# Hint Cards

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.

1. If he produces 1000 lures in one month, what is the average production cost per lure?
2. Create a function, C(x), that models the average production cost per lure.
3. Calculate the average production cost per lure if he produces 4000 lures in one month? 8000 lures? 12000 lures? 420000 lures?
4. As he produces more lures what price does the average cost of production approach? Why?
5. If he wants the average cost of production to be \$1, how many lures would he have to produce in one month?
6. 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.

# Mr Cornelius’ Desmos Lesson

This lesson on graphing conic sections rocked on multiple levels. For the students, it involved concrete mastery of standards, conceptual understanding of several topics, higher order thinking skills, student autonomy and intellectual need. For the teacher, Mr. Cornelius of Great Oak High School, it was a week’s worth of experimenting with new software and pedagogy. The genesis of the lesson was a combination of an email and a diagram. I had sent to my Math Department a link to the free online graphing calculator Desmos.com; a mutual colleague, Michael White, shared the idea of having students use their knowledge of equations to graph a smiley face. Mr. Cornelius merged these ideas into a new 5-day lesson in the computer lab. That week produced a multitude of pleasant surprises.

Michael started with a whole-class demonstration of Demos at the end of the period on a Friday. He posed the Smiley Face graph (shown above) as the minimal requirement for passing the assignment. The strength of this lesson is two-fold: 1) There are a variety of equations involved (circle, ellipse, parabola, absolute value, as well as linear), and 2) repeated restriction of the domain and range.

Michael invited students to create their own designs for a higher grade. He expected only a few takers, but in the end only a few decided to produce the Smiley Face, and this is where the richness of the lesson was truly found. During the week-long lab session, I observed one of the days and took a few pictures of some works-in-progress.

As you can see, the students independently chose to include inequalities in order to produce the shading. Here was my favorite use of shading.

What really impressed me about the lesson was the examples of students who asked to learn something new in order to produce something they chose to create. In the example below, a student wanted a curly (wavy) tail for her pig. Mr. Cornelius taught her how to graph sine and cosine waves. Granted, this was a superficial lesson, but to see someone wanting to learn a skill from next year’s course was a treat.

The rigor that the students imposed upon themselves, again as demanded by their creative idea, was remarkable. Look at the detail of the door handle on this house.

My favorite moment was this one with Michael and a handful of students. It is not as sexy as the pictures that the students were producing, but it was far more significant. Three students all had a similar question, so Mr. Cornelius conducted a mini-lesson on the board while the rest of the class worked away on their graphs. The topic on the board was not part of Michael’s lesson plan. It was sheer improvisation. For me, this interaction was the treasured gem of the lesson experience: A teachable moment generated by an intellectual need.

This was the first run of Michael’s lesson and in a conversation that we had while he was grading the assignments he conceded that he needed a scoring rubric. We also discussed how this idea could be woven throughout both Algebra 1 and 2 courses. The idea of Graphing Designs could span linear, exponential, quadratic and conic equations. I smell a lesson plan brewing!

(P.S. For those of you that get hooked on Desmos, I suggest you also check out the Daily Desmos Challenge)

# Lesson: The Tortoise and The Hare

### Who wins this race depends on careful manipulation of data, algebraic equations and graphs.

SUBJECT: Algebra
TOPICS: Writing, graphing and solving systems of equations. Rate and unit conversion
PAGES: 3