I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Modeling Volume with Polynomials Day 1, which asks students to model the volume of a rectangular prism with a polynomial.
I also use this time to correct and record the previous day's Homework.
In this lesson, each pair of students needs 10 pieces of quarter inch graph paper, scissors, a ruler, and tape. This entire lesson engages students in mathematical modeling (Math Practice 4). Our goal is to create a box with the largest possible volume given the paper provided. To construct the box, students cut out congruent squares from each corner of the box and fold up the sides. Modeling the technique is important as many students still lack the spatial sense to visual this from the written directions or are inexperienced with building things.
To get the conversation rolling, I take the model and ask, "What size cut out square will give us the largest volume?". Hopefully someone suggests building models and measuring the volume. Someone who suggest writing an equation (which is skipping ahead) – in which case I'd let them know that we are going to start with a physical model first, bringing up the fact that scientists use physical models to investigate before jumping right to equations.
The students then create boxes with cut out squares at 1/2 inch intervals. So the first box will only have a 1/2 x 1/2 square cut out each corner, the second will have 1 x 1 cut out of each corner, and so on. Using 8.5" by 11" paper the provides 8 models. I walk around and help/encourage students. I thought long and hard whether I wanted my students to make all of these in pairs or whether they should break up the work and decided that it is worth it to have students physically build these models for themselves (Math Practice 1).
Inspiration for the modeling problem came from:
Lial, Margaret L., E. John. Hornsby, and David I. Schneider. "3.4 Polynomial Functions: Graphs, Applications, and Models." College Algebra. Boston: Pearson/Addison-Wesley, 2005. 331-49. Print.
Here we jump from the geometry to the algebra. The students will now create a chart relating the side length of the cut out square to the volume of the box. I put both the name and the algebraic expression onto the chart since this will help students make the connection between the physical model and the graph they are about to make. This portion shouldn’t take very long. The students check their charts with the people sitting near them when they finish ensuring accuracy. We then share as a class.
At this point, each student will create a graph which is the purpose of the final two pieces of graph paper. I encourage tidiness and clarity in their graphs (Math Practice 6). I also ask that they fill the whole page. The students may ask to be given the domain and range but I encourage them to reason through it (Math Practice 1) by giving hints like “What does the x-axis represent?” or “How large can the square be?”.
Once they have drawn their curve through the points, we discuss the contextual meaning of the graph. The goal is that they realize that the curve represents ALL of the possible volumes for the box. I may also ask them to identify the total number of volumes it represents. This can lead to some good discussion on the infinite nature of numbers as well as the real verses hypothetical situations.
In addition to modeling with polynomials, another goal of this lesson is to identify some major features of a polynomial graph. I struggled with how to introduce relative minimum or maximum and some of the other features in a meaningful way in this unit. This lesson fortunately does just that. This section may run in to the second day of this lesson which is perfectly acceptable.
First the students identify the x- and y- intercepts as well as the domain and range and describe how they are meaningful to our volume problem. Students do a think-pair-share for each of these (Math Practice 2).
Next they identify the maximum for the graph. The maximum is super important to this entire problem. It represents the largest volume. Some students may ask WHY it is a relative maximum. I let them know that with the domain restriction, we are only analyzing a small part of the graph and there may be other places on either side that are larger.
Finally, I lead a discussion about increasing and decreasing the intervals. The students use arrows on the graph itself and use interval notation to identify these segments. Again, they need to determine how this is meaningful to our model.
Please see the PowerPoint for detailed presentation notes.
I use an exit ticket each day to provide a quick formative assessment to judge the success of the lesson.
This Exit Ticket asks students to predict the degree of the polynomial that can be created from our volume model. This gives me a good idea of the level of the preparedness students will come with in the second part of this lesson .