Quantitative Reasoning
Student Pages 1.A, Data for life
Lesson 1, Part A
Data for Life
This class is a community of learners. Having
shared goals strengthens communities. Our
shared goal for this course is to maximize the
learning experience for everyone. To do that,
we will begin by working together to collect
data that will be organized and used in later
lessons. We will also reflect on activities that
influence our learning.
1) Complete question 1 on the Data
Recording Sheet (Mandatory Categories).
Credit: iStockphoto
Objectives for the lesson
You will understand that:
¨ Data can be collected from many real-life sources and used for future
mathematical study.
¨ Analysis of data can provide information to help make decisions about positive
changes in your study habits and lives.
¨ Working together builds a positive learning community.
You will be able to:
¨ Collect data from your daily life.
¨ Work positively in a group to make a decision.
2) Now look at question 2 on the Data Recording Sheet (Optional Categories). We
need to narrow down the list to three categories. Rate the categories by putting them
in order from most likely to affect your ability to learn (#1) to least likely to affect your
learning (#9).
Consensus is an agreement. It is not a vote. When we vote, some people win and
others lose. With consensus, the group comes to an agreement that everyone supports.
3) Discuss your priorities with your group and come to consensus on your group’s top
three choices. A consensus means that you all agree. In a community of learners,
everyone wins.
4) How did your group make your choices and reach consensus?
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Quantitative Reasoning
Student Pages 1.A, Data for life
5) As we begin the course and create a positive learning environment, let’s reflect on
activities that can positively affect your ability to learn. List three activities that would
have a positive effect on your learning in this class.
6) Now reflect on activities that could negatively affect your ability to learn. List three
activities that would have a negative effect on your learning in this class.
7) List one action that you could take to minimize the negative effects on your learning.
8) Collect your data on daily water consumption for the next 10 days on the Daily Water
Consumption Data Recording Table below. You will need this information for a later
lesson.
Daily Water Consumption Data Recording Table
Day
Number
of cups of
water
consumed
per day
1
2
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4
5
2
6
7
8
9
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Quantitative Reasoning
Student Pages 1.A, Data for life
Data Recording Sheet
1) Complete the information for the following Mandatory Categories table below.
Mandatory Categories
Record your personal
information.
Gender (male or female)
Height, expressed as a decimal in feet and inches
(For example: 5’ 7” = about 5.58 feet)
Number of hours per week working on math with others
outside of class (tutoring, study group)
Number of hours per week working on math alone outside of
class
Number of hours per week spent in math class
Number of cups of water consumed per day
Number of hours per week commuting to work and college
2) Consider the Optional Categories table below. Number the categories from most
likely to affect your learning (#1) to least likely to affect your learning (#9). Discuss
your ordering with your group members. Then the class will come to consensus on
the top three categories for the class. These categories will be used as data sets in
future lessons. When the top three categories are identified, record your own data
for those three categories. Turn in this sheet to your instructor at the end of class.
Optional Categories
Number
your
choices.
Record your personal
information for the top
three categories
chosen by the class.
Number of alcoholic beverages consumed per
week
Number of caffeinated beverages consumed per
week
Number of daily servings of fruits and vegetables
Number of daily servings of protein
Number of hours exercising (e.g., walking, yoga,
biking, sports) per week
Number of hours using technology for fun per
week (e.g., TV, email, video games, texting,
Facebook)
Number of hours working per week
Number of hours sleeping per week
Number of hours preparing meals and eating per
week
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Quantitative Reasoning
Student Pages 1.A, Data for life
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Quantitative Reasoning
Suggested Instructor Notes 1.A, Data for life
Lesson 1, Part A
Data for Life
Overview and student objectives
Overview
This lesson begins to create a learning community through a
conversation about data and also provides a means to
stimulate conversation about health topics that are relevant to
the students in this class. Students submit data for a class
spreadsheet to be maintained by the instructor. The instructor
keeps the data in an Excel spreadsheet, creating the graphs
and charts and distributing data and/or graphs, as needed, for
different lessons and activities. In later tasks, students will
study data plots, lines of best fit, measures of center, etc. using
the data collected from the class.
Lesson Length: 25
minutes
Prior Lesson: None
Next Lesson: Lesson
1, Part B, “Our Learning
Community” (25
minutes)
Constructive
Perseverance Level: 1
Theme: Health Literacy
Goal: Communication
Quantitative
Reasoning Outcome:
S1
Objectives
Students will understand that:
Related Foundations
Outcome: N8
•
Data can be collected from many real-life sources and
used for future mathematical study.
•
Analysis of data can provide information to help make decisions about positive
changes in their study habits and lives.
•
Working together builds a positive learning community.
Students will be able to:
•
Collect data from their daily lives.
•
Work positively in a group to make a decision.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Student Pages for Lesson 1, Part A
•
Excel spreadsheet with mandatory and optional categories
Prerequisite assumptions
Students from the Foundations course should be familiar with a learning community.
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Suggested Instructor Notes 1.A, Data for life
Background context
In this activity, baseline data are collected, which will be discussed and analyzed in
subsequent lessons.
Suggested instructional plan
Frame the lesson
(4 minutes)
Whole
Class
Introduction
•
Talk about data collection and how it will be studied throughout the
term. The lessons in the course are centered on real-world contexts.
This activity is designed to focus on data that are immediately
available and relevant to the students.
•
Ask students to read the introductory paragraph and fill in the
“Mandatory Categories” table with their personal information.
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
Lesson activities
(18 minutes)
Group
Work
Whole
Class
Discussion
Questions 2–4
•
Ask students to proceed to question 2, ranking the optional
categories and discussing with their group to choose three.
•
Let students know that they will need to be as honest as possible
when submitting their data to make this activity work well. Individual
data will be anonymous to one another. The instructor will collect the
data and post it to an Excel spreadsheet.
o Use the accompanying spreadsheet (excerpt shown at the end of
these notes) to record student data for later distribution.
o You may choose to include average course grade and average
test/quiz grades as dependent variables sometime during the
term.
•
Be prepared to diminish conflict if consensus cannot be reached
about the top three optional categories. Work toward maintaining a
positive classroom environment with the option for students to “opt
out” if he/she is uncomfortable. Beware if the class size is small. A
small class could sabotage anonymity. Instructor should be mindful
and sensitive to student privacy.
•
As a class, determine the top three optional categories on which you
will collect data. Ask students to share the methods their group used
to come to consensus.
•
Ask students to complete their data for the top three categories.
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Suggested Instructor Notes 1.A, Data for life
Questions 5–7
•
Ask students to complete 5–7 and share with the class as time
permits.
Question 8
•
Ask students to be diligent about filling out the table in question 8. If
they try to fill it all out on the last day, their data will likely be
inaccurate.
Wrap-up/transition
(3 minutes)
Wrap-up
Transition
•
Collect the Data Recording Sheet with the mandatory and optional
categories. See the Follow-up for instructor section below.
•
Ask students to reflect on this data collection activity. Ask them to
write their initial reactions about working in groups for the first class
meeting on a 3x5 inch card. How did they feel about sharing
personal information?
•
Have students refer back to the Objectives for the lesson and
check the ones they recognize from the activity. Alternatively, they
may check objectives throughout the lesson.
•
As an introduction to future lessons, discuss data collection in
general, pitfalls in data collection, and how data might be used.
Illustrate with one of the mandatory categories.
Suggested assessment, assignments, and reflections
•
Give the Preview Assignments, if any, for the lesson activities you plan to
complete in the next class meeting.
Follow-up for instructor
Complete the provided Excel sheet with mandatory category data and those categories
decided by the class.
Caution: Rather than deleting the unused categories on the master sheet, save a
master copy. Subsequent classes may decide on different categories.
Print a copy of the compiled data for each student.
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Suggested Instructor Notes 1.A, Data for life
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Suggested Instructor Notes 1.A, Data for life
Lesson 1, Part A
Data for Life – ANSWERS
This class is a community of learners. Having
shared goals strengthens communities. Our
shared goal for this course is to maximize the
learning experience for everyone. To do that,
we will begin by working together to collect
data that will be organized and used in later
lessons. We will also reflect on activities that
influence our learning.
1) Complete question 1 on the Data
Recording Sheet (Mandatory Categories).
Credit: iStockphoto
Objectives for the lesson
You will understand that:
¨ Data can be collected from many real-life sources and used for future
mathematical study.
¨ Analysis of data can provide information to help make decisions about positive
changes in your study habits and lives.
¨ Working together builds a positive learning community.
You will be able to:
¨ Collect data from your daily life.
¨ Work positively in a group to make a decision.
2) Now look at question 2 on the Data Recording Sheet (Optional Categories). We
need to narrow down the list to three categories. Rate the categories by putting them
in order from most likely to affect your ability to learn (#1) to least likely to affect your
learning (#9).
Consensus is an agreement. It is not a vote. When we vote, some people win and
others lose. With consensus, the group comes to an agreement that everyone supports.
3) Discuss your priorities with your group and come to consensus on your group’s top
three choices. A consensus means that you all agree. In a community of learners,
everyone wins.
Answers will vary.
Sample answers: They all had the same top three; everyone had at least one of their
top three choices in the group’s top three list.
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Suggested Instructor Notes 1.A, Data for life
4) How did your group make your choices and reach consensus?
Answers will vary.
5) As we begin the course and create a positive learning environment, let’s reflect on
activities that can positively affect your ability to learn. List three activities that would
have a positive effect on your learning in this class.
Answers will vary.
Sample answer: Attending every day; doing homework; seeking help when needed.
6) Now reflect on activities that could negatively affect your ability to learn. List three
activities that would have a negative effect on your learning in this class.
Answers will vary.
Sample answers: Being a single parent; number of hours have to work; lack of
exercise; concerns about finances; math or test anxiety; not enough time to study.
7) List one action that you could take to minimize the negative effects on your learning.
Answers will vary.
8) Collect your data on daily water consumption for the next 10 days on the Daily Water
Consumption Data Recording Table below. You will need this information for a later
lesson.
Daily Water Consumption Data Recording Table
Day
Number
of cups of
water
consumed
per day
1
2
The Charles A. Dana Center at
The University of Texas at Austin
3
4
5
10
6
7
8
9
10
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Quantitative Reasoning
Suggested Instructor Notes 1.A, Data for life
Data Recording Sheet
1) Complete the information for the following Mandatory Categories table below.
Mandatory Categories
Record your personal
information.
Gender (male or female)
Height, expressed as a decimal in feet and inches
(For example: 5’ 7” = about 5.58 feet)
Number of hours per week working on math with others
outside of class (tutoring, study group)
Number of hours per week working on math alone outside of
class
Number of hours per week spent in math class
Number of cups of water consumed per day
Number of hours per week commuting to work and college
2) Consider the Optional Categories table below. Number the categories from most
likely to affect your learning (#1) to least likely to affect your learning (#9). Discuss
your ordering with your group members. Then the class will come to consensus on
the top three categories for the class. These categories will be used as data sets in
future lessons. When the top three categories are identified, record your own data
for those three categories. Turn in this sheet to your instructor at the end of class.
Optional Categories
Number
your
choices.
Record your personal
information for the top
three categories
chosen by the class.
Number of alcoholic beverages consumed per
week
Number of caffeinated beverages consumed per
week
Number of daily servings of fruits and vegetables
Number of daily servings of protein
Number of hours exercising (e.g., walking, yoga,
biking, sports) per week
Number of hours using technology for fun per
week (e.g., TV, email, video games, texting,
Facebook)
Number of hours working per week
Number of hours sleeping per week
Number of hours preparing meals and eating per
week
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Suggested Instructor Notes 1.A, Data for life
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Student Pages 1.B, Our learning community
Lesson 1, Part B
Our Learning Community
Building a strong learning community can help
everyone achieve at a higher level in this
class.
1) What are some things each of us can do to
contribute to our learning community?
Credit: iStockphoto
Objectives for the lesson
You will understand:
¨ Your importance in the learning community.
¨ The course policies and procedures you need to know and follow.
You will be able to:
¨ Seek and give help to one another inside and outside of class. 2) Use your syllabus and conversations with classmates to complete the information on
these pages. Use them as a resource throughout the semester, adding to them
when you find new avenues of support.
Instructor’s Name
Instructor’s Office Location
Instructor’s Office Hours
Office Phone Number
Email Address
Tutoring Lab Location and Phone
Number
Computer Lab Location
Other On-campus Resources
Classmate(s) Contact Information
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Student Pages 1.B, Our learning community
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Suggested Instructor Notes 1.B, Our learning community
Lesson 1, Part B
Our Learning Community
Overview and student objectives
Overview
Lesson Length: 25 minutes
Although most students will have formed learning
communities in the feeder math course, Foundations of
Mathematical Reasoning, they may not have all been in the
same course section. Therefore, it is important to spend time
building this new learning community. The suggested
strategies are optional and may be customized, but bear in
mind that the goals of this class meeting are to:
Prior Lesson: Lesson 1,
Part A, “Data for Life”
Next Lesson: Lesson 1,
Part C, “Instant Runoff”
(25 minutes)
Constructive Perseverance
Level: 1
Theme: Student Success
•
Lay the foundations for a positive learning
environment.
•
Establish a sense of community and shared
responsibility.
•
Help students form connections with one another and
with the instructor.
•
Give students key information about course content, policies, and structures.
Goal: Communication
Quantitative Reasoning
Outcomes: None
Related Foundation
Outcomes: None
Objectives
Students will understand:
•
Their importance in the learning community.
•
The course policies and procedures they need to know and follow.
Students will be able to:
•
Seek and give help to one another inside and outside of class.
Instructors will:
•
Gather information about students and/or initiate activities to help to make a
personal connection with each student.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Poster paper for class norms and group supports. Save responses and bring to
class for the first few class meetings, or type up and make them available to the
students to add to their binders.
•
Index cards
•
Student Pages for Lesson 1, Part B
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Quantitative Reasoning
Suggested Instructor Notes 1.B, Our learning community
o Student Information Sheet
o Syllabus Quiz
•
Preview Assignment 1.C
Prerequisite assumptions
There are no mathematical prerequisite assumptions for this lesson. There is the
assumption that most students will have taken Foundations of Mathematical Reasoning
during a prior semester.
Making connections
This lesson:
•
Connects back to the learning community and student success strategies in
Foundations of Mathematical Reasoning.
•
Connects forward to the strategies for working groups that will be needed
regularly throughout the course.
Background context
A learning community is a group of people who share common emotions, values, or
beliefs and are actively engaged in learning together from one another.
Community psychologists such as McMillan and Chavis (1986) state that there are four
key factors that define a sense of community: 1) membership, 2) influence, 3) fulfillment
of individuals needs, and 4) shared events and emotional connections.1 The participants
of a learning community must feel some sense of loyalty and belonging to the group
(membership) that drive their desire to keep working and helping others. Also, the things
that the participants do must affect what happens in the community—that means, an
active and not just a reactive performance (influence). Besides a learning community
must give the chance to the participants to meet particular needs (fulfillment) by
expressing personal opinions, asking for help or specific information, and sharing
stories of events with particular issue included (emotional connections) emotional
experiences.
Suggested instructional plan
Frame the lesson
(6 minutes)
Student
Page
•
In the previous activity, students worked in groups to collect data
about themselves. In this lesson, students will analyze
characteristics that contribute to a productive, whole class learning
community as well as to productive work groups.
1
McMillan, D. W., & Chavis, D. M. (1986). “Sense of community: A definition and theory.” Journal of
Community Psychology, 14, 6–23.
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Suggested Instructor Notes 1.B, Our learning community
•
Take a moment to talk about the definition of a learning community
(see Background context of this lesson).
•
Refer students to the first question on the Student Page. Give
students a couple of minutes to jot down a response to the question.
•
Solicit student responses, validating those responses by noting
them on the poster paper. Continue to probe for participation
responses such as “contribute to our group discussion,” “be polite,”
etc., and more policy-focused responses such as “come to class,”
“be on time,” “bring materials,” etc.
o Ask, “What if you are having difficulty?”
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
Lesson activities
(15 minutes)
Caution:
•
Timing is important with this lesson as there are many components.
If this lesson is the first day of class, it will be important to have
enough time to discuss the syllabus at the end of the class.
•
Be aware that there may be some students who were not in the
Foundations of Mathematical Reasoning class. They may need
additional support in acclimating to this classroom environment.
•
This course will challenge students both in terms of content and in
terms of expectations of them as learners. You can expect a variety
of emotional reactions: excitement about a “different” way of
learning math; anxiety about something that is unfamiliar;
resentment that you are not “teaching” them in the way to which
they are accustomed.
•
Students are most willing to work hard and try new things when they
understand the reason for it. Therefore, throughout the course, look
for opportunities to explain to students why you make certain
instructional choices or why activities are designed in a certain way.
Such explanations are valuable when you see signs of frustration.
Strategy: Define a “positive learning environment”
Lead a class discussion. It is important to acknowledge and honor both
positive and negative reactions.
Interject information in the discussion. Important points to make:
Group
Work
•
Group work will be an important part of the course. Research shows
that people learn and remember information best when they
experience learning in a variety of ways (e.g., listening, talking,
writing, reading, doing). This course is structured to incorporate
these different ways of learning. In an active, collaborative learning
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Suggested Instructor Notes 1.B, Our learning community
environment, students are responsible for their own learning and for
supporting the learning of others.
Class
Discussion
•
Write the group task on the board:
o “In your group, create two lists: 1) Things that support my
learning when working in a group; and 2) Things that inhibit (or
interfere with) my learning when working in a group.”
•
Give groups about 5 minutes to generate their lists. After about 3
minutes, prompt them to start the second list if they have not done
so (although they may be working on both lists at once).
•
Take one “supports learning” comment from each group, creating a
list on poster paper. Continue until all groups feel their responses
have been represented.
•
Repeat the process for the “inhibits learning” comments.
o Be sure the “supports learning” list includes: Be positive about
yourself and working with your peers; be respectful of one
another; work hard to be a productive member of a group.
•
Hold a whole class discussion about the two lists.
o “How can these two lists help us this semester?”
o “What can I do as the instructor to help you have successful
working groups?”
Strategy: Student information sheet
(sample included in this document)
Ask students to complete a short form that summarizes key information
about them. This information sheet can also be a way to gather contact
information if you do not have easy access to that information. Collect the
Student Information Sheets.
Optional Strategy: Quick reference index cards
Give each student an index card. Ask them to write their names, the
contact information they wish to share with you, and responses to two or
three quick prompts. For example:
•
“When did you take your last math class?”
•
“Is this your first time in college?”
•
An excellent prompt is “Tell me something interesting about
yourself.”
Responses to these questions give you something to talk about with
students and can be an aid in learning names. You can also use the cards
in class to randomize calling upon students.
Optional Strategy: Math autobiography
•
You might also choose to ask students to complete a math
autobiography in more detail than the information sheet or a math
journal to be completed throughout the term.
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Quantitative Reasoning
Suggested Instructor Notes 1.B, Our learning community
Wrap-up/transition
(4 minutes)
Wrap-up
Transition
•
Have students refer back to the Objectives for the lesson and
check the ones they recognize from the activity. Alternatively, they
may check objectives throughout the lesson.
•
Suggest that students move their Student Page with question 2 to
the front of their binders for easy reference throughout the
semester.
o If you plan to type and distribute the class and group norms, tell
students the method of distribution. (Example: “I will bring them
to class next time” or “I will distribute via the online course
website.”)
•
Alternatively, note that you will bring the lists to the first few class
meetings to help the class and groups learn to work productively.
•
Move on to a discussion of the syllabus.
o “Now we’ll get more specific about the course requirements.”
o A strategy called “crowd sourcing” could be used to engage the
students in a discussion of the syllabus. Students report to the
class an item that is important on each page.
•
Alternatively, distribute the syllabus and ask students to read it at
home and bring it to the next class for discussion.
Suggested assessment, assignments, and reflections
•
Give assignments for any activities that you want students to complete outside of
class.
o Setting classroom norms
o Syllabus quiz (could be given at the end of this class or beginning of the next
class)
o Student information sheet
o Math autobiography
•
Prepare students for the next class.
o Determine which QR lessons you will cover during the next class meeting,
and assign the Preview Assignments for those lessons.
o Suggestion for students who missed this class: Make personal contact with
any students who miss this information. Ideally, the students should meet with
you during office hours. At a minimum, try to talk to the student before or after
the next class. Depending on the norms and routines that you have
established, you might also ask other students to help the absent students
understand the role of community and the expectations that the class has
established.
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Suggested Instructor Notes 1.B, Our learning community
Syllabus quiz
Student Name ________________________________________________
1)
What is the course instructor’s office phone number?
2)
On what day(s) and time(s) are office hours held?
3)
What is the attendance policy for this course?
4)
What overall percentage will result in a C grade for the semester?
5)
True or False? For any false statement, please correct the statement.
a) Attendance is crucial for success in this course.
b) You will complete the entire midterm and the entire final exam in class.
c) Late assignments can be handed in up to three weeks after the due date.
d) Any student who attends class, takes good notes, completes assignments, and
studies outside of class should receive a good grade in this course.
6)
What are the criteria for receiving maximum points for participation?
7)
Homework assignments account for what portion of your final grade?
8)
When are the tests scheduled for this class?
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Suggested Instructor Notes 1.B, Our learning community
Student information sheet
Personal
Full name:
Current address:
Contact information (please complete at least one method of contact):
Home phone:
Cell phone:
Email:
Other:
Do you work?
(yes / no)
If yes: full-time or
part-time?
How many
hours?
Are there other outside obligations or factors that might influence your performance in
this class that you would like for me to know?
Academic
What is your major?
(put “undeclared” if relevant):
Where did you go to high school?
What year did you graduate
from high school?
Do you plan to transfer from this college? circle one:
yes / no
If yes, where do you plan to transfer?
What other colleges (if any)
have you attended?
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Suggested Instructor Notes 1.B, Our learning community
Course-specific
What math courses have you taken at
the community college level?
List the courses you are enrolled in this
semester.
What grade do you intend to earn in this
course?
What strengths or assets do you have
that will help you achieve your goal in
this course? These strengths/assets can
include things about you, such as being
a hard worker or things about your life
such as having a supportive family or
having a flexible work schedule.
Course Date College What is your level of comfort with math?
1
2
3
4
Not comfortable
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5
6
7
8
9
10
Very comfortable
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Suggested Instructor Notes 1.B, Our learning community
Lesson 1, Part B
Our Learning Community – ANSWERS
Building a strong learning community can help
everyone achieve at a higher level in this
class.
1) What are some things each of us can do to
contribute to our learning community?
Answers will vary.
Sample answers: Be on time, contribute to
discussion; bring materials; do homework;
be polite; seek help when needed; offer
help to others.
Credit: iStockphoto
Objectives for the lesson
You will understand:
¨ Your importance in the learning community.
¨ The course policies and procedures you need to know and follow.
You will be able to:
¨ Seek and give help to one another inside and outside of class. 2) Use your syllabus and conversations with classmates to complete the information on
these pages. Use them as a resource throughout the semester, adding to them
when you find new avenues of support.
Instructor’s Name
Instructor’s Office Location
Instructor’s Office Hours
Office Phone Number
Email Address
Tutoring Lab Location and Phone
Number
Computer Lab Location
Other On-campus Resources
Classmate(s) Contact Information
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Suggested Instructor Notes 1.B, Our learning community
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Preview Assignment 1.C
Preview Assignment 1.C
Monitoring your readiness
In preparing for the next math class and lessons, reflect on the discussions from the first
class on building a learning community, working collaboratively with your classmates,
and how you can manage your personal time to work effectively in this class.
1) For each of the following, rate how confident you are to proceed in this class. Use
the following descriptions to rate yourself:
5—I am extremely confident
4—I am somewhat confident
3—I am not sure how confident
2—I am not very confident
1—I am definitely not confident
Skills needed for Lesson 1, Part C
Skill or Concept: I . . .
Rating from 1 to 5
Understand the expectations of this course.
Understand what a learning community is.
Can manage my time effectively to be
successful in this course.
Can work in a group during class to discuss
and solve a math problem.
Can read mathematics assignments and
begin the problems.
Use your ratings to prepare for future lessons. If your rating is a 3 or below, you
should get help early in this course.
Ways to get help:
• see your instructor for help
• ask your instructor for on-campus resources
• set up a study group with classmates so you can help each other
• work with a tutor
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Preview Assignment 1.C
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Student Pages 1.C, Instant Runoff
Lesson 1, Part C
Instant Runoff
Elections are held every day to choose
persons for public office. Election methods
can also be used to select between multiple
scenarios or issues, such as the menu for an
office party.
1) In an election involving two people, when
looking at the votes cast, what criteria
would you use to determine who should
win? What about an election involving
three people?
Credit: iStockphoto
Objectives for the lesson
You will understand:
¨ That earning the most votes may not be sufficient to win an election.
¨ That there are multiple considerations and methods for ranking candidates in an
election.
¨ That multiple ranking methods can be employed to make decisions about other
issues.
¨ The difference between the terms plurality and majority in an election.
You will be able to:
¨ Create a first-degree equation involving percentages and solve for the variable.
¨ Employ the “Instant Runoff” method to determine the winner of an election.
¨ Apply and justify selection strategies to election results and decisions about other
issues. “The term plurality, also called relative majority used in the context of voting, refers to
the largest number of votes to be received by any candidate (or any proposal in a
referendum). It is contrasted with an absolute majority, or simple majority, which is
more than half of the votes . . . In an election contested by more than two candidates,
plurality occurs when one candidate receives the most votes, but not necessarily more
than half of the votes while in a majority election, a candidate wins if they receive over
half of the votes. When no candidate receives a majority in the first round of voting, a
two-round system or ranked voting system can be used to choose a winner.”1
1
Source: http://en.wikipedia.org/wiki/Plurality_(voting)
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Student Pages 1.C, Instant Runoff
Joe Agris
Curt Cleaver
Ted Cruz
David Dewhurst
Ben Gambini
Craig James
Tom Leppert
Lela Pittenger
% of
Vote
Glenn Addison
Candidate
2) In 2012, one of Texas’ U.S. Senate seats was up for grabs. Nine candidates were on
the ballot for the Republican Party primary. The results of the primary are shown
below.2 Who do you think should have won? Why?
2%
0%
0%
34%
45%
1%
4%
13%
1%
3) Texas election rules state that elections are majority elections. Who is the winner of
this election?
4) The number of votes received by Cruz and Dewhurst are listed below.
Part A: How many total votes were cast in the election?
Part B: How many votes would be needed to win in a majority election?
Candidate
Original Primary
Votes (%)
Ted Cruz
477,428 (34%)
David Dewhurst
621,850 (45%)
5) Since no candidate received a majority in original primary election, the election rules
required a runoff election. The results of the runoff election are shown below. The
runoff is also a majority election. Who won the runoff election? How?
Candidate
Runoff
Votes (%)
Ted Cruz
628,336 (57%)
David Dewhurst
477,888 (43%)
2
Source: http://www.thepoliticalguide.com/Elections/2012/Senate/Texas/1/
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Student Pages 1.C, Instant Runoff
Another election method called the “Instant Runoff” method, or elimination method, is to
ask voters to rank the candidates in order of preference. Consider this new scenario
and election method. The compiled results are shown in a table, called a preference
schedule.
6) Three candidates (Alex, Blake, and Charlie) applied for a position at a company. The
interviewing committee ranked their choices in order of preference in the table
below. Notice that three people ranked Alex in 1st place and Blake in 2nd place, and
so on.
st
1 choice
2nd choice
3rd choice
3 voters
Alex
Blake
Charlie
1 voter
Alex
Charlie
Blake
2 voters
Blake
Alex
Charlie
1 voter
Blake
Charlie
Alex
2 voters
Charlie
Blake
Alex
Part A: How many voters were on the interviewing committee?
Part B: A majority is still needed to win in his scenario. How many votes are needed
for a majority?
Part C: How many 1st place votes did each candidate receive? Do we have a
winner?
Part D: Which candidate received the fewest 1st place votes? To employ the Instant
Runoff method, mark out or eliminate that person’s name in each column of
the table. Any name below that one in the table will now move up.
Part E: How many 1st place votes do we now award to the two remaining
candidates? Who wins the election?
7) The method shown in question 5 is known as the Instant Runoff method. What are
some pros and cons of this method?
8) Can you think of another election method for choosing the winning candidate in an
election? What are pros and cons of your method? Can any process guarantee that
the choice made (or winner) in an election is correct and fair to all candidates?
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Student Pages 1.C, Instant Runoff
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Suggested Instructor Notes 1.C, Instant Runoff
Lesson 1, Part C
Instant Runoff
Overview and student objectives
Overview
Lesson Length: 25 minutes
In this lesson, students begin a sequence of activities around
civic responsibility by exploring some concepts of social
choice. Specifically, they investigate the first of several ways
to determine the winner of an election. Lesson 1, Part D
extends the discussion with the Borda Count method.
Prior Lesson: Lesson 1,
Part B, “Our Learning
Community”
An important point of this lesson is to encourage discussion
as the students in the class get to know one another.
Constructive Perseverance
Level: 1
Objectives
Theme: Civic Responsibility
Students will understand:
Goals: Reasoning,
Evaluation
Next Lesson: Lesson 1,
Part D, “Borda Count” (25
minutes)
•
That earning the most votes may not be sufficient to
win an election.
•
That there are multiple considerations and methods for Related Foundations
Outcome: N8
ranking candidates in an election.
•
That multiple ranking methods can be employed to make decisions about other
issues.
•
The difference between the terms plurality and majority in an election.
Quantitative Reasoning
Outcomes: N1, N2
Students will be able to:
•
Create a first-degree equation involving percentages and solve for the variable.
•
Employ the “Instant Runoff” method to determine the winner of an election.
•
Apply and justify selection strategies to election results and decisions about other
issues.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Preview Assignment 1.C (for review; students will have completed before class)
•
Student Pages of Lesson 1, Part C
•
Practice Assignment 1.C
•
Syllabus Quiz from Lesson 1, Part B, if desired
•
Determine a grouping method, such as a deck of cards.
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Suggested Instructor Notes 1.C, Instant Runoff
Prerequisite assumptions
Students should be able to determine the original amount, given the percentage that a
given number is of the original.
Making connections
This lesson:
•
Connects back to work that students encountered in the Foundations of
Mathematical Reasoning course, reading, interpreting, and making decisions
about data summarized in tables and graphical displays.
•
Connects forward to the remainder of the course in which decisions are based on
data rather than emotion or anecdotal evidence.
Background context
None.
Suggested instructional plan
Frame the lesson
(5 minutes)
Student
Page
•
Give Syllabus Quiz from Lesson 1, Part B, if desired.
•
Be mindful of the timing of each component in this lesson. It will be
necessary to stay on task to complete the lesson.
•
Separate the class into groups of 3 or 4 students by whatever
process makes sense to you. Possible methods:
o Distribute playing cards to students, and put those who have
Kings in one group, all who have Queens in another group, etc.
o Create random groups, counting off (1, 2, 3, 4).
•
Ask students to consider question 1 on the Student Page. Students
may respond by saying, “The person with the most votes.” Ask
follow-up questions designed to lead to more specificity, such as:
o “What if all three candidates each win 33% of the vote?”
o “What if two candidates are very close and one candidate is far
behind?”
o “What if one candidate has almost all of the votes and the other
two candidates have very few votes?”
o “What if there are four or more candidates? Would your answers
to any of the earlier questions change?”
o “What if a voter can submit a first choice and a second choice?”
•
Note: The names for the voting example were purposefully chosen to
reflect gender-blindness and can be either a male’s or female’s
name.
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Suggested Instructor Notes 1.C, Instant Runoff
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
Lesson activities
(17 minutes)
Group
Work
then
Debrief
Questions 2–5
•
Allow about 5 minutes for students to answer questions 2–4. Remind
them that they need to always justify their responses in this class.
•
Students may realize from the discussion of question 1 that just
having the most votes may not be sufficient. If not (if students are
saying Dewhurst should have won because he had the most votes),
ask the following:
o “What percent of the voters in this election did not support
Dewhurst? What might this indicate about his future ability to lead
his constituents?”
•
For question 3, students often think that a majority of the votes is the
same as having the most votes.
o Ask, “If majority means ‘more than 50% of the votes,’ how many
votes are needed to win a majority in an election with 20 voters?
How many votes are needed in an election with 27 voters?”
[Answer: With 20 voters, 20 x 0.50 = 20/2 = 10 votes are needed
to win exactly 50%, so 11 votes are needed to capture more than
50% of the votes. With 27 voters, 27/2 = 13.5 votes, so 14 or
more votes would constitute a majority.]
•
Students will need to realize that only 79% of the votes are shown in
the table and that they need to find 79% of total votes shown.
Questions 6–8
Group
Work
then
Debrief
•
Point out that the table in question 6 is called a preference schedule,
as it shows the preferences of the voters.
•
Allow about 5 minutes for group work on these questions. Circulate
throughout the room and ask questions to ensure that students are
reading the table correctly.
o For example, ask, “Is it clear to you how many first-choice votes
were received by Charlie?”
•
Question 6, Parts A and B: Indicate to students that it is always a
good practice to first determine the total number of voters and the
number needed for a majority. There may be a majority winner, and
work done in eliminating the lowest vote getter will be wasted.
•
Question 6, Part C: Students determine that Alex does not have
enough votes for a majority.
o Ask, “Hypothetically, what if we give the position to Alex? How do
the people who didn’t put Alex in 1st place feel about him or her?”
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Suggested Instructor Notes 1.C, Instant Runoff
[Answer: Three people put Alex in last place, which is almost as
many who put Alex in first place.]
•
Question 6, Parts D and E: Charlie has the fewest 1st place votes.
Eliminating Charlie only affects the last column, where Charlie was
first. Blake would move up into the 1st place slot, giving Blake a total
of five 1st place votes. Blake now has a majority and is the winner.
Wrap-up/transition
(3 minutes)
Wrap-up
•
The overall goal is to give students awareness that there are multiple
methods to run elections and to count votes and that there is not
necessarily a “right” method. The next lesson will explore the Borda
Count method.
•
Ask students to reflect on this lesson and write their reflections on a
3x5-inch card. Sample questions and prompts include:
o “Was the lesson interesting? How did the discussion go in your
group? Did everyone contribute? How can we ensure that group
discussions are productive?”
o “List one concept that is not clear after the lesson.”
o “Is there anything you would like to share with your instructor?”
•
Have students refer back to the Objectives for the lesson and check
the ones they recognize from the activity. Alternatively, they may
check objectives throughout the lesson.
•
Mention that the next lesson will explore another method of choosing
a winner in an election.
Transition
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 1.C.
•
Give the Preview Assignments, if any, for the lesson activities that you plan to
complete in the next class meeting.
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Suggested Instructor Notes 1.C, Instant Runoff
Lesson 1, Part C
Instant Runoff – ANSWERS
Elections are held every day to choose
persons for public office. Election methods
can also be used to select between multiple
scenarios or issues, such as the menu for an
office party.
1) In an election involving two people, when
looking at the votes cast, what criteria
would you use to determine who should
win? What about an election involving
three people?
Answers will vary.
Credit: iStockphoto
Objectives for the lesson
You will understand:
¨ That earning the most votes may not be sufficient to win an election.
¨ That there are multiple considerations and methods for ranking candidates in an
election.
¨ That multiple ranking methods can be employed to make decisions about other
issues.
¨ The difference between the terms plurality and majority in an election.
You will be able to:
¨ Create a first-degree equation involving percentages and solve for the variable.
¨ Employ the “Instant Runoff” method to determine the winner of an election.
¨ Apply and justify selection strategies to election results and decisions about other
issues. “The term plurality, also called relative majority used in the context of voting, refers to
the largest number of votes to be received by any candidate (or any proposal in a
referendum). It is contrasted with an absolute majority, or simple majority, which is
more than half of the votes . . . In an election contested by more than two candidates,
plurality occurs when one candidate receives the most votes, but not necessarily more
than half of the votes while in a majority election, a candidate wins if they receive over
half of the votes. When no candidate receives a majority in the first round of voting, a
two-round system or ranked voting system can be used to choose a winner.”1
1
Source: http://en.wikipedia.org/wiki/Plurality_(voting)
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Suggested Instructor Notes 1.C, Instant Runoff
Joe Agris
Curt Cleaver
Ted Cruz
David Dewhurst
Ben Gambini
Craig James
Tom Leppert
Lela Pittenger
% of
Vote
Glenn Addison
Candidate
2) In 2012, one of Texas’ U.S. Senate seats was up for grabs. Nine candidates were on
the ballot for the Republican Party primary. The results of the primary are shown
below.2 Who do you think should have won? Why?
2%
0%
0%
34%
45%
1%
4%
13%
1%
Answers will vary.
3) Texas election rules state that elections are majority elections. Who is the winner of
this election?
Answer: There is no winner. To have a majority, a candidate must win more than
half of the votes. Although Dewhurst received more votes than anyone else, a
plurality, he didn’t receive more than 50%.
4) The number of votes received by Cruz and Dewhurst are listed below.
Part A: How many total votes were cast in the election?
Sample answer: To find the total number of votes, create an equation and solve for
x:
(0.34+0.45)x = (477,428+621,850)
0.79x = 1,099,278
x = 1,391,491 total votes cast
Part B: How many votes would be needed to win in a majority election?
Candidate
Original Primary
Votes (%)
Ted Cruz
477,428 (34%)
David Dewhurst
621,850 (45%)
Answer: Votes needed to win a majority election: 1,391,491/2 = 695,745
2
Source: http://www.thepoliticalguide.com/Elections/2012/Senate/Texas/1/
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Suggested Instructor Notes 1.C, Instant Runoff
5) Since no candidate received a majority in original primary election, the election rules
required a runoff election. The results of the runoff election are shown below. The
runoff is also a majority election. Who won the runoff election? How?
Candidate
Runoff
Votes (%)
Ted Cruz
628,336 (57%)
David Dewhurst
477,888 (43%)
Answer: Ted Cruz won the runoff. He received a majority of the votes in the runoff
election. It appears that he not only received many of the votes of the losing
candidates from the original primary, but he may have also captured some of David
Dewhurst’s votes. (Either that or many of Dewhurst’s supporters in the original
primary didn’t show up for the runoff.)
Another election method called the “Instant Runoff” method, or elimination method, is to
ask voters to rank the candidates in order of preference. Consider this new scenario
and election method. The compiled results are shown in a table, called a preference
schedule.
6) Three candidates (Alex, Blake, and Charlie) applied for a position at a company. The
interviewing committee ranked their choices in order of preference in the table
below. Notice that three people ranked Alex in 1st place and Blake in 2nd place, and
so on.
1st choice
2nd choice
3rd choice
3 voters
Alex
Blake
Charlie
1 voter
Alex
Charlie
Blake
2 voters
Blake
Alex
Charlie
1 voter
Blake
Charlie
Alex
2 voters
Charlie
Blake
Alex
Part A: How many voters were on the interviewing committee?
Answer: There were 9 voters.
Part B: A majority is still needed to win in his scenario. How many votes are needed
for a majority?
Answer: At least 5 votes are needed (9/2 = 4.5, then round up).
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Suggested Instructor Notes 1.C, Instant Runoff
Part C: How many 1st place votes did each candidate receive? Do we have a
winner?
Answer: Alex received 4 first place votes, Blake received 3 and Charlie received 2.
No one received a majority, so no winner has been found.
Part D: Which candidate received the fewest 1st place votes? To employ the Instant
Runoff method, mark out or eliminate that person’s name in each column of
the table. Any name below that one in the table will now move up.
Answer: Charlie received the fewest 1st place votes and is eliminated.
Part E: How many 1st place votes do we now award to the two remaining
candidates? Who wins the election?
Answer: Alex received four 1st place votes and Blake received five 1st place votes.
Blake now has a majority and wins the election. (The table below is not required but
illustrates the rationale for Blake.)
3 voters
1st choice Alex
2nd choice Blake
3rd choice Charlie
1 voter
Alex
Charlie
Blake
2 voters
Blake
Alex
Charlie
1 voter
Blake
Charlie
Alex
2 voters
Charlie
Blake
Alex
7) The method shown in question 5 is known as the Instant Runoff method. What are
some pros and cons of this method?
Answers will vary.
Sample answer: Pro: The Instant Runoff method saves time and money, and
reduces the amount of effort required (e.g., people don’t have to come back and
vote again). Con: This method could be cumbersome if there is a large number of
voters or candidates.
8) Can you think of another election method for choosing the winning candidate in an
election? What are pros and cons of your method? Can any process guarantee that
the choice made (or winner) in an election is correct and fair to all candidates?
Answers will vary.
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Practice Assignment 1.C
Practice Assignment 1.C
1) During the 2013–2014 academic year, Wichita Falls Independent School District in
Wichita Falls, Texas, began planning for new and renovated high school facilities. A
committee of community members was formed to explore various options for a bond
proposal to pay for the plan. The committee presented three options to the Board of
Trustees.1
Plan
Estimated Minimum Cost
Plan A
$132 million
Plan B
$168 million
Plan C
$127 million
Major Features
Build one new school
Renovate two old schools
Build two new schools
Close all old schools
Build one new school
Renovate one old school
The Board of Trustees needed to decide on which plan to put on the ballot in the
bond election. Suppose that the board members ranked their choices, with the
results shown in the preference schedule below. The election method used to
determine the winner is the elimination method.
1st Choice
2nd Choice
3rd Choice
3 votes
A
B
C
3 votes
B
A
C
1 votes
C
A
B
Part A: How many board members voted?
Part B: How many votes are needed for a majority?
Part C: Did one plan receive a majority of the votes? If so, which one?
Part D: Now use the Instant Runoff method to eliminate one of the plans and look
for a winner. Does one plan have a majority of the votes? If so, which one?
1
Source: http://www.wfisd.net/CFAT
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Practice Assignment 1.C
2) Suppose a different preference schedule of the voting for the school plan resulted in
the election. A new preference schedule is shown below. The election is a majority
election, using the Instant-Run-Off method, if needed. Which plan is the winner?
3 votes
A
B
C
st
1 Choice
2nd Choice
3rd Choice
2 votes
B
A
C
2 votes
B
C
A
3) Suppose the board members decided not to recommend a plan, but instead put all
of the plans on a ballot for a community-wide ranking election. Voters were also
given the choice of “Keep” to designate keeping all the schools the way they are.
The resulting preference schedule may look something like this:
1st Choice
2nd Choice
3rd Choice
4th Choice
Column 1
3,294 votes
A
B
Keep
C
Column 2
2,461 votes
B
A
Keep
C
Column 3
1,127 votes
Keep
C
B
A
Column 4
2,014 votes
C
Keep
A
B
Column 5
Part A: How many votes were cast in the election shown above?
Part B: How many votes were needed for a majority?
Part C: Which option had a majority?
Part D: Using the Instant Runoff method, which option is eliminated? Plan A?
Plan B? Plan C? Option “Keep”?
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Practice Assignment 1.C
Part E: Once the option from Part D is eliminated, how many 1st place votes does
each option have?
Part F: None of the remaining options has a majority. You would need to perform
another elimination. Which option should be eliminated next, and which
option is the ultimate winner?
Part G: Look back at Parts A–F. Compare how each plan began with how the plan
ended in the results. For example, how popular was Plan A in Part D?
Where did Plan A finish? Perform this analysis for each plan.
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Practice Assignment 1.C
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Student Pages 1.D, Borda Count
Lesson 1, Part D
Borda Count
Recall that in Lesson 1, Part C, an
interviewing committee was trying to select
the appropriate applicant to fill a job
position. In this lesson, we will explore
another election method called the Borda
Count method.
1) Suppose the committee’s preference
schedule looks like the one shown
below.
Credit: iStockphoto
Part A: How many people are on the committee? How many votes are needed for a
majority?
Part B: Does anyone have a majority of the votes?
1st choice 2nd choice 3rd choice Column 1 1 voter Alex 3 voters Alex 3 voters Blake 1 voter Blake Blake Charlie Alex Charlie Charlie Blake Charlie Alex Column 2 Column 3 Column 4 Column 5 Objectives for the lesson
You will understand:
¨ Earning the most votes may not be sufficient to win an election.
¨ There are multiple considerations and methods for ranking candidates in an
election.
¨ Multiple ranking methods can be employed to make decisions about other
issues.
You will be able to:
¨ Employ the Borda Count method to determine the winner of an election.
¨ Apply and justify selection strategies to election results.
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Student Pages 1.D, Borda Count
2) In the Borda Count method, voters’ choices are given weights. In an election with 3
choices, a voter’s 1st choice is worth 3 points, the 2nd choice is worth 2 points, and
the 3rd choice is worth 1 point.
Part A: Write these point values in the appropriate cells of the first column.
Part B: Column 2 represents one voter. How many points did that voter give Alex?
Blake? Charlie? Write the appropriate points next to each applicant’s name
in Column 2.
Part C: In Column 3, how many total points did Alex receive? (Notice that three
voters put Alex in 1st place, each giving him 3 points.) How many points did
Charlie and Blake receive? Fill in Column 3, then continue to Columns 4
and 5.
Part D: How many total points did each candidate receive? Who should be offered
the position?
Part E: If that person already took another job and turns down the offer, should the
committee offer the job to someone else? If so, who and why?
Part F: How many points does each voter award to candidates? How many total
points are then available to candidates? How can you use this information to
cross-check your work?
3) Does the “point” method of the Borda Count election method provide advantages or
disadvantages over the elimination method in Lesson 1, Part C? Explain your
answer.
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Suggested Instructor Notes 1.D, Borda Count
Lesson 1, Part D
Borda Count
Overview and student objectives
Overview
Lesson Length: 25 minutes
In this lesson, students continue their investigation of voting
and election methods by awarding points to their favorite
candidate or issue using the Borda Count method. It is
important to note that students should know that election
organizers should not manipulate an election in order to
elect their favorite candidate, but instead choose the method
in advance of the election.
Prior Lesson: Lesson 1,
Part C, “Instant Runoff”
Students should continue to work in the same groups in
order to develop a high level of comfort with one another and
the class as a whole and continue to build a learning
community.
Objectives
Next Lesson: Lesson 2,
Part A, “Graphical Displays”
(25 minutes)
Constructive Perseverance
Level: 1
Theme: Civic Responsibility
Goals: Reasoning,
Evaluation
Quantitative Reasoning
Outcomes: N1, N2
Related Foundations
Outcome: N8
Students will understand that:
•
Earning the most votes may not be sufficient to win an election.
•
There are multiple considerations and methods for ranking candidates in an
election.
•
Multiple ranking methods can be employed to make decisions about other
issues.
Students will be able to:
•
Employ the Borda Count method to determine the winner of an election.
•
Apply and justify selection strategies to election results.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Student Pages for Lesson 1, Part D
•
Practice Assignment 1.D
Prerequisite assumptions
None.
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Suggested Instructor Notes 1.D, Borda Count
Making connections
This lesson:
•
Connects back to work that students encountered in the Foundations of
Mathematical Reasoning course, reading, interpreting, and making decisions
about data summarized in tables and graphical displays.
•
Connects forward to the remainder of the course in which decisions are based on
data rather than emotion or anecdotal evidence.
Background context
None.
Suggested instructional plan
Frame the lesson
(4 minutes)
Student
Page
•
Students should be able to immediately begin to answer question 1
without an introduction from the instructor. In the previous example,
there were an odd number of voters, so students had to round up to
determine the number needed for a majority. In this case, there is an
even number (eight) of voters. Watch for students who may simply
divide by two and believe that four votes are sufficient for a majority.
o Reminder: The names for the voting example were purposefully
chosen to reflect gender-blindness and can be either a male’s or
female’s name.
•
Ask:
o “Does any applicant have more than 50% of the 1st place votes?”
[Answer: No, Alex and Blake each have exactly 50%.]
o “If four votes are insufficient for a majority, how many are
needed?” [Answer: We have to increase to the next whole
number. Five votes are needed for a majority.”] “Why?”
•
Transition to the lesson activities by briefly discussing the Objectives
for the lesson. These objectives are similar to the objectives for
Lesson 1, Part C. If you are completing the two lessons on the same
day, you may wish to simply point out to students that the objectives
are similar.
Lesson activities
(16 minutes)
Group
Work
then
Debrief
Questions 2 and 3
•
Allow about 12 minutes for students to answer questions 2 and 3.
Circulate throughout the room to ensure that students understand the
instructions in the introduction to question 2.
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Suggested Instructor Notes 1.D, Borda Count
•
Since this lesson is early in the course, you may wish to begin by
providing more structure, displaying the table on the document
camera and illustrating how to complete the first two columns.
However, you may wish to consider whether illustrating the process
will undermine your efforts to develop perseverance.
•
During the debriefing discussion, ask students to consider what
impression that interviewing committee members might have about
each applicant. For example, in Column 2, one voter ranked the
applicants in the following order: Alex, Blake, and Charlie.
o Ask, “What impression does this voter have about Charlie?”
[Answer: We don’t know. That person may really feel that Charlie
is not a good fit for the company. However, he or she may feel
that Charlie is a good candidate, just not as strong as Alex or
Blake.]
•
Question 2, Part F: Encourage students to always check their work
when possible: “It is very easy to lose your place while computing the
points in the Borda Count method, so the cross-check method is
important.”
•
Refer students back to question 1. Remind them to always check for a
majority winner at the beginning, rather than jumping to use other
methods.
Wrap-up/transition
(5 minutes)
Wrap-up
•
The overall goal is to give students awareness that there are multiple
methods for running elections and to count votes.
o Ask students to reflect on question 3. Compare and contrast the
methods in Lesson 1, Parts C and D. Point out that either method
is fine (and there are other methods as well) and that there is not
a “right” method.
o The important points are that decisions on which method is
appropriate and how to count the votes must be made in
advance.
•
Ask students to reflect on this lesson and write their reflections on a
3x5-inch card. Sample questions and prompts include:
o “Was the lesson interesting? How did the discussion go in your
group? Did everyone contribute? How can we ensure that group
discussions are productive?”
o “Is there anything you would like to share with your instructor? “
o “List one concept that is not clear after the lesson.”
•
Have students refer back to the Objectives for the lesson and check
the ones they recognize from the activity. Alternatively, they may
check objectives throughout the lesson.
Transition
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Suggested Instructor Notes 1.D, Borda Count
•
Remind students about Lesson 1, Part A, when student data were
collected. Lesson 2, Part A, will explore graphical displays, including a
graph of the Lesson 1, Part A data.
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 1.D.
•
Give the Preview Assignments, if any, for any lesson activities that you plan to
complete in the next class meeting.
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Suggested Instructor Notes 1.D, Borda Count
Lesson 1, Part D
Borda Count – ANSWERS
Recall that in Lesson 1, Part C, an
interviewing committee was trying to select
the appropriate applicant to fill a job
position. In this lesson, we will explore
another election method called the Borda
Count method.
1) Suppose the committee’s preference
schedule looks like the one shown
below.
Credit: iStockphoto
Part A: How many people are on the committee? How many votes are needed for a
majority?
Answer: There are 8 people voting. Half of 8 is 4, so bump that up to 5 votes to
constitute a majority.
Part B: Does anyone have a majority of the votes?
Answer: No one has a majority.
1st choice (3 pts) 2nd choice (2 pts) 3rd choice (1 pt) Column 1 1 voter Alex (3 pts) 3 voters Alex (9 pts) 3 voters Blake (9 pts) 1 voter Blake (3 pts) Blake (2 pts) Charlie (6 pts) Alex (6 pts) Charlie (2 pts) Charlie (1 pt) Blake (3 pts) Charlie (3 pts) Alex (1 pt) Column 2 Column 3 Column 4 Column 5 Objectives for the lesson
You will understand:
¨ Earning the most votes may not be sufficient to win an election.
¨ There are multiple considerations and methods for ranking candidates in an
election.
¨ Multiple ranking methods can be employed to make decisions about other
issues.
You will be able to:
¨ Employ the Borda Count method to determine the winner of an election.
¨ Apply and justify selection strategies to election results.
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Suggested Instructor Notes 1.D, Borda Count
2) In the Borda Count method, voters’ choices are given weights. In an election with 3
choices, a voter’s 1st choice is worth 3 points, the 2nd choice is worth 2 points, and
the 3rd choice is worth 1 point.
Part A: Write these point values in the appropriate cells of the first column.
Answer: Noted in table in red.
Part B: Column 2 represents one voter. How many points did that voter give Alex?
Blake? Charlie? Write the appropriate points next to each applicant’s name
in Column 2.
Answer: Noted in table in red.
Part C: In Column 3, how many total points did Alex receive? (Notice that three
voters put Alex in 1st place, each giving him 3 points.) How many points did
Charlie and Blake receive? Fill in Column 3, then continue to Columns 4
and 5.
Answer: Noted in table in red.
Part D: How many total points did each candidate receive? Who should be offered
the position?
Answer:
Alex: 3 + 9 + 6 + 1 = 19 points
Blake: 2 + 3 + 9 + 3 = 17 points
Charlie: 1 + 6 + 3 + 2 = 12 points
Alex should be offered the position.
Part E: If that person already took another job and turns down the offer, should the
committee offer the job to someone else? If so, who and why?
Answers may vary.
Sample answer: Blake’s total points were close enough to Alex’s that Blake should
be offered the position next; a new preference schedule should be created, which
eliminates Alex (in which case, Blake wins anyway).
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Suggested Instructor Notes 1.D, Borda Count
Part F: How many points does each voter award to candidates? How many total
points are then available to candidates? How can you use this information to
cross-check your work?
Answer: Each voter awards 6 points (3 points for 1st, 2 points for 2nd and 1 point for
3rd). Since there are 8 voters, each awarding 6 points, there are a total of 48 points
to be awarded. Since Alex’s, Blake’s, and Charlie’s points total 48 (19 + 17 + 12), we
can be fairly confident that we didn’t make any computational errors.
3) Does the “point” method of the Borda Count election method provide advantages or
disadvantages over the elimination method in Lesson 1, Part C? Explain your
answer.
Answers will vary.
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Suggested Instructor Notes 1.D, Borda Count
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Practice Assignment 1.D
Practice Assignment 1.D
Recall that during the 2013–2014 academic year, the school board in Wichita Falls,
Texas, began planning for new and renovated high school facilities. A committee of
community members was formed to explore various options for a bond proposal to pay
for the plan. The committee presented three options to the Board of Trustees.1
Plan
Estimated Minimum Cost
Plan A
$132 million
Plan B
$168 million
Plan C
$127 million
Major Features
Build one new school
Renovate two old schools
Build two new schools
Close all old schools
Build one new school
Renovate one old school
1) The Board of Trustees needs to decide on which plan to put on the ballot in the bond
election. Suppose that the board members ranked their choices, with the results
shown in the preference schedule below.
st
1 Choice
2nd Choice
3rd Choice
3 votes
A
B
C
3 votes
B
A
C
1 votes
C
A
B
Part A: Using the Borda Count method, how many total points will be distributed?
Part B: How many points does each option receive?
Part C: Which plan is the winner?
1
Source: http://www.wfisd.net/CFAT
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Practice Assignment 1.D
2) Suppose that the preference schedule was as shown below.
3 votes
A
B
C
st
1 Choice
2nd Choice
3rd Choice
1 votes
B
A
C
2 votes
B
C
A
Part A: Is there a majority winner?
Part B: Using the Borda Count method, which plan is the winner?
3) Suppose the board members decided not to recommend a plan, but instead put all
of the plans on a ballot for a community-wide ranking election. Voters were also
given the choice of “Keep” to designate keeping all the schools the way they are.
The resulting preference schedule may look something like this:
1st Choice
2nd Choice
3rd Choice
4th Choice
3,294 votes
A
B
Keep
C
2,461 votes
B
A
Keep
C
1,127 votes
Keep
C
B
A
2,014 votes
C
Keep
A
B
Part A: How many points should be awarded for “1st choice” when using the Borda
Count method with 4 options?
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Practice Assignment 1.D
Part B: Use the Borda Count method to determine the winning plan for the school
facility plan.
Part C: Show the cross-check of your work.
Part D: Which option would win if the “Instant Runoff” elimination method from
Lesson 1, Part C were used to determine the winner?
4) Consider the scenario presented in the table in question 3 again. What if, before the
election, you wanted Plan B to win? Would it be better to promote Plan B or
sabotage Plan A in some way such as supporting Plan C? Present other ways to
improve the rating of Plan B.
5) Go to the internet and research other real-life uses of the Borda Count method (e.g.,
http://en.wikipedia.org/wiki/Borda_count). Find and present an example of the use of
the Borda Count method. Discuss and explain the use of the method and how the
winner is determined.
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Practice Assignment 1.D
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