Wow! What a big and open topic. I look at the examples given at Illustrative Mathematics
Don't forget to PIN this page for future reference!
and it was not what I would have
come up with.
Katy is told that the cost of producing x DVDs is given by C(x)=1.25x+2500. She is then asked to find an equation for C(x)x, the average cost per DVD of producing x DVDs.
She begins her work:C(x)x=1.25x+2500xand finishes by simplifying both sides to get:
C=1.25+2500xIs Katy's answer correct? Explain.
Dig Deeper
So I dig deeper into the meaning and I reread. I see how their example fits the standard, but I want to be able to see this and create it myself.
I believe that all most teachers are in this same boat. We are immobilized by the unknown. For so long we have understood exactly what our kiddos need to do well. We know what the tests look like. We understand the standards. And now.... There is too much unknown. It is overwhelming.
Breaking it Down
And so I begin breaking everything down. By Unit, by week, by standard, by assessment question.It gives me order and control and I feel a bit better. But I want to be careful not to forget the big picture of the Mathematical Practices. That is why I added them to my main unit pages. It is an instant reminder of the bigger picture.
I do not want to fall back into task teaching, although kids need to understand tasks. More importantly, I want to teach kids how to think, analyze and problem solve for their lives, not just in my class. If we can remember the big picture and focus on our big goals for our kids and apply it to Interpreting Functions we can achieve success for ourselves and most importantly our students. So lets jump in....
Interpreting Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
First comment. Regarding the original Katy problem. I either do not understand the notation used OR I disagree with the Math. If the cost of producing x DVD's is given by the function C(x) = 1.25x + 2,500, then I believe the average cost of producing x DVD's should be represented by the function (C(x))/x = (1.25x + 2,500)/x which could be simplified to
1.25 + 2,500/x. (Sorry for improper symbols and notation.)
Second, I believe students should master this concept as described in the CCSS F.1F.1, but I do not think too much emphasis should be placed on proper notation. If a student understands the concept, can explain it and use it solve real world problems, then why should they have to know the notation or symbols?
First of all, when you simplified, it was incorrect.
Second of all, students need to understand the notation is they want to move on to calculus in the future. In the real world scenarios, the notation may not be used, but on a literal level students need to understand that certain things need to be written in a way that other people can understand. Many professions have short hand which others in their profession understand, but if they were to write out their description of the function, it could be misinterpreted, confusing or just plain lengthy. Math uses notation to express precise shorthand, which is used in many professions.
1) Katy is not correct; Mathmanmm's equation is correct.
2) Knowing the proper notation allows students to understand and solve questions on standardized tests.
3) If x was zero, there could not be an average cost.
Which makes sense, because if you don't make any, you shouldn't have any cost...
I think it would be good if Quinn explained why the simplification was incorrect. I think it is because you can't divide by x because x could be zero.
I always showed my students that a function is where you take the relationship (i.e. (x,y) which is really x=input and y=output) and mention that the relationship could be a function. A function is where for every input (x) there can be one output (y).
I use the analogy of putting a dollar in a vending machine. That is the input. There can be only one output for this (i.e. one soda). Another example I give is if two students were working at the same job (say $10.00/hour) and they worked the same hours that week. It would not be fair (nor a function) if one person made more than the other (i.e. two different outputs).