Rabbit Run -- Day 1 of 2

I love using maximizing area problems to talk about quadratics. The focus of today's lesson is for students to write a formula for area if they know the length of a rectangle with a fixed perimeter.  I love this task, and others like it, because students can practice MP 8: Look for and express regularity in repeated reasoning.  This activity pairs nicely with the previous lesson, A Fireworks Display, as it gives students the opportunity to work with a quadratic function in a completely different context.

I begin class by reading the problem, Rabbit Run, out loud with students and letting them ask clarifying questions.  I also introduce some graphic organizers that might help them keep track of their work on the problem.  I like to emphasize that the organizers are optional, and they may, in fact, find a way that works better for them to organize their numbers.

Note: The IMP Year 2 curriculum also has a similar task as this one on page 284 of their Year 2 textbook.

Next, students begin working on today’s task.  I let them work in small groups or alone on this activity.  As they work, I circulate, looking for students that have "ah ha" moments and providing support whenever necessary.  

Issues to watch for:

• If students are having difficulty getting started, I usually ask them:

What are some possible dimensions for a rectangle that has a perimeter of 72 feet?  Let's start by choosing the length.  If the length is (example), how can you find the width?

What if you changed the side that is 20 feet to another amount.  Have the student choose an amount.  Ask what would the width of the pen then have to be? What would be the resulting area?

• I encourage students to notice the repetition in their calculations as they try out different dimensions.  I try to get them focused on starting with a length of their choice and then working with that length to find the width.  This is where the repeated calculations come in.  I try to get students to notice the steps that they are doing over and over again.  In some cases, I write out the steps as the student narrates them orally.
• Once students have tried three different sets of dimensions, you can ask them, what if the length of the rectangle was x? What would you do to x to find the width of the other side? What would do to find the area?
• Once students think they have found the maximum area, I ask them how they can be sure and ask them to prove it to me.   I can make the link back to the previous lesson where they were trying to find the maximum height of the rocket. If they look at values on both sides of the maximum, they should decrease.
• I emphasize working on finding the formula today, rather than making the graph.  If students who finish early have time, they can start the graph, but I usually do that as a whole group at the start of the next lesson.  This allows me to ask students what they think the graph would look like based on the context of this problem.

Note: Instead of Question 5 on the Rabbit Run handout, I ask students to compare this problem to work they did on the Fireworks problem in the previous lesson.

DIFFERENTIATION: I like to provide a A Rabbit Run Organizer for struggling students to help them keep track of their dimensions.  

Students who are more comfortable with this work can focus on comparing and contrasting the puppy pen problem to the rocket problem.