Note: Students began their work on the Rabbit Run activity in the previous lesson.
I ask students to take out their work from yesterday and share out some of the possible dimensions they found for Rabbit Run problem. I have students enter some of their possible dimensions in a whole class table on the board. I use the A Rabbit Run Discussion PowerPoint to guide the discussion.
A key piece to starting the discussion right is getting students to notice some things about the table results. I motivate this part of the discussion by focusing on where the maximum area of the run occurs. Students will see that when the length is 18 feet, the run will have the largest area. I like to ask them how they can be sure that this is the maximum. From there, they should be able to articulate that increasing the length to a value greater than 18 decreases the width, and vice versa.
I expect some of my students to notice the symmetry in the areas around the maximum. If they do, I will highlight this point so that we can make the same connection with the graph of the parabola later.
This is a great lesson for students to practice SMP8: Look for and express regularity in repeated reasoning. When students are working on the Rabbit Run problem, as they change the length of the side of the rectangle, they are repeatedly going through the same steps to find the resulting width of the rectangle and, ultimately, the area. I will both observe and guide students mathematical practices as they undertake the investigation.
There are different ways to guide students toward the expression. You might try:
Ultimately, I find this content lends itself well to a Silent Lesson. The reason this lesson is so great to explore “silently” is because of the repetition that is inherent in the steps. Instead of talking about this repetition, a silent lesson enables students to see the repetition and the pattern that emerges. It is then easier to make the jump to using a variable for the initial width instead of a number. Students have already seen they do the same thing each time with a number, now they are ready to generalize to a variable and write an algebraic expression.
My students have responded very positively to Silent Lessons. I often solicit feedback from them after a silent lesson to get a sense of how it went. Overwhelmingly, students report that they are better able to concentrate and focus when the lesson is silent. They also report that they like to follow the lesson on the board and look for the pattern. I find they take pride in coming up to the board and writing the next step. I am careful not to let the silent lesson run too long (no more than 25 minutes or so). Students seem to find silent lessons new and exciting!
To begin the Silent Lesson, I divide my board in two: one side showing a diagram of the rabbit run and the other side showing the calculations I did to get the area. I start off with an example, and then label a new rectangle with a new length. Then I pass the marker to a student and motion for him/her to show the calculations. I do that one or two more times so everyone gets the hang of it. Then I go back to the board, draw a rectangle and label the length "L". Make a thinking expression and see if any students can come to the board and begin to write an expression for the width in terms of the length.
Check out this video to see this silent lesson in action: Rabbit Run Silent Lesson.mp4
(Please note that there is not much audio to this video as it is a “silent” lesson!)
After the Silent Lesson, I continue with the Rabbit Run Discussion PowerPoint. My goal is for my students to identify the similarities between the Rabbit Run problem and the Fireworks Display problem. I try to elicit as many similarities as I can from students before giving them the notes and a basic introduction to quadratic terminology.
Students have spent the last two days working on similar problem situations that involve quadratics. You’ll want to give them some time at the end of class to let this new content sink in. You could have them write individual reflections or you could have them “turn and talk” to a partner. Some reflection questions you might consider are:
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