This is a companion discussion topic for the original entry at http://studio.code.org/s/algebraFacilitator/stage/2/puzzle/1
Because the CS in Algebra lessons have a fairly easy learning curve, I haven’t had many problems with students understanding. The biggest challenge is getting students to think about the definition in terms of the variable. My students can often write 2 perfect examples, but they have a hard time abstracting those changes and inserting their variable. Those students that struggle, we make a quick function machine. I don’t want the answer, I want the problem.
For example:
Me: Your example says timesten (30), what’s the problem?
Student: 30 * 10
M: How about timesten (43)?
S: 43 * 10
M: timesten (976)?
S 976 * 10?
M: timesten (x)?
S: x * 10
M: That’s your definition, write that down.
 Students sometimes have difficulty understanding the reasoning behind why f(x) is equal to y if they lack the understanding of the fact that if y depends on the value of x, it is a function of x.
 Students also often struggle with the reason why or why not a specific relation is a function. Teachers spend a lot of time discussing rules such as the Vertical Line Test, etc. but this hinders the students from understanding the meaning behind a relation being a function or not. Overall, I would like to encourage teachers to focus more on the conceptual understanding of functions, and how that focus can increase student achievement.
 Students should practice more in modeling contexts that allow them the opportunity to view applications of the content in order to deepen their understanding.

When functions are introduced students will sometimes struggle to make the connection between
f(x) = and y =. They struggle to join the fact that y is a function of x, or dependent on x. 
Another conceptual piece that students struggle with is the fact that for every x value there is only one y value. It is important to help students make the connection with the fact that if there was more than one value of y per x we would never be able to tell what the value of y would be for a specific value of x (for example when x = a there is only one value for y). Teaching the procedure of checking functions graphically using the vertical line test, leaves gaps in the conceptual understanding of functions.

When teaching functions in a realworld context, students sometimes struggle to see when a function may no longer be effective in making projections for the future. In the case of the relationship between investing in advertising and its effect in revenue, there will eventually be a maximum yield in profit when considering realworld situations.
A little cheesy but… “The x’s and y’s are in a relationship. In order for their relationship to function, work out, the x’s must relation loyal to the y’s. In order words, the x’s cannot cheat on the y’s. Look at this tables and tell me if their relationships will function…”
If I teach the vertical line test without having the students write the coordinates where their vertical line intercepts, they have a hard time understanding why it works. If they write the coordinates, they’ll see that the x’s repeat in the domain.
I thought this article hit a nail on the head for teaching Algebra in 8th grade. It has always been a frustration to help a student, only to have them say “What is a function?”
I completely agree with 95999 Tabatha that we need to focus on conceptual understanding of functions.
how will we teach the kids a lesson