I added two components to a lesson task on rational equations.The first was an idea called, Hint Cards, shared by Michael Pershan at Shadow Con at NCTM 2015. The second, which I will discuss in another post, was having students use Desmos to confirm answers that they found algebraically. There were so many surprising positives from this lesson that I have to share.
As always, lesson awesomeness starts with a good task. The class had studied the simplifying, solving and graphing of rational functions. It was time to write and apply them. My school’s Algebra 2 team decided on a common task created by our colleague, Jake Paino, titled Optimum Bait Company. The task offered the following context followed by six prompts.
My brother Matt owns Optimum Bait Company. Optimum Bait Company manufactures fishing lures. The monthly cost to run the factory is $4200 and the cost of producing each lure is an additional $0.25 per lure.
- If he produces 1000 lures in one month, what is the average production cost per lure?
- Create a function, C(x), that models the average production cost per lure.
- Calculate the average production cost per lure if he produces 4000 lures in one month? 8000 lures? 12000 lures? 420000 lures?
- As he produces more lures what price does the average cost of production approach? Why?
- If he wants the average cost of production to be $1, how many lures would he have to produce in one month?
- If he wants to make a profit of at least $4000 per month, what is the minimum number of lures he would have to produce if he sells every lure he produces for $4?
I was thinking that the students would need a lot of help on this, so I created a set of six Hint Cards. Each card gave some assistance for one of the prompts.
Front of Card
|Back of Card|
#1: Average Cost of 100 lures
|Average = Total Cost/Total Number|
#2: Create C(x)
|Let x = number of lures|
#3: Average Cost per Lure
C(4000) = (4200 + 0.25(4000))/4000
|#4: Limit of Average Cost||
The Ratio of the Leading Coefficients
#5: Average Cost of $1
C(x) = 1, instead of x = 1
|#6: Profit of $4000||
Profit = Income – Expenses
As an incentive, I announced the following scoring system.
- Like all other tasks, this will be worth 5 points.
- There are 6 prompts. Every wrong answer to a prompt costs a point.
- There are also 6 hints. Every hint used costs a point.
- Yes, that means you either have one free pass on a wrong answer, or a free hint.
- The only thing that you may ask of the teacher is for a hint card to a specific prompt.
- 30 minutes will be allotted to complete the task.
I was pleasantly surprised how little they needed or wanted help. Many groups didn’t even take advantage of their free hint. In fact, for all of eight groups, I gave out only seven Hint Cards… total. The most common hints asked for were for #1 and #5. The most commonly missed questions were the last two. I suspect that if I had given more time on the task, students may have ask for more hints and given more effort on what I consider to be the hardest questions for the task. That’s a lesson for next time; there will definitely be a next time because of the benefits that resulted from the Hint Cards technique:
- The time crunch spurred a hyper-focus in the students.
- The level and intensity of the student discourse was heightened tremendously.
- A common dynamic was having one student raise a hand for a hint, while another group member protested, “we don’t need it, yet.”
- The average score on the task was 4.2 out of 5.
Hint Cards delivered a terrific learning experience for my students. One that I will be sure to give them again.