Name: Becky Rubenstein

email: [email protected]

School:

Position: h.s. math teacher

Title: Related Rates

Subject: Math

Topic: Related rate, implicit differentiation, regression equations

Grade: High

Comments:

Invitation

Frames the teacher's curriculum design challenge. Let’s determine the rate at which our class can perform the "Wave". We’ll time ourselves using 2-second intervals and after performing several trials will collect this data in a table. Our independent variable will be time and the number of people up and waving will be dependent on time. Then, let’s plot the points using our calculators and calculate a linear regression. With this information, we now can use the derivative to find our class rate of doing the "Wave". Now, individually, let’s see if you can calculate a rate of your own.

Tasks

Actions the students will be asked to undertake. Consume a Tootsie Roll Pop (we’ll assume it’s a sphere) and measure its radius before you begin and every 30 seconds thereafter until you reach the soft center. Try to be consistent in your efforts to eat the TRP. Using your graphing calculator, plot these points and calculate a regression line. This is your own personal rate of consumption of the TRP. Next, find an equation relating volume of a sphere with its radius. Use your knowledge of calculus to differentiate this equation. You objective is to find the rate at which the volume of the TRP is changing when the radius is .75 of its original size.

_000200000C7E00000627_C78,Assessment

Criteria by which the students work will be evaluated.

Prepare a report that will include your data, graphs, equations, and all calculations that you performed. Discuss the reasonableness of your conclusion. Were you surprised at the rate at which the TRP gave up volume? Was it more or less than you had anticipated? How might you have changed the outcome?

Standards

Frameworks developed by the school, district, or state as guidelines in the development and assessment of curricula. Since this is an Advanced Placement course, the topic of Related Rates is part of the College Board’s curriculum and will be included on the AP Exam in the spring. My goal is to prepare my students for this exam as well as for success in a subsequent college calculus class. The State standards that are addressed in this UOP include: 1.Using numbers in problem-solving situations and communicating reasoning. 2.Algebraic methods are used to explore, model and describe patterns and functions involving numbers. 3.Students use data collection and analysis in problem solving situations. (Using appropriate technology) 4. Students use geometric concepts and relationships in problem-solving situations. 5. Students use appropriate methods for computing with real numbers in a problem solving situations and determine if the results are reasonable.

Situations

Places where the activity will take place, and the amount of time and the periods of time that the students will have to work on the activity.

The initial activity will take place in the classroom with all students participating in the "Wave". Students will record data on their calculators and calculate a regression line. Consumption of the TRP and preparation of a report will take place outside of the classroom. The report will be due the following week.

Interactions

Way the students will work, the ways the teacher will work with the students, and the ways the students will interact with others. The way students will work, the ways the teacher (and aides, et al.) will work with the students, the ways the students will interact with others. Students will work together in class while performing the wave. Calculations will be made as a group with the teacher also modeling the steps necessary to calculate a line of regression. Discussions will take place in class on approaches to solving the problem. Students can determine the best way to find the radius of a sticking moist TRP.

Tools

Tools that the students will use to approach their tasks. The tools that the students will use to approach their tasks (pens and notebooks, computers, the encyclopedia, specific software). Students will need to have a graphing calculator, some measuring device (ruler and string, or calipers). Students can either neatly write their reports or can word-process their results. Graph paper will also be helpful or a computer program can be used to create their graphs. Math textbooks can be used to help with formulas and dimension analysis.