Mathematics Lesson Study: Sequences and SeriesWhitney George and Nathan Warnberg

Lesson Study

The idea behind a lesson study is simple. Instead of individuals workingon a lesson plan, a group of fellow teachers collaboratively plan, teach,observe, reflect, and revise a single lesson.

Lesson Study Design

Backwards Design with collaborators

Calculus II Subject Matter

Sequences and Series

{1 14

142

143

144

145

14

142

143

144

145

What is1

4+

1

42+

1

43+

1

44+

1

45+ · · · =

∞∑

n=1

(1

4

)n

Purpose of Lesson Study

Understand how do students compare andcontrast sequences and series.

Time Table

• Hold 1-2 meeting per week for the first 6 weeks of class.

• Develop and revise lesson plan

• Implement lesson study March 9, 10, 11, 2015

• Reflect and analyze comments and observations from lesson study

• Future: Revise lesson study for Spring 2016

Instructor: Nathan Warnberg. Observers: Whitney George and Chad Vidden

Sample Lesson Study

Section 11.1: Sequences Lesson Plan

Day 1 Objective: Introduce notation and vocabulary, practice manipulating sequences

1 Motivating Material

1. Suppose at the beginning of the semester I give you two options. Which of these two deals would you take?

(a) Option 1: On the first day of class that you attend (there are 56 total) I give you 1 billion dollars, thesecond day of class you attend I give you 2 billion dollars, the third day of class you attend I give you 3billion dollars and so on.

(b) Option 2: On the first class you that attend I give you 1 penny. The second class you attend I takethe 1 penny back and give you 2 pennies. The third day of class I take the 2 pennies back and give you4 pennies, then 8 pennies and so on.

2. Have them vote for their choice immediately after hearing the problem.

3. Ask the following questions:

(a) How much money will you have after 2 days, 4 days, 5 days?

Solution: Day 1: 1 penny vs 1 billion dollarsDay 2: 2 pennies vs 3 billion dollarsDay 3: 4 pennies vs 6 billion dollarsDay 4: 8 pennies vs 10 billion dollarsDay 5: 16 pennies vs 15 billions dollarsDay 20: 219 = 5242.88 dollars vs 210 billion dollarsDay 54: 253 = 90, 071, 992, 547, 409.92 (90 trillion) vs 1,485,000,000,000 (1.48 trillion)

(b) How much after 20 days?

(c) How much money will you have after 54 days?

This will be used for motivation for developing an

(d) Conclude this activity with pointing out that it’s easier to develop a formula for the pennies than thedollars. But, in the next few days they will learn how to sum up the dollars in an efficient way usingseries.

Estimated Time: 10 minutes

1.1 Goals for Activity

1. Students will begin to think about a generating function an for a sequence.

2. Students will begin to think about series.

3. Students will begin to see a sequence as a list of numbers and a relationship between an−1 and an.

2 Introduction to Sequences and Notation

1. Define a sequence as an ordered list of numbers and provide examples:

(a) {1,−1, 1,−1, 1,−1, 1, . . . }(b) {1, 1, 2, 3, 5, 8, 11, . . . }(c) {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, . . . }

Page 1 of 4

Section 11.1: Sequences Lesson Plan

(d)

{1

1,

1

2,

1

3,

1

4, . . .

}

2. Comment on ‘nice’ sequences and not nice sequences. Have the students yell out random numbers and generatea non-nice sequence and compare this to 1, 1/2, 1/3, 1/4, . . .

3. For example, in the sequence we say the first term is 1, the second term is 1/2, the third term is 1/3, etc.What is the 147th term?

4. Introduce notation: We may write this as: {an}∞n=1 = {a1, a2, a3, . . . } = {1/n}∞n=1.

5. Have students generate the first 5 terms of the following sequences.

(a) an =1

n

(b)

{n

n+ 1

}

(c) {(−1)n cos(nπ)}∞n=0

Mention the different notation. In particular in (b) if no bounds are present we assume n = 1.

6. Give them a sequence

{1

3,

1

9,

1

27,

1

81,

1

243, . . .

}and have develop a generating sequence.

7. Repeat the above exercise with the following sequence:−3

1,

5

3,−7

7,

9

15,−11

31,

13

63,−15

127, · · · = (−1)n(2n+ 1)

2n − 1.

Here we will do a Think-Pair-Square activity. Have students pair up and one work on the topnumerator and one work on the denominator (≈ 1 minute alone). Compare and discuss their resultswith each other (≈ 1 minute). Now have the pairs form pairs and have everyone compare and discusstheir results (≈ 2 minutes) Have a class discussion.

8. (If time remains) Graph a few sequences using Desmos and hint at idea of limits:

https://www.desmos.com/calculator/feu2wxrk7c

(a)

{1

1,

1

2,

1

3,

1

4, . . .

}

(b)

{n

n+ 1

}

(c) {(−1)n cos(nπ)}∞n=0

(d)(−1)n(2n+ 1)

2n − 1

Estimated Time: 45 minutes

2.1 Goals for Section

By the end of this section,

1. Students will be able to understand the importance to the order within a sequence,

2. Students will be able to find specific terms given a generating sequence,

3. Students will be able to develop their own generating function for a sequence.

Page 2 of 4

Sample Day Agenda:

• GOALS: Introduce limits of sequences and develop skills for determining if a sequenceconverges or diverges

• TEACHING TECHNIQUES: Introduction to definitions and theorems, along withteacher-worked examples, student-worked examples and teacher-lead discussion interspersed

• OBSERVATIONS: Review material took much more time than was expected. The lessonfelt rushed and students had little time for the corresponding worksheet.

• REFLECTIONS: Revise time line. Spend less time on review of material and more time onworksheet where students can work with peers on developing an intuition for problem-solvingusing sequences.

Lesson Learned

Reflection and Analysis

• “Describe how a sequence, the corresponding series, and thecorresponding sequence of partial sum are related to each other,”

• Students were able to identify patterns quite readily.

• Mixed results with summation notation:∞∑

n=0

(−1)n2n

2n + 1= 1− 2

3+

4

5− 8

7+

16

9∓ .....

The Future

• Emphasis of Material: Success with sequences and series reliesheavily on precise definitions and theorem statements. Emphasizethis throughout the lesson study.

• Student Discussions:

• Revise Timing: Spend more time on technical manipulations ofmath expressions.

Acknowledgments

• CATL and Bill Cerbin

Department of Mathematics | University of Wisconsin–La Crosse | La Crosse, WI Mail: {wgeorge, nwarnberg}@uwlax.edu