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 titled Optimum Bait Company*.* I’m not sure where the task came from, but it 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.