Differentiated Lesson: Systems of Equations

Today in class I anticipated I'd have a range of abilities with what we are doing in wrapping up a multi-day lesson. I have become fully dependent on the half class set of chromebooks on days like today. I would even say that I am happy that I do not have a 1:1 ratio in my classroom because of the conversations rooted in the need to share and discuss their thinking. 

Regular access to classroom technology is crucial for this day to happen in the way that it does.

We have been working toward  solving systems of equations in algebra, the last 2-3 days have been spent on this algebra-like investigation.

As part of the review day, students were checking graphical solutions to a system of equations on this worksheet ("websheet"?) from http://www.mathematicsvisionproject.org. I asked students to do their algebra check on paper but I wanted them to graph (mostly) on Desmos (yay!) and at least once on a graphing calculator (...for future college board tests...meh). 

On the second page of the MVP math sheet, #5 says:

"A theater wants to take in at least $2000 for a certain matinee. Children’s tickets cost $5 each and adult tickets cost $10 each. The theater can seat up to 350 people. Find five combinations of children and adult tickets that will make their goal."

I challenged the class to also come up with some equations that represented the situation and to think about what could be replaced by variables. I did not expect everyone at this point in our learning cycle to do so but was very pleased with some results I will share below. Other students in the class that weren't ready just found the 5 possible combinations, still laying the groundwork for next class when we discuss the problem as a group. Students could also get additional practice on this web app if they felt they needed it.
I was most pleased with a group that not only went to desmos, I was able to start talking to them about the inequalities they had written, which regions created contained points that were considered solutions and if they needed to add any (constraints) to better define the solution region. At the end of class they were excited and were going to continue working on it tonight even though they did not have to. (Here is what they made as an extension to the word problem shown above.)
Thank you Desmos for making free tools like this for math teachers. I really like that students can graph equations in standard form (not just in the form y=mx+b). It helped in building understanding for these students as they were looking at the solution area of their graph. On a related note, you should also check out their classroom activities page.