Paging Doctor Math: Tile Floor Task

I'd like to start this post with two disclaimers:

1. I don't pretend to know anything about laying tiles or carpentry of any sort. If anything in this problem is inaccurate to how it would be done in real life, feel free to let me know and I will update it...

2. I normally give credit to the creators of these tasks in all of my posts. I appreciate everything these math professionals do to make my job easier and my students' education better. That being said, I have used this task for a few years and I do not know where it came from. I know I modified it in some way, but I don't know its origin. Please fill me in if you know where this came from.

Back to business...of laying tiles. The idea of this task is for students to come up with an answer that would serve as an approximation for how much it would cost to tile the basement in the diagram provided. Here is what the students received at the start of class...

Tile Floor Task Info

I did this task a few times previously where I did not provide a floor map that students could draw on...I just gave them the picture on the computer. I learned that many of my students had trouble accurately drawing the diagram for use (as would I), so I made a few copies for each student. This task for the most part has one answer, but the variation comes in when the students try to decipher the square footage. There are multiple ways to divide up the floor, and students exhausted every possibility.

We had not yet discussed how to find the area of a triangle or trapezoid, so it was interesting to see how the students maneuvered around the roadblock at the bottom of the basement diagram. Some students took one of the triangles and placed it on the opposite side forming a rectangle. Others put the two triangles together to make a square. I enjoyed the creative aspect of something I haven't explicitly taught that can still be accomplished. Many math teachers have the debate as to whether these tasks should introduce new concepts or stick with applying previous knowledge...I say both.

One interesting misconception that I see common in 6th graders was on full display in this task..."should I multiply or divide?" This seems so easy to me that it's almost hard to explain to the students...

Example where you should multiply: 176 tiles multiplied by $17.50 per tiles to get the total price of the tiles. My advice to students...1 tile is 17.50, 2 tiles are $35.00, 3 tiles are $52.50 and so on...

Example where you should divide: 176 tiles divided by 22 tiles per hour to get the number of hours worked by the guy laying the tiles. My advice...start with 176 and see how many group of 22 are available (some students used repeated subtraction for this)...

Both of these methods take forever...but I felt most students came away with which of the two operations should be used before they had to completely finish the longer method.

Another aspect of these tasks that is probably overlooked from a teaching perspective is the amount of preparation the teacher must make to be ready to discuss all of the possibilities (correct or incorrect) a student may present. I like to hope that every student comes up with the correct path on attempt one, but that is rarely the case. I probably have 1000 different conversations during the hour given during class for these tasks, and all the conversations are a little different. As a teacher, you really have to be a flexible and quick thinker to keep up. This gets easier with experience! Here is what I did to plan for this lesson to make things easier...you'll never think of everything but it certainly helps (I do this for all my tasks and store the pics on my iPad Mini which I carry around during class):

Note: Ignore the bottom of the first picture. I accidentally wrote that the 396 was the tiles instead of the square feet, and I noticed my mistake mid-task and had to quickly get a replacement page ready on the go...

Finally, here are the Common Core standards this task focused on along with student work. Notice none of the standards list learning extensive knowledge of the carpentry profession :-)

6.RP.A.1

6.RP.A.3.D

6.G.A.1