5. Since I know these values I can "complete" my function f(x) by subtituting in these values for b and c. 

From here I can easily figure out the value for f(4) by substituing 4 in for x in my function f(x). 

So I have finally finished my problem and found out that f(4)=33 when f(1)=9, and f(3)-f(2)=8. 

If I were doing this problem in my classroom here are some follow up questions I might ask:

1. Do the values of b and c change if we were given that f(1)=3? What if f(1)=-9?

2. How does the value for f(4) change if we were given that f(2)-f(3)=8?

3. Graph the functions f(x)=x^2+3x+5 and f(x)=x^2-3x-5 on the same graph. What happens to our original graph when b and c are both negative? 

The Meaning Behind the Numbers

It took me a long time before I realized I wanted to become a teacher. I should start out by saying I come from a whole family of teachers, and not just any teachers they are all math teachers. My grandparents are retired college math professors, my mom was also a college math professor and now  is teaching preschool in Ohio, and my dad is a 5th grade math teacher. So the oblivious choice for me to make would be to become a math teacher right? Wrong. Until last year I wanted to do everything but teach math. I wanted to go to med school, I wanted to be a psychiatrist, I wanted to be a biomedical engineer and design artificial organs and most recently I wanted to be an occupational therapist. After five major changes in two years I called my grandma and told her I had no idea what I wanted to major in and more importantly what I wanted to do for the rest of my life. She gave me some of the best advice I have had to this day, she said "stop running from your calling". I knew that she meant my call to be a math teacher, she always said it was "in my blood". So long story short, listen to your grandma because she always knows what's best for you. 

Once I did decided I wanted to be a teacher and found my passion in education I started thinking about what type of teacher I wanted to be and what I wanted to value in my classroom. Since I want to be a math teacher I knew that math absolutely had to be the highest priority in my class. Every teacher wants to change the lives of their students, they wants to make a difference. I knew, and still know, that I want to make a difference in my students lives through the math I teach. I want to make math approachable and realistic to my students. Everyone has heard (or maybe even been the one to say) "when will I use this?" in a math class. My goal as a teacher is to show my students how the math they are learning is important and that there is a reason why we are learning this. I want my students to see through projects and activities there is more to math that just numbers on a lined sheet of paper, or solving for some unknown, that math is about problem solving and reasoning. 

In my classroom I want to use activities like "101 Questions" by Dan Meyer or "Would You Rather Math" to  engage my students. I don't want them to just give me an answer I want them to tell me how they got there and why they believe that is the right answer. I want my students to learn how to problem solve and collaborate. I want them to learn how to explain their thinking, and what to do if they are struggling with a problem. If my students do a problem incorrectly I want them to be able to go back to what they know and try the problem again a different way. Life is so much more complex than simply solving problems from a textbook and therefore my classroom will be more complex than solving problems from a textbook.