Keystone Algebra I - The Bowl Problems

There is this problem in our Algebra 1 state exam, well it's in the sampler I'm not allowed to actually see the exam.  Yesterday my Algebra 1B students took their assessment on writing linear equations, just the boring stuff no applications.  So for XP points (read about those here) I gave them the bowl problem.  I explained that it was something I didn't teach directly yet and that it was challenging, but I wanted to see what they could do with it.

My Method (very typical):

I create two ordered pairs (1, 2) and (5, 5) where x is the number of bowls and y is the height of the stack in inches.

I determine the slope: 3/4

and I find the y-intercept: 5/4

So my equation is y = 3/4x + 5/4

Then the height of a stack of 10 bowls is y = 3/4 (10) + 5/4 or 8.25 inches.  

Lily's Method:

The point of this post is to share the work of one of my students, Lily.  

I saw that her equation was y = 3/4x + 2 with no work, she says that she did it all in her head (I did see her working on it without writing anything down if that makes sense).  

So, I marked her y-intercept incorrect.

Then I saw how she labeled her variables:

x - variable: "The number of bowls on top of the first bowl."

y - variable:  "The height of the bowls."

And I crossed off her "on top of the first bowl" part because it wasn't the way I did it.

Then her answer for the height of 10 bowls was correct and I was like, "Wait? What?"  She plugged 9 into her equation instead of 10 and that's when it finally sunk into my thick skull.  She was composing functions!!!  Here is her work (and my comments) if you are interested.