Week+3+-+draft+for+performance+task

 Week 6 - performance task - ALYSSA KRAUSS


 * STAGE ONE: Desired Results **

TOPIC: Optimal forms for supermarket packaging – the relationship between the radius and the surface area of cylindrical containers.

This task is intended for high school calculus students. The task involves having students create a table of values to see which values for height and radius minimize surface area for a fixed volume. Next students solve for surface area in terms of radius (by using the volume formula for a right circular cylinder and making a substitution). Using the equation they just found they then use graphing software to plot different values for radius and find the optimal radius. I would start this assignment in class, by reviewing the concepts needed to solve it, and then have students work on it (at home) for a week – at the end of which selected students would present their solutions to the class.

ESTABLISHED GOALS:

The California standards that apply to this lesson are the following:

CA Standards for Calculus 11.0 Students use differentiation to solve optimization - maximum  and minimum - problems in a variety of pure and applied contexts. CA Standards for Algebra II 25.0 Students use properties of number systems to justify steps in  combining and simplifying equations.

 The task serves as motivation for the Calculus Standard 11.0, and reviews the content from Albegra II Standard 25.0 (which is a necessary skill for Calculus Standard 11.0).

ESSENTIAL QUESTIONS: Can we minimize surface area for a given volume? How? Can we maximize surface area for a given volume? <span style="font-family: Georgia,serif;">What are some real-world applications of optimization problems?

<span style="font-family: Georgia,serif;">ENDURING UNDERSTANDINGS: <span style="font-family: Georgia,serif;">Students will gain experience in using technology (graphing calculator) as an aide in problem solving. <span style="font-family: Georgia,serif;">Students will understand relationships between two formulas and how to solve for one in terms of the other. <span style="font-family: Georgia,serif;">Students will gain experience working with volume and surface area and learn how the two measurements are related to each other. <span style="font-family: Georgia,serif;">Students will construct charts to understand how surface area changes as radius changes. <span style="font-family: Georgia,serif;">Students will have a strong understanding of maximum and minimum problems work – this understanding will be the foundation for extrema problems in calculus and the mean value theorem.


 * <span style="font-family: Georgia,serif;">STAGE TWO - Planned Assessments (acceptable evidence) **

<span style="font-family: Georgia,serif;">In order to assess learning of the important concepts listed above, the following assessment is proposed as a homework assignment:

<span style="font-family: Georgia,serif;">Many people are invloved in deciding how to package the products yoou see in the grocery store. Packaging engineers select materials and package shapes to adequately protect the product through shipping at a resonable cost. <span style="font-family: Georgia,serif;">A container's shape, as well as its material, is important in determining it's strength. From an engineering perspective, the sphere is the strongest form, followed by the circular cylinder. The rectangular box comes in a poor third. From a perspective of cost, it is preferable to use the smallest amount of material possible. <span style="font-family: Georgia,serif;">The table gives approximate measurements, in inches, of several common items packed in cylindrical containers.

An infiinite number of dimensions can be used to constuct a right circular container of a given volume.
 * Product || Radius (inches) || Height (inches) || Volume (cubic inches) ||
 * coffee creamer || 1.50 || 6.85 || 48.42 ||
 * cleanser || 1.45 || 7.50 || 49.54 ||
 * coffee || 1.95 || 5.20 || 62.12 ||
 * pineapple juice || 2.10 || 6.70 || 92.82 ||
 * frosting || 1.63 || 3.60 || 30.05 ||
 * soup || 1.30 || 3.80 || 20.18 ||
 * tomato puree || 1.95 || 4.40 || 52.56 ||
 * baking powder || 1.25 || 3.65 || 17.92 ||

QUESTIONS 1, Create a table of values for the dimensions of a cylinder with a volume of 49.54 cubic inches. Does it appear that the cleanser container minimized surface area? 2. Suppose that you are designing a coffee creamer container that has a vloume of 48.42 cubic inches. Use the equations for the surface area of a cylinder and the volume of a cylinder to develop an equation relating the radius, r and the surface area S. //S = 2(pi)r^2 + 2(pi)rh// V = (pi)r^2h 3. Repeat question 2 for each of the other containers in the table. Use a graphing utility to plot each equation. Determine whether the radius of each container is larger than, smaller than, or equal to the "optimal" radius. 4. Suppose, in order to fit more writing on the cylinder, you wanted to maximize the surface area of a cylinder that holds 49.5 cubic inches. Can you do this? Explain. +++++This problem was taken from //Calculus, 6th Edition, by Larson, Hostetler, and Edwards//.

<span style="font-family: Georgia,serif;">This assessment will serve as good evidence of learning, and it is very closely anchored to the learning goals. The assessment will be graded using a scoring rubric (which follows below) where students will be able to earn a maximum of 12 (10 points plus 2 extra credit) points. The scoring rubric provides a fair and consistent means by which a student can be assigned a grade. The rubric will be provided to the students along with a copy of the assessment, and ensures that they know what is expected of them; it also gives them the power to succeed. Students are expected to complete this assessment alone, for homework. The teacher will be available by email to answer any questions that students may have when they are working at home. The given timeframe for this assessment is one week. The assessment will be assigned on a Monday and will be due the following Monday in class. It serves as a review for a variety of skills taught in Algebra II, but also allows students to see and utilize real-world applications of mathematics.

<span style="font-family: Georgia,serif;"> || <span style="font-family: Georgia,serif;">Some work was done to derive the formula, but the formula is incorrect. || <span style="font-family: Georgia,serif;">Formula is either absent, or no work was shown. ||
 * <span style="font-family: Georgia,serif;">SCORING RUBRIC || <span style="font-family: Georgia,serif;">2 points || <span style="font-family: Georgia,serif;">1 points || <span style="font-family: Georgia,serif;">0 points ||
 * <span style="font-family: Georgia,serif;">Question 1 – table constructed || <span style="font-family: Georgia,serif;">Table is labeled correctly and shows correct computations. || <span style="font-family: Georgia,serif;">Table is incorrectly labeled but calculations appear to be correct. || <span style="font-family: Georgia,serif;">Calculations are incorrect. ||
 * <span style="font-family: Georgia,serif;">Question 1 – question answered? || <span style="font-family: Georgia,serif;">Correctly with explanation || <span style="font-family: Georgia,serif;">Correctly, no explanation || <span style="font-family: Georgia,serif;">Incorrectly ||
 * <span style="font-family: Georgia,serif;">In question 2 – developed appropriate formula. || <span style="font-family: Georgia,serif;">Formula is present and correct. Work is shown.
 * <span style="font-family: Georgia,serif;">In question 3 - are calculations correct? || <span style="font-family: Georgia,serif;">All calculations are correct. || <span style="font-family: Georgia,serif;">Some calculations are correct, but some are not. || <span style="font-family: Georgia,serif;">Calculations are either not present or are altogether incorrect. ||
 * <span style="font-family: Georgia,serif;">Question 3 - are the graphs complete? || <span style="font-family: Georgia,serif;">Yes, the graphs are present, complete and correctly identified as larger than, smaller than, or equal to. || <span style="font-family: Georgia,serif;">Some issues with the graphs, but overall a good effort. || <span style="font-family: Georgia,serif;">Graphs not present or are present but do not answer the question posed. ||

<span style="font-family: Georgia,serif;">***BONUS: Up to two extra points for answering the 4th question correctly and providing a reasonable explanation.