This is Part III of my course design for Calculus II. Here are Part I and Part II. As usual, you were not the audience for this, so please pardon cryptic parts and typos.

This is Steps 6 through 8, known as the Intermediate Design Phase. Basically, the idea is to create a schedule for the course. I feel like I have a much better handle on the course after doing this. I finally figured out the organizational structure for the course, which I needed to do in order to do Steps 6 through 8. I tried to organize it more by application than mathematical topic, for better or worse. I am also not sure if I should have combined the probability and physics applications for integration, but I wanted fewer topics. :

TOPICS:

1. ODEs: SIR with Covid 19

a. Differential Equations (Modeling, exponential and SIR)

b. Differential Equations (separable) (initial exponential estimation of covid 19 cases)

c. Differential Equations (Euler’s Method/CAS)

d. Series and Sequences (Taylor series, series solutions)

e. Fourier Series solutions to PDE

2. Applications of Integrals: Work in Physics and Probability/Normal distribution/joint probabilities/conditional probability of continuous random variables (virus load given antibody counts)

*a. Integration (slicing/applications)

*b. Integration (parts)

*c. Integration (improper)

d. Integration (numerical)

*e. Series and Sequences (Taylor series)

f. Multivariable Calculus (iterated integrals over general regions)

3. Optimization: No context (but applied)

*a. Multivariable Calculus (optimization)

4. Series No context, just to help with SIR

a. Partial sum definition, problematic series (\sum (-1)^n, -1/12)

*b. Series and Sequences (ratio, alternating series test, comparison test, harmonic series/p-test, integral test–estimation!)

*c. Series and Sequences (N-epsilon)

d. Fourier Series

5. Error Analysis

a. Series (alternating series error estimation)

b. Series and Sequences (Taylor Series Error)

c. Euler’s Method (geometric series for Euler’s Method)

d. Integration (improper)

Here are Steps 6 through 8, noting that I left Days 13–15 blank because I want a buffer/time for presentations:

STEP 6: COURSE STRUCTURE

Action:

-Identify 4 to 7 major concepts, issues, or topics in your course.

1. ODEs: SIR with Covid 19

a. Differential Equations (Modeling, exponential and SIR)

b. Differential Equations (separable) (initial exponential estimation of covid 19 cases)

c. Differential Equations (Euler’s Method/CAS)

d. **Repeat**Series and Sequences (Taylor series, series solutions)

e. Fourier Series solutions to PDE

2. Applications of Integrals: Work in Physics and Probability/Normal distribution/conditional and joint probabilities of continuous random variables (virus load given antibody counts)

*a. Integration (slicing/applications)

*b. Integration (parts)

*c. Integration (improper)

d. Integration (numerical)

*e. **Repeat**Series and Sequences (Taylor series)

f. Multivariable Calculus (iterated integrals over general regions)

3. Optimization: No context (but applied)

*a. Multivariable Calculus (optimization)

4. Series No context, just to help with SIR

a. Partial sum definition, problematic series (\sum (-1)^n, -1/12)

*b. Series and Sequences (ratio, alternating series test, comparison test***estimation***, harmonic series/p-test, integral test–estimation!)

*c. Series and Sequences (N-epsilon)

d. Fourier Series

5. Error Analysis

a. Series (alternating series error estimation)

b. Series and Sequences (Taylor Series Error)

c. Euler’s Method (geometric series for Euler’s Method)

d. Integration (improper)

-What is the appropriate sequence for instroducing these to the students?

Due to the block plan, I am going to interleave these. However, I want to do selected ODEs and integration prior to series in order to motivate series, and I want to do integral slicing problems before iterated integrals in multi, I want to do geometric series before Euler’s Method error.

-What initial ideas do you have for assignments or problems that would reflect the increasing complexity of the subject as students move from topic to topic?

Generally: start with a standard problem, then move to a messy real-world project

This might be it. Calculus II is just a pile of somewhat unrelated topics, so they don’t build much. I think the building will come from the projects.

STEP 7: INSTRUCTIONAL STRATEGY

Topics:

Differential Equations (separable and numerical solutions)

Integration (parts, applications, numerical, improper)

Series and Sequences (N-epsilon, various tests for convergence, Taylor series)

Multivariable Calculus (optimization, iterated integrals over general regions)

Error Approximation

Differential Equations (separable and numerical solutions)

Day 1 In-Class: Set up models for coronavirus; basic exponential (with separable solution) and SIR

Between 1 and 2: Practice separable solutions

Between 2 and 3: Do prep work for Euler’s Method (think about best guess, read, watch), Review tangent line approximations

Day 3 In-Class: Euler’s Method Practice, Solve with Taylor series

Between 3 and 4: Euler’s Method practice, Practice with Taylor Series

Day 4 In-Class: Practice with Taylor series

Between 4 and 5: Practice Taylor series

Between 7 and 8: Review Taylor series by practicing

Day 8 In-Class: Review practice with Taylor series

Between 9 and 10: Review Euler’s Method

Day 10 In-Class: Practice Euler’s Method

Day 13 In-Class: Show Fourier solutions to PDEs

Between 13 and 14: Practice Fourier solutions to PDEs

Day 14 In-Class:

Between 14 and 15:

Day 15 In-Class:

Integration (parts, applications, numerical, improper), Work and Probability

Day 1 In-Class: Integration (slicing/applications), Integration (parts)

Between 1 and 2: Practice with Integration (slicing/parts), watch intro to probability stuff (no multivariable), watch intro to improper and numerical

Day 2 In-Class: Intro to probability (improper, exponential model, problematize with normal model), numerical to deal with normal

Between 2 and 3: Practice improper and numerical, watch Taylor series motivator video

Day 3 In-Class: Practice with Taylor Series intregration of normal model

Between 3 and 4: Practice integration with Taylor Series (do for both sin x both ways, etc), conditional probability introduction

Day 4 In-Class: Practice double integrals with probability

Between 4 and 5: Multiple integral practice (setting up double integrals in two ways)

Day 5 In-Class: Double integral practice

Between 7 and 8: Review slicing, parts, improper

Day 8 In-Class: Practice slicing, parts, improper

Between 10 and 11: Review Numerical Integration

Day 11 In-Class: Numerical integration practice

Between 11 and 12: Practice work/parts

Day 12 In-Class: Practice work/parts

Between 12 and 13:

Day 13 In-Class:

Between 13 and 14:

Day 14 In-Class:

Between 14 and 15:

Day 15 In-Class:

Series and Sequences (N-epsilon, various tests for convergence, Taylor series)

Between 4 and 5: Taylor series practice

Day 5 In-Class: Transition to series (plug in numbers into Taylor series), talk about partial sums, convergence

Between 5 and 6: Videos on convergence tests

Day 6 In-Class: Practice convergence tests (geometric, ratio, alternating, comparison, p-series, integral test)

Between 6 and 7: Practice convergence tests (geometric, ratio, alternating, comparison, p-series, integral test), video on N-epsilon

Day 7 In-Class: Practice convergence tests (geometric, ratio, alternating, comparison, p-series, integral test), practice N-epsilon

Between 7 and 8: Practice N-epsilon

Day 8 In-Class: Practice N-epsilon

Between 8 and 9: Practice N-epsilon, Review alternating series, Taylor Series

Day 9 In-Class: Practice Alternating series, Taylor Series

Between 11 and 12: Review N-epsilon

Day 12 In-Class: Practice N-epsilon

Between 12 and 13:

Day 13 In-Class:

Between 13 and 14:

Day 14 In-Class:

Between 14 and 15:

Day 15 In-Class:

Multivariable Calculus (optimization, iterated integrals over general regions)

Between 1 and 2: Optimization Intro video

Day 2 In-Class: Optimization practice

Between 2 and 3: Optimization practice

Between 5 and 6: Optimization practice

Day 6 In-Class: Optimization practice

Between 6 and 7: Optimization practice

Between 12 and 13:

Day 13 In-Class:

Between 13 and 14:

Day 14 In-Class:

Between 14 and 15:

Day 15 In-Class:

Error Estimation

Between 9 and 10: Series error estimation video

Day 9 In-Class: Series error estimation practice

Between 9 and 10: Series error estimation practice, Euler’s Method Error Estimation video

Day 10 In-Class: Euler’s Method error estimation practice

Between 10 and 11: Euler’s Method error estimation practice, Numerical Integration error estimation video

Day 11 In-Class: Numerical Integration Error Estimation practice

Between 11 and 12: Numerical Integration Error Estimation Practice

Between 12 and 13:

Day 13 In-Class:

Between 13 and 14:

Day 14 In-Class:

STEP 8: CREATING THE OVERAL SCHEME OF LEARNING ACTIVITIES

Day Xa means the first part of the day (roughly 1 hour), and Xb is the second

Day 1a In-Class: Set up models for coronavirus; basic exponential (with separable solution) and SIR

Day 1b In-Class: Integration (slicing/applications), Integration (parts)

Between 1 and 2: Practice with Integration (slicing/parts), watch intro to probability stuff (no multivariable), watch intro to improper and numerical Practice separable solutions, Optimization Intro video,

Day 2a In-Class: Intro to probability (improper, exponential model, problematize with normal model), numerical to deal with normal

Day 2b In-Class: Optimization practice

Between 2 and 3: Practice improper and numerical, watch Taylor series motivator video, Do prep work for Euler’s Method (think about best guess, read, watch), Review tangent line approximations, Optimization practice

Day 3a In-Class: Euler’s Method Practice

Day 3b In-Class: Practice with Taylor Series intregration of normal model, Solve ODE with Taylor series

Between 3 and 4: Practice integration with Taylor Series (do for both sin x both ways, etc), conditional probability introduction, Euler’s Method practice, Practice with Taylor Series

Day 4a In-Class: Practice with Taylor series

Day 4b In-Class: Practice double integrals with probability

Between 4 and 5: Multiple integral practice (setting up double integrals in two ways), Taylor series practice

Day 5a In-Class: Double integral practice

Day 5b In-Class: Transition to series (plug in numbers into Taylor series), talk about partial sums, convergence

Between 5 and 6: Double integral practice, Videos on convergence tests, Optimization practice

Day 6a In-Class: Practice convergence tests (geometric, ratio, alternating, comparison, p-series, integral test)

Day 6b In-Class: Optimization practice

Between 6 and 7: Practice convergence tests (geometric, ratio, alternating, comparison, p-series, integral test), video on N-epsilon, Optimization practice

Day 7a In-Class: Practice convergence tests (geometric, ratio, alternating, comparison, p-series, integral test)

Day 7b In-Class: practice N-epsilon

Between 7 and 8: Practice N-epsilon, Review slicing, parts, improper,Practice N-epsilon, Review Taylor series by practicing

Day 8a In-Class: Practice N-epsilon

Day 8b In-Class: Practice slicing, parts, improper

Between 8 and 9: Practice N-epsilon, Review alternating series, Taylor Series, Series error estimation video

Day 9a In-Class: Practice Alternating series, Taylor Series

Day 9b In-Class: Series error estimation practice

Between 9 and 10: Practice Series Error Estimation, Review Euler’s Method, Euler’s Method Error Estimation video

Day 10a In-Class: Practice Euler’s Method

Day 10b In-Class: Euler’s Method error estimation practice

Between 10 and 11: Euler’s Method error estimation practice, Review Numerical Integration, Numerical Integration error estimation video

Day 11a In-Class: Numerical integration practice

Day 11b In-Class: Numerical Integration Error Estimation practice

Between 11 and 12: Practice work/parts, Review N-epsilon, Numerical Integration Error Estimation Practice

Day 12a In-Class: Practice work/parts

Day 12b In-Class: Practice N-epsilon

Between 12 and 13: ??

Day 13 In-Class: Show Fourier solutions to PDEs

Between 13 and 14: Practice Fourier solutions to PDEs

Day 14 In-Class: ??

Between 14 and 15: ??

Day 15 In-Class: ??