I modified the worksheets we were given at the workshop to create two foldable booklets that could be glued in our interactive notebooks. I was hoping to get through both booklets in one 50-minute class period, but that was super optimistic. In reality, it took 1.5 class periods to complete these problems.
|Oil and Natural Gas Production Profit Booklet Foldables|
Some students offered insights into the oil and natural gas industry that were way over my head. They were throwing around jargon that meant nothing to me. These were the students who have actually spent time on rigs and worked out in the field alongside family members. Other students offered insights such as "Oil smells funny." or "You put gas in your car."
The Essential Question for this lesson was "How do oil and natural gas producers determine if wells are feasible to drill?" Students were to come back and answer this question AFTER completing the lesson.
We also needed to review some important vocabulary before beginning to work with numbers and equations.
As we worked through each of the questions together as a class, we would continually refer back to this vocabulary box. If a question mentioned profit, we would review the definition of profit before even attempting an answer to the question. These terms were new to almost all of my students. They had a really hard time differentiating between operating costs, profit, and revenue.
Day One focused on answering questions related to this data:
Here's the first batch of questions we explored together:
After reviewing the vocabulary, students were able to complete this section with relatively few problems. The function notation proved tricky for several of my lower level students. Many students would just wait until someone in the class provided the answer. I hate this, but I'm not sure what to do about it. I guess I could give everybody in the class a different set of data to work with, but that would probably drive me INSANE!
Having written functions for revenue and operating costs, students could now tackle the topic of profit.
This is where we ended Day One.
Day Two. Your company is now ready to drill a well!
Now, we've moved from discussing one wellsite to comparing two wellsites.
Again, we're practicing function notation. This time, students are told what to name their functions.
Now comes the fun part. Graph the two functions. Figure out where they cross.
This was really hard for me to do, but I did it anyway. I didn't tell my students how to number their axes. I let each class pick. I was risking that some classes might not find a graphical solution. Why do this? Well, this is math. This is real-life. In real-life, every decision has consequences. My students need to learn this. They need to learn that math isn't always pretty. It can be messy. Sometimes you have to try multiple approaches to something before you find the way that works.
Handing my kids a graph with the axes already marked cheats them and me out of a learning experience. My kids need to see the consequences to their decisions. If I choose for them, I'm cheating them out of an opportunity to think critically. To be an effective teacher, I need to know what my students are thinking. I need to know how well they can think critically. If I choose for them, I'm cheating myself out of learning more about where my students are. The more I know about where my students are, the more I can help them get to where they need to be.
In this class period, students decided to let each line on the y-axis represent $1,000. And, each line on the x-axis represented 5 days. We started to graph A(x), the cost of drilling wellsite A for x days. My students quickly identified the y-intercept of the equation, and we marked that on our graphs. The slope was a little trickier. How in the world do you graph a slope of 200? First, I asked my students to rewrite 200 as a fraction. 200 was soon expressed as 200/1. The numerator of our slope represents the change in y of our function. What is the change in y of our y-axis? The distance between each line on our y-axis represents $1,000. The denominator of our slope represents the change in x of our function. What is the change in x on our x-axis? The distance between each line on our x-axis represents 5 days. Now, how can we "unreduce" our slope so that the numerator is a multiple of $1,000 and the denominator is a multiple of 5? Students soon realized that 200/1 is equivalent to 1000/5. Instead of spending a ton of time trying to estimate how to go up 200 dollars and over 1 day on the graph, we were able to quickly and efficiently move up 1000 dollars and over 5 days.
This is a strategy I learned at the OERB workshop. This was a completely new strategy to me, and I kinda sorta fell in love with it. We're always asking our students to reduce the slope. But, I don't think I've ever asked students to "unreduce" their slope before this lesson.
I let the students take more charge on graphing B(x). Again, we had to "unreduce" our slope of 250/1. This class ended up multiplying the numerator and denominator by 10. Going up 2500 dollars and over 10 days was a little trickier since every other dot was not on the intersection of two grid lines. In retrospect, we could have corrected this by multiplying the numerator and denominator by 20.
We found the point of intersection of the two graphs and wrote this solution to the system as an ordered pair.
Next, I asked students to algebraically determine the solution. Since we were only beginning our study of systems of equations, I had to basically set up this problem for my students. Once I set up the equation, they were able to easily manipulate the variables to solve for x. Finding the corresponding cost by substituting the found value of x into one of the cost functions did not occur to them.
Next, students were required to write exactly what this found ordered pair meant. My students struggled with the writing which is proof that I need to require a lot more writing in my classroom. With Common Core, my students need to get used to justifying their answers. Even more, they need to get used to writing in complete sentences. Always using units. Showing their work. We've got a lot of work to do!
Their answers to the essential question were disappointing. But, I think that's mostly my fault for just expecting them to write when I haven't required that of them for the entire year. My students will rise toward my expectations. But, first I must set expectations. This is one of the major things I am working on for this summer. Procedures and expectations. Clearly spelled out. Practiced Daily.
I have uploaded the files for this activity here. Credit for the lesson goes to the OERB. I have slightly modified some of the wording and structure to make it work for my students.
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