Jill Manning
Course: Precalculus
Information for these lesson plans came from the following sources:
Functioning in the Real World, A Precalculus Experience by Sheldon P.
Gordon, Florence S. Gordon, B.A. Fusaro, Martha J. Siegel, and Alan C.
Tucker.
http://www.homepages.dsu.edu/palmerj/Evan/newtons/NewtonsLaw.htm
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html
1. Unit: Newton’s Law of Heating and Cooling
The following lesson plans will take 7 days to complete. The lesson
begins with a brief review of logarithms so that students are not struggling
with that skill as they are solving difference equations. Students will then
be introduced to sequences and general equations for sequences. An
introduction to Newton’s Law of Heating and Cooling will follow the
discussion of sequences. The students will discuss the model equation,
the equilibrium point and the graph associated with Newton’s Law. The
unit will culminate with an experiment. Data from this experiment will be
graphed on a TI-82 (or similar graphing calculator).
2. General Instructional Procedures: To introduce student to an experiment
that teaches students how to use their math skills and apply them to a
problem.
3. Specific Learning Objectives: Students will be first taught the concepts
and as a class we will model an experiment. Then the students will
perform an experiment on their own. They will be responsible for
gathering accurate data and modeling this data using their graphing
calculators.
4. Rational: To teach students how to solve problems and apply the skills
that they have to those problems.
5. Instructional Procedures:
Day 1: Students will be given a handout that reviews the rules of logarithms and
contains a few problems for the students to work through. Students will be given
about 5-10 minutes to complete the problems after which the instructor will
review the answers and answer any questions the students may have at that
time. After the logarithm review is completed, students will be provided a
worksheet on sequences. The first page of the worksheet introduces students to
the general formula for a geometric sequence. The second sheet is a worksheet
that tests students understanding of the information. Students will be provided
the rest of the class period to work on the handout and ask questions. Any
remaining problems will be for homework.
Assessment: The worksheets will be collected and correct answers will be
provided to the students through a discussion at the beginning of the class
period. Students receive a + for completed homework with all correct answers, a
check for completed worksheets that contain mistakes and a – for incomplete
worksheets.
Day 2 and 3: Introduction to the concept of Newton’s Law of Heating and
Cooling. Students will be provided with a handout that details what the law is
and what it can be used for. The idea of the worksheet is to introduce the topic
before the students are exposed to the math behind the equation. After
discussing the concept, students will then discuss the formulation of the equation
and will work with a few examples. The students will then be provided a
worksheet that they must complete independently. The worksheet will be due at
the beginning of day 4.
Newton’s Law worksheet: Students are asked to write down the pattern that
the equation has when the initial temperature is T 0 at the time n = 0.
The following is the answer to that question:
T0 = T0
T1 = (1+r)T0 – r(S)
T2 = (1+r)T1 – r(S) plugging in from above:
T2 = (1+r)[(1+r)T0 – r(S)] – r(S) multiplying through
T2 = (1+r)2T0 – (1+r)r(S)-r(S)
T3 = (1+r)T2 – r(S) plugging in from above
T3 = (1+r)[(1+r)2T0 – (1+r)r(S)-r(S)]-r(S)
T3 = (1+r)3T0 – (1+r)2r(S) – (1+r)r(S) – r(S)
With the assistance of the instructor, the pattern that the students should
conclude is
Tn = (1+r)nT0 – (1+r)n-1r(S) – (1+r)n-2r(S) - … - (1+r)r(S) – r(S)
This can also be written as:
Tn = (1+r)nT0 – [(1+r)n-1r(S) – (1+r)n-2r(S) - … - (1+r)r(S) – r(S)]
with the terms in brackets being a geometric series.
Students will be provided with the following information about the sum of this
geometric series. The information is from the following website:
http://www.homepages.dsu.edu/palmerj/Evan/newtons/NewtonsLaw.htm
sum = [r(S)(1-(1+r)n)]/[1-(1+r)]
sum = [r(S – S(1+r)n)]/[-r]
sum = -S + S(1+ r)n
Then Tn = (1+r)nT0 – (-S + S(1+ r)n)
and Tn = (1+r)nT0 + S - S(1+ r)n
The general solution is:
Tn = S + (T0 – S)(1+r)n
Assessment: The worksheets will be collected and correct answers will be
provided to the students through a discussion at the beginning of the class
period. Students receive a + for completed homework with all correct answers, a
check for completed worksheets that contain mistakes and a – for incomplete
worksheets.
Day 4: Introduction to equilibrium points. Students will be provided a worksheet
and asked to consider certain situations in which an object either heats up or
cools down (questions 1 and 2). Question 3 (When will an object reach
equilibrium? When will an object’s temperature stop changing?) is designed to
be a group discussion/brainstorming activity. Students will then be lead through
a discussion that will find the equilibrium point of the equation we developed the
previous day. Students now have all the skills necessary to complete a
worksheet of problems.
The answers to the discussion questions from the worksheet are as follows:
1. 350 degrees
2. 40 degrees
3. When the object has reached the temperature of its surroundings.
4. Tn+1 = Tn call this T’ Instruct students to substitute T’ into the difference
equation and simplify.
The problems on the worksheet are from the following test:
Functioning in the Real World, A Precalculus Experience by Sheldon P.
Gordon, Florence S. Gordon, B.A. Fusaro, Martha J. Siegel, and Alan C. Tucker
Assessment: The worksheets will be collected and correct answers will be
provided to the students through a discussion at the beginning of the class
period. Students receive a + for completed homework with all correct answers, a
check for completed worksheets that contain mistakes and a – for incomplete
worksheets.
Day 5: Students will be lead through an experiment (Coffee Experiment) that has
been modeled for them so that they will acquire the calculator skills they will need
in order to perform their own experiments the following day.
The directions for the data input are from the tutorial provided by the U of I
guidelines for Module 5 adapted to suit a TI-83 calculator.
Day 6: Experiment.
Equipment:
One can of pop per student
Thermometer (one per student)
Plastic disposable cup (one per student)
Worksheet – to be completed during the experiment and for
homework.
Each student will be provided with a can of pop that has been kept cool in a
cooler. Students will begin by pouring a portion of the pop into a cup that has
also been provided to them. Student should immediately take the temperature of
their soda and write it down on the sheet provided. Students will then be
instructed to take the temperature of the soda every two minutes for the duration
of the class period. The instructor will remind the students every two minutes
that it is time to take the temperature of the pop to help ensure that the data is
accurate. Following the instructions on the Coffee Experiment handout that the
students received the previous day, the students are instructed to graph their
data and to find the equation for the line of regression.
Assessment: The worksheets will be collected and correct answers will be
provided to the students through a discussion at the beginning of the class
period. Students receive a + for completed homework with all correct answers, a
check for completed worksheets that contain mistakes and a – for incomplete
worksheets.
Unit Assessment: The end of the unit is characterized by a large quiz/small test.