Unit Plan: Measures of Central Tendency and...

UNIT PLAN: MEASURES OF CENTRAL TENDENCY AND SPREAD

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Kaci Cohn Probability and Statistics

Mrs. Cathy Plowden, Cooperating Teacher

Mrs. Jennifer Cribbs, Professor

Unit Plan: Measures of Central Tendency and spread

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Table of Contents

UNIT INTRODUCTION .......................................................................................................... 2

RATIONALE FOR UNIT ............................................................................................................................. 2

UNIT OBJECTIVES ..................................................................................................................................... 2

STATE STANDARDS .................................................................................................................................. 3

NCTM NATIONAL CURRICULUM STANDARDS ................................................................................... 4

INSTRUCTIONAL STRATEGIES ................................................................................................................ 4

ASSESSMENT STRATEGIES ..................................................................................................................... 5

RESOURCE LIST ............................................................................................................... 6-10

LONG RANGE PLAN ..................................................................................................... 11-13

HISTORY AND TECHNOLOGY COMPONENT .................................................................... 14

DEVELOPMENT OF THE TOPIC THROUGH K-12. .......................................................... 15-16

LESSON ONE. ................................................................................................................ 17-18

LESSON TWO. ............................................................................................................... 19-22

LESSON THREE. ............................................................................................................. 23-25

LESSON FOUR. .............................................................................................................. 26-29

LESSON FIVE ................................................................................................................. 29-32

APPENDIX .................................................................................................................... 33-49

TO MAKE THE VOCABULARY BOOK: ............................................................................................... 33

EXIT SLIP .................................................................................................................................................. 33

LESSON TWO .................................................................................................................................. 32-35

LESSON THREE ................................................................................................................................. 36-39

LESSON FOUR.................................................................................................................................. 40-41

LESSON FIVE .................................................................................................................................... 42-49


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Unit Introduction

RATIONALE FOR UNIT

This unit is important for several reasons, first and foremost because it aligns with both state and national

standards, which are heavily emphasized in today’s classrooms. Basic probability and statistics are critical

subjects with real-world application which are used daily: the chance of rain, whether you make a green light,

political party affiliation, and the odds that your car gets stolen. Every time you say ―what are the odds‖ or

―the chance that that happens…‖, you’re talking about probability. It is important that students be able to

understand the population they live in and are surrounded by, and that they can appropriately collect,

represent, and understand the meaning of data. I chose this lesson because it fit well into the time that I

taught. I taught the final lesson in the plan as a test review, before their unit test. I also decided to do a unit

in the Probability and Statistics class because I didn’t take it in high school and therefore it can sometimes be

a difficult subject for me to understand. I figured that by creating a unit with several lessons, I would

understand the material better and be more helpful in answering questions during my teacher’s lessons. There

is also a good chance that I teach a probability and statistics class and by creating and teaching a unit in that

class now, I’m becoming more familiar with the material and required standards, and will feel more

comfortable if I do end up teaching that class.

UNIT OBJECTIVES

Upon completion of this unit, students will be able to:

1. Find the number of ways that ―r‖ objects can be selected from ―n‖ objects, using the permutation rule

2. Find the number of ways that ―r‖ objects can be selected from ―n‖ objects without regard to order,

using the combination rule

3. Describe sets of measurements in the face of variability

4. Describe a distribution of data from its graph

5. Understand the term outliers by observing graphs

6. Compute measures of center and spread

7. Relate summary statistics (mean, median, standard deviation) to graphs of data

8. Explain and compute the meanings of each element of a box plot

9. Relate box plots to histograms and sketch one after seeing the other

10. Differentiate between the mean of a distribution from the median

11. Understand the difficulty in measuring accurately

12. Measure variability in data and identify causes for it


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STATE STANDARDS

Standard DA-1: The student will understand and utilize the mathematical

processes of problem solving, reasoning and proof, communication,

connections, and representation.

DA-1.2 Execute procedures to find measures of probability and statistics by using tools such as

handheld computing devices, spreadsheets, and statistical software.

DA-1.5 Apply the principles of probability and statistics to solve problems in real-world contexts.

DA-1.6 Communicate a knowledge of data analysis and probability by using mathematical

terminology appropriately.

DA-1.8 Compare data sets by using graphs and summary statistics.

Standard DA-3: Through the process standards the s tudent will

demonstrate an understanding of how to collect, organize, display, and

interpret data.

DA-3.2 Organize and interpret data by using pictographs, bar graphs, pie charts, dot plots, histograms,

time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots.

Standard DA-4: Through the process standards the student will

demonstrate an understanding of basic statistical methods of analyzing

data.

DA-4.4 Use procedures and/or technology to find measures of central tendency (mean, median, and

mode) for given data.

DA-4.6 Use procedures and/or technology to find measures of spread (range, variance, standard

deviation, and interquartile range) and outliers for given data.

DA-4.7 Use procedures and/or technology to find measures of position (including median, quartiles,

percentiles, and standard scores) for given data.

DA-4.9 Explain the significance of the shape of a distribution.

Standard DA-5: Through the process standards the student will

demonstrate an understanding of the basic concepts of probability.

DA-5.2 Use counting techniques to determine the number of possible outcomes for an event.


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NCTM NATIONAL CURRICULUM STANDARDS

Data Analysis and Probability Standard for Grades 9 -12

For univariate measurement data, be able to display the distribution, describe its shape, and select and

calculate summary statistics

Understand the concepts of sample space and probability distribution and construct sample spaces and

distributions in simple cases

Understand histograms, parallel box plots, and scatterplots and use them to display data

Measurement Standard for Grades 9-12

Analyze precision, accuracy, and approximate error in measurement situations

Problem Solving Standard for Grades 9-12

Build new mathematical knowledge through problem solving

Solve problems that arise in mathematics and in other contexts

Reasoning and Proof Standard for Grades 9-12

Make and investigate mathematical conjectures

INSTRUCTIONAL STRATEGIES

Students are arranged in groups of three, randomly chosen by Mrs. Plowden. I really like this setup because I

am an avid supporter of group work and believe that, when done correctly, it will greatly improve student

learning. All of my lessons are inquiry-based learning tasks, in which the students begin by exploring a topic

and through the activity, the topic is learned and/or connected to previous content. This groupwork structure is

also very convenient because it allows both Mrs. Plowden and I to be available to help people/groups that

are confused and explain a concept in more detail. Also, groups can work at varying paces, depending on

how much prior knowledge and understanding they bring to the table. Groupwork requires that students think

individually to muster their thoughts and then understand their ideas and the material well enough to be able

to explain it to their peers. It allows students to discover alternate ways of solving a problem and also

creates a bond among the students that may not have been there before the groupwork began.

Depending on the lesson, there are either guided notes for them to follow along, or we discuss things as a

class that they are expected to remember, thus should take notes. Alongside of my lessons, the students will

also have a small book that they created from computer paper for just this unit, in which they’ll write terms

and definitions with formulas for each new vocabulary word that we learn.

During class discussions, students will be randomly called upon to explain their thoughts or reasoning behind

their opinions and answers, forcing the students to stay actively engaged in the lesson. The questioning of

students’ thought will be persistent, because students always need to justify their thoughts; ―I don’t know‖ is not

an acceptable reason for any answer, ever.


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At the beginning of each week, I am going to go over any lingering questions from the week before

(identified by the exit slips discussed in the Assessment Strategies section), and discussing the main concepts we

learned the previous week. This will quickly refresh students and get their head back into my classroom and

into our math world. I had a teacher here at Clemson who did this, along with the exit slips, and I found it to

be incredibly helpful in my success in the course. He was incredibly well-tuned in to his classes and what

students were thinking because of this method.

ASSESSMENT STRATEGIES While students work on their activities, I will be informally assessing students by walking around the room,

observing and listening to student conversation. Occasionally, I’ll stop at a group and ask them about a part

of the activity and briefly discuss it, asking for their reasoning and justification for why they did what they

did.

During class discussions, I’ll have students think for themselves, then discuss with their groups, and then we can

discuss the topic as a class. This encourages students to think for themselves and gives them more confidence

to share their ideas. Frequently, students are nervous to share their opinions or answers in fear of being

wrong or judged, but by using this tactic that we learned in READ 498, students are more prone to sharing

their opinions because they only have to tell their initial thoughts to two other people, and they can modify

their thoughts depending on what their peers share. Students’ responses during these discussions serve as an

informal formative assessment for both the student and I to reflect upon.

Worksheets and guided notes will be handed out and collected for a grade the class after they’re completed

(that way they can be used to complete the homework). This will help me formally gauge how well the

students understand the material and potentially adjust my teaching timeline to include further instruction on a

topic if necessary. Homework will be handed out periodically, which will review the past day or two’s

material, to help assess how well students can retain the information they learned in class. If students are

succeeding with flying colors on the class work assignments and then barely scraping by on the homework, it is

a red flag to me that maybe that student is not utilizing groupwork in an appropriate manner; the student

may just be copying other students’ work and turning it in as his own, with little or no understanding of the

material. I would conference with this student if such a situation were to occur, to both help them understand

the material and ensure they use their group members as resources for help, not just for copying work. These

assignments will act both as formative and summative assessments.

At the end of the unit, I will administer a test to act as a summative assignment. By this point, the students

should know all of the material and understand it, An important thing we’ve learned this semester in several

classes is that is it critical to assess students in the way that they’re taught. Because of this, I am going to have

a two-part test: individual and an inquiry-based group task. This way, I can formally assess the students’

knowledge. This will also act as both a formative and summative assessment because it will provide me with

feedback on how I should change the unit in the future, but it is also assessing the students’ proficiency in

measures of central tendency and spread.

My final form of assessment is at the end of each week, I am going to have students fill out an exit slip (See

Appendix for detailed questions). By having students fill this out at the end of the week, I’m able to use the

weekend to read over them and modify my plans for the coming week accordingly. This slip will act as a

formal formative assessment.


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Resource List BANNISTER, NICOLE. "LECTURE NOTES AND RESOURCES." MTHSC 408.

CLEMSON, SC. FALL 2011. LECTURE. In MTHSC 408, covered a variety of topics. We discussed moral issues in classrooms, how to

implement specific strategies, specific inquiry-based mathematics tasks, and more. We had to read

several articles and chapters from books and reflect upon these with our thoughts and how our view of

the classroom changed because of it. She provided us with blackline activities for us to use in our

classroom, so that we would have some interesting tasks right from the beginning.

BIUMAN, ALLAN G. ELEMENTARY STATISTICS: A STEP-BY-STEP APPROACH: INSTRUCTORS EDITION. NEW YORK, NY: MCGRAW HILL, 2004. PRINT.

This is the instructor’s edition of a common textbook that high schools use for probability and statistics

classes. It can be used to help guide specific lessons and giving examples/homework.

BOCK, R.K., AND W. KRISCHER. THE DATA ANALYSIS BRIEFBOOK. N.P., MAR 1999. WEB. 26 NOV 2011. <HTTP://PHYSICS.WEB.CERN.CH/PHYSICS/DATAANALYSIS/BRIEFBOOK/>.

This website is mainly centered on The Data Analysis BriefBook, which is essentially an extensive

glossary of probability and statistics vocabulary. I like it because of the extensiveness of it, therefore

it can be used for several different levels, and each definition is the basic information students need to

know, along with a formula if the word has a corresponding one. Also on this website, there are links

to outside resources related to probability and statistics, which are also quite readable.

BURRILL, GAIL, CHRISTINE A. FRANKLIN, LANDY GODBOLD, AND LINDA J. YOUNG. NAVIGATING THROUGH DATA ANALYSIS IN GRADES 9–12. 2003.

The activities in this book help students understand simple random sampling and also comes with a

CD-ROM that has different interactive activities to help both students master the concepts and

additional readings for teachers.


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GEOGEBRA DOWNLOAD. GEOGEBRA, N.D. WEB. 26 NOV 2011. <HTTP://WWW.GEOGEBRA.ORG/CMS/EN/DOWNLOAD>. Geogebra is a really cool program because it’s both free and you can either download it to your

computer or you can just use it online. It’s available in several languages so for students whose first

language is not English, they can still benefit from using Geogebra. Geogebra is dynamic

mathematics software that can be used for any mathematics class beginning at the elementary level

and going through the collegiate level, including probability and statistics.

HALTIWANGER, LEIGH. "POWERPOINT PRESENTATIONS." READ 498. CLEMSON, SC. FALL 2011. LECTURE.

The slides from READ 498 are very descriptive and have a lot of information regarding different

learning strategies that you can implement in your classroom. From these I learned how to assess what

students already know, find out what they want to learn, where their confusion is, and so much more.

They are strategies I plan to implement in my classroom.

HORTON, ROBERT. "LECTURE NOTES AND RESOURCES." EDSEC 326. CLEMSON, SC. SPRING 2011. LECTURE. In this class, I was taught to think about mathematics differently. We were provided with several

challenging mathematics tasks that I have since used and/or seen in my classroom observations. Dr.

Horton was the first professor who taught us how critical it is for students to explore concepts and

discover more concrete formulas on their own, through inquiry-based learning.

"MICROSOFT MATHEMATICS 4.0." DOWNLOAD CENTER. MICROSOFT, 12 JAN 2011. WEB. 26 NOV 2011. <HTTP://WWW.MICROSOFT.COM/DOWNLOAD/EN/DETAILS.ASPX?ID=15702>. Microsoft Mathematics is a very cool tool we learned about in ED F 425. It is a free program

available for download that will solve several different types of problems. It is essentially a

graphing calculator, which is incredibly convenient for students who maybe have a computer at home

of computer access somewhere and don’t own a graphing calculator, so that they can reap the

benefits of owning one.

NCTM, FIRST. PRINCIPLES AND STANDARDS FOR SCHOOL

MATHEMATICS. RESTON, VA: NCTM, 2000. PRINT.

This book is incredibly helpful because it includes the national standards for math education. It gives

examples on how teachers can complete these standards to their fullest and how teachers can change

students’ views of mathematics.


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O'CONNOR, JOHN J., AND EDMUND F. ROBERTSON. HISTORY OF MATHEMATICS. N.P., AUG 2011. WEB. 8 DEC 2011. <HTTP://TURNBULL.MCS.ST-AND.AC.UK/HISTORY/>.

This is an interesting website simply because it has several different parts of the history of

mathematics. It talks about the history of mathematics during different time periods. It is always

good for both students and teachers to understand the history behind the mathematics they’re learning

and how it developed and changed over time.

ONLINE MATH HELP & LEARNING RESOURCES. N.P., N.D. WEB. 26 NOV 2011. <HTTP://WWW.ONLINEMATHLEARNING.COM/>. This website will be incredibly helpful to teachers, students and parents. There are several resources

for many classes, including both Probability and Statistics. Each course’s page differs depending on

the subject, but there are lesson plan resources, definitions and explanations, videos and subject-

related games on the different pages.

POLLETT, PHIL. THE PROBABILITY WEB. N.P., 17 JAN 2011. WEB. 26 NOV 2011. <HTTP://PROBWEB.BERKELEY.EDU/>.

This is an awesome resource because it has a wide variety of information including research abstracts,

books, probability conference information, job websites, quotes, software, and teacher resources. I

think this is my favorite resource that I’ve found so far because of the plethora of information that is

included on the site.

"PROBABILITY AND STATISTICS." MATH FORUM. DREXEL UNIVERSITY, 2011. WEB. 26 NOV 2011. <HTTP://MATHFORUM.ORG/PROBSTAT/>.

This is a helpful website because it has four major headings, Classroom Materials for Teachers and

Students, Software for Probability and Statistics, Internet Projects, and Public Forums, which all then

have several links on their pages with corresponding websites. There are so many resources available

just from this one resource that it is an easy one-stop website to navigate to.

PROBLEM WITH SPREADSHEETS." MATHEMATICS TEACHER. 92.5 (1999): 407. WEB. 29 SEP. 2011. <HTTP://WWW.NCTM.ORG/PUBLICATIONS/ARTICLE.ASPX?ID=17870>. This article addresses students’ tendency to underestimate the probability of an even occurring more

than once, and the importance of students committing to a prediction before determining the results.


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SCHEAFFER, RICHARD, ANN WATKINS, JEFFREY WITMER, AND MRUDULLA GNANADESIKAN. ACTIVITY-BASED STATISTICS: INSTRUCTOR RESOURCES. 2ND ED. EMERYVILLE, CA: KEY COLLEGE PUBLISHING, 2004. PRINT.

This book is incredibly helpful for teachers in planning their lessons because it consists of different

activities to use for several different topics within statistics.

"SECONDARY MATHEMATICS BENCHMARKS PROGRESSIONS, GRADES

7–12: PROBABILITY AND STATISTICS (PS)." MATHEMATICS

BENCHMARKS, GRADES K-12. THE DANA CENTER, N.D. WEB. 26 NOV

2011.

<HTTP://WWW.UTDANACENTER.ORG/K12MATHBENCHMARKS/SECO

NDARY/PROBSTAT.PHP

This website has standards that are addressed in a Texan Probability and Statistics course, and

corresponding lessons and tasks for each one. While the standards themselves vary from ours, they

are very similar and the lessons and tasks can definitely be remapped to our standards. This is also a

great site for other strands of math because it has the same thing for math courses beginning in

Kindergarten. It is also very well-organized, because with the elementary levels, you can look through

the benchmarks either by grade or by subject through the grades, which I thought was helpful and

unusual.

SEPPÄLÄINEN, TIMO, AND BÁLINT TÓTH, EDS. ELECTRONIC JOURNAL OF PROBABILITY. N.P., 2011. WEB. 26 NOV 2011. <HTTP://128.208.128.142/~EJPECP/>. This website publishes research articles in probability theory. This is a good resource if a teacher

needs to get into a lot of detail behind a subject or needs to find very explicit proof of something,

then it will most likely be able to be found on this website.

SHAUGHNESSY, J. MICHAEL, AND BETH CHANCE. STATISTICAL QUESTIONS FROM THE CLASSROOM. 2005.

This book discusses some of the most frequently asked statistics questions among both students and

teachers, offering insight as well as examples and visual representations.


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SIEGRIST, KYLE. VIRTUAL LABORATORIES IN PROBABILITY AND STATISTICS. N.P., 1997. WEB. 26 NOV 2011. <HTTP://WWW.MATH.UAH.EDU/STAT/>

This website is incredibly helpful to math teachers teaching probability and statistics. There are 15

generic topics that when you click on one, it gives an explanation of what it is, has links to specific

problems involving the topic, and has different applets that can also be used to understand the topic.

Also, at the bottom of the page, there is link with external resources all regarding probability and

statistics for people to use as a resource to find more information

THE ROPER CENTER FOR PUBLIC OPINION RESEARCH C JOURNAL OF PROBABILITY. N.P., 2011. WEB. 26 NOV 2011.

<HTTP://WWW.ROPERCENTER.UCONN.EDU/>. This website is really cool because it is essentially a library of data. It was founded in 1947 and has

been collecting public opinion data ever since. While most of the data relates to the United States,

there is data from over 50 different nations represented. This is a great resource to find interesting

data that will intrigue students, but that you don’t necessarily have to collect yourselves in the

classroom.


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Long Range Plan Probability and Statistics Long Range Plan

Mrs. Plowden

Big ideas that we will cover this year:

How to develop research questions

How to gather information

How to analyze and evaluate information

How to report information

How to use data to predict outcomes

How to calculate the probability of a particular outcome

The purpose of this class is to make you a better consumer and a research-based decision maker.

Course Outline:

1ST NINE WEEKS

Data Display Types

o Histogram

o Frequency Polygon

o Ogive

o Pareto Chart

o Time-Series

o Pie Chart

o Bar Chart

o Scatterplot

o Stem & Leaf Plot

o Box & Whisker Plot

o Frequency Table


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o Dot Plot

Gathering and reporting survey data

o Good survey techniques

o Types of data gathering

Random

Systematic

Stratified

Cluster

o Measurement Bias

o Reporting findings

o Analyzing Findings

2ND NINE WEEKS

Basic Probability

o Counting Principles

o Permutations

o Combinations

o Tree Diagrams

o Sample space

o Theoretical v. Empirical

o Subjective Probability

o Conduct probability experiments and analyze data

o Calculate probabilities


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3RD NINE WEEKS

Statistical Analysis

o Read and interpret articles from magazines with statistical information

o Conduct basic statistical research using commercial data bases and statistical publications

o Analyze and report on findings

Statistical Inference

o Identify normal curve properties

o Calculate and use z-values in problem solving

o Use central limit theorem to solve problems

o Compute normal curve probabilities

o Calculate and interpret confidence intervals

4TH NINE WEEKS

Hypothesis Testing

o Write null and alternative hypotheses

o Test Hypotheses using confidence intervals

o Test observations using chi square testing

Final Project

o Design and conduct a final statistical or probability project using:

Experiments

Observations

Surveys

o Produce a final report on findings


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History and Technology Component

Conceptually, probability has been around for thousands of years. Things were estimated on intuitive risks,

but there were no set ways to calculate the odds of one situation occurring over another. In 1494, Fra Luca

Paccioloi, an Italian mathematician and mathematics teacher, wrote the first printed work on probability,

Summa de Arithmetica, Geometria, Proportions et Proportionalita (David 1962). In 1526 (not printed until

1663) Geronimo Cardano wrote a book, A Book on Dice Playing, which studied problems of both probability

and combinatorics as well as included some observations on gamblers and their psychology (Gindikin 1994).

The first ―sighting‖ of modern probability was in 1654, when Chevalier de Méré asked Blaise Pascal a

question regarding his gambling. Chevalier de Méré bet on a roll of a die that at least one 6 would appear

during a total of four rolls. He had won several times in the past, so he felt fairly confident in his bet. After

several games, he decided switch things up and change the number of rolls. He bet that he would get a total

of 12, or a double 6, on twenty-four rolls of two dice, but he soon realized that his new game was not

winning him any more money. Frustrated with this new discovery, he asked Pascal why his new approach was

not as profitable, specifically wanting to know whether it was profitable to bet on rolling a double six

appearing at least once in 24 rolls. Pascal wrote a colleague, Pierre de Fermat, asking his advice on the

problem, despite having solved the problem to his relative satisfaction. There were still more complicated

aspects that he struggled with, such as increasing the number of gamblers. Through several letters, Fermat

and Pascal built arguments off each other’s previous letter and in the end, developed general rules of

probability and are considered the founders of probability theory (Devlin 2010). Another important name in

the history of probability is Jakob Bernoulli, who developed a strategy for games of chance. After his death,

his work, Ars Conjectandi, was published and despite not being fully understood at its time of publication, was

a great stepping stone in understanding probability (Shafer 1996).

In my unit on probability, the only technology I think I would be using is a graphing calculator, so that students

can use the random number generator function. The random number generator can take the place of several

things, including a die, spinner and other tools we may not have to actually use. At this level, I would still want

them to compute the mean/median/min/max by hand, just for reinforcement, along with creating a table of

values and the corresponding graph. After they become comfortable with finding these elements and

graphing by hand, I will teach them how to use these functions on their calculator. Mrs. Plowden does have a

SMARTBoard in her classroom, which I will use to project materials and also record student data on, but I will

not be using many of the unique features of the SMARTBoard during this unit.

Sources:

David, F. N. (1962). Games, gods, and gambling. London: Charles Griffin & Co. Ltd. Devlin, K. (2010). The unfinished game: Pascal, Fermat, and the seventeenth-century letter that made the world modern. New York, NY: Basic

Books. Gindikin, S. (1994). The great art: the controversial origins of "cardano's formula". Quantum: The Magazine of Math and Science, 5(1), 40-

45. Shafer, G. (1996). The significance of Jacob Bernoulli's Ars Conjectandi for the philosophy of probability today. Journal of Econometrics,

75(1), 15-32. Retrieved from http://www.sciencedirect.com/science/article/pii/0304407695017666


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Development of the Topic through K-12 You will trace the topic of your unit across the K-12 mathematics curriculum as stated in both state and national

mathematics standards. In addition to a narrative discussion of the origination and development of your topic

through K-12 mathematics schooling, you will also include an image of this tracing in the form of a timeline which

shows the development of your topic. If you and your instructor deem your particular topic too focused for this

portion of your unit plan, you will then articulate the Big Mathematical Idea from which your unit topic stems and

trace the development of the Big Idea. This portion is worth a total of 20 possible points.

Data analysis begins in kindergarten. From kindergarten through second grade, the main focus of it is to organize

some set of data into a graphical display. In kindergarten the displays expected are simply drawings and

pictures. In first grade, students are expected to have advanced to using picture graphs, object graphs bar

graphs and tables. In second grade they’re expected to organize the data in more concrete charts, pictographs,

and tables. In each year, they’re asked to interpret their data in some form, which would entail simply describing

what their displays represent.

From third grade through fifth grade, students are expected to learn more specific graphs and be able to use fully

functional, accurate tables. In third grade, students are expected to be using bar graphs, dot plots, and

pictographs to represent their data. In fourth grade, line graphs and double bar graphs whose scale increments

are greater than or equal to 1 are added onto the list of displays they should be utilizing. Through both third and

fourth grade, students are expected to interpret their data, which again entails being able to describe their graph

and what the elements in it represent. In fifth grade, they steer away from focusing on graphing specifically and

lean more towards the analyzing of the graphs. They look at what can affect the nature of the data set, what

mean, median and mode are and how to calculate them, and learn to interpret what these measures of central

tendency mean in terms of the data set.

In sixth grade, students are also expected to learn to use frequency tables, histograms, and stem-and-leaf plots,

and be able to determine when each one is appropriate to use. They explore deeper into the measures of central

tendency, learning how to determine which measure is the most appropriate for a specific situation. In seventh

grade, students learn and use box plots and circle graphs. They also are introduced to the concept of interquartile

range and are asked to calculate and interpret it. In eighth grade, the focus is not as heavy on data analysis as it

is on other strands of mathematics, but students are continued to be asked to interpret data represented through

both graphs and tables using the measures of central tendency.

In ninth through eleventh grade, students build upon other math skills mainly used in other math strands such as

algebra. These skills are incredibly helpful, however, in analyzing data because algebra is frequently needed to

solve the equations for spread (i.e. standard deviation and variance) along with interpreting specifically what

different parts of a graph means (i.e. the meanings of increasing, decreasing, concavity, continuous, smooth curve,

etc.).

In the probability and statistics class, the meanings of the measures of central tendency are applied to more types

of data and represented on more types of graphs. They learn formulas for concepts they may have been

previously, vaguely introduced to. They also learn about distributions and the spread of data and the meaning

behind distributions.


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THE DEVELOPMENT OF PROBABILITY THROUGH K-12

Kindergarten through 2nd Grade

K-6.1 Organize data in graphic displays in the form of drawings and pictures. K-6.2 Interpret data in graphic displays in the form of drawings and pictures. 1-6.2 Organize data in picture graphs, object graphs, bar graphs, and tables. 1-6.3 Interpret data in picture graphs, object graphs, bar graphs, and tables by using the comparative terms more, less, greater, fewer, greater than, and less than. 2-6.2 Organize data in charts, pictographs, and tables. NCTM: Sort and classify objects according to their attributes and organize data about the objects; NCTM: Represent data using concrete objects, pictures, and graphs.

3rd through 5 th Grade

3-6.2 Organize data in tables, bar graphs, and dot plots. 3-6.3 Interpret data in tables, bar graphs, pictographs, and dot plots. 3-6.4 Analyze dot plots and bar graphs to make predictions about populations.

3-6.5 Compare the benefits of using tables, bar graphs, and dot plots as representations of a given data set. 4-6.1 Compare how data-collection methods impact survey results. 4-6.2 Interpret data in tables, line graphs, bar graphs, and double bar graphs whose scale increments are greater than or equal to 1. 4-6.3 Organize data in tables, line graphs, and bar graphs whose scale increments are greater than or equal to 1. 5-6.2 Analyze how data-collection methods affect the nature of the data set. 5-6.3 Apply procedures to calculate the measures of central tendency (mean, median, and mode). 5-6.4 Interpret the meaning and application of the measures of central tendency. NCTM: Collect data using observations, surveys, and experiments NCTM: Represent data using tables and graphs such as line plots, bar graphs, and line graphs

6th through 8 th Grade

6-6.2 Organize data in frequency tables, histograms, or stem-and-leaf plots as appropriate. 6-6.3 Analyze which measure of central tendency (mean, median, or mode) is the most appropriate for a given purpose. 7-6.2 Organize data in box plots or circle graphs as appropriate. 7-6.3 Apply procedures to calculate the interquartile range. 7-6.4 Interpret the interquartile range for data. 8-6.1 Generalize the relationship between two sets of data by using scatterplots and lines of best fit. 8-6.8 Interpret graphic and tabular data representations by using range and the measures of central tendency (mean, median, and mode) NCTM: Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots

9th through 11 th Grade

Students are building on their algebra and reasoning skills they will be needed to think through and calculate statistic problems. They learn more about the nature of graphs which will help them analyze the graphs of data sets.

Probability and Statistics Class

Students learn more complex ways to analyze data such as variation and standard deviation, and need to be able to apply all of the skills learned in previous classes to solve problems and analyze data. They explore data sets with different levels of complexity and compare 2+ sets of data. They use technology to help them visualize graphs for large samples and populations and use it to help them quickly calculate measures of central tendency and spread. NCTM: Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable NCTM: Understand histograms, parallel box plots, and scatterplots and use them to display data NCTM: Compute basic statistics and understand the distinction between a statistic and a parameter


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Lesson One Title of Lesson: Permutations and Combinations

Subject: Probability and Statistics

Grade level: 12

Teacher: Kaci Cohn (Cooperating teacher: Mrs. Cathy Plowden)

Objective(s): (APS 4)

For students to understand the difference between permutations and combinations through a hands-on activity, discussion, and explicit instruction.

SCSDE Curriculum Standard(s) Addressed: (APS 4, 6)

DA-5.2 Use counting techniques to determine the number of possible outcomes for an event.

NCTM National Curriculum Standard(s) Addressed:

Make and investigate mathematical conjectures

Prerequisites:

Students need to have basic vocabulary skills, enough to rearrange a four letter word into other words

Materials/Preparation: (APS 6)

o Sets of index cards with the letters R, A, T, E written on them

o Notebook paper for note-taking

o Writing utensil for note-taking

Procedures: (APS 4, 5, 6, 7, 8, 9)

o Introductory Activity- Students will be in groups of 2

o Start by giving the students the task to individually figure out how many 3-letter words they can list in 20 seconds.

o Students discuss among their groups how they formed the words (if there was any pattern or logic to it, i.e. rhyming, alphabetically).


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o Give each group a set of index cards and see how many words they can form by rearranging the cards in different orders. Encourage students to use the technique of starting with one letter and see how many words they can make starting with that one letter, and continue with the rest of the letters.

o Now have each student open a pack of m&m’s. Have students determine and record how many different combinations of 2 colors they can make, then 4, then 6. Have students discuss reasoning behind each combination, and if yellow-red is different from red-yellow with their group.

o Main Activity

o Have students discuss in groups what the difference between the first activity and the second. Bring the class together as a whole to discuss their thoughts. What made the groups of letters different than the groups of colors? Discuss other examples of permutations and combinations, without actually talking about the vocabulary words themselves.

o Introduce vocabulary words ―permutation‖ and ―combination‖ and have students record their own definition of each along with an example. Start by using tree diagrams to count out different permutations and combinations. Then, move to discuss the formulas and practice finding different permutations and combinations.

Assessment: (APS 3)

o Through discussion and asking the students their opinions, the teacher can then determine the students’ level of understanding

Adaptations: (APS 6, 7)

o The students are placed in groups so that if there is a struggling student, his/her peer can assist them in clearing up the confusion. If this doesn’t help, the teacher can then help

o Have guided notes for struggling or slow note-taking students

Follow-up Lessons/Activities: (APS7)

o In the next couple lessons, we will discuss how we can use permutations and combinations of objects to compare both the objects themselves and with other objects through different types of graphs.

REFLECTION—PEER-TAUGHT LESSON:

I taught my lesson with Maryah on permutations and combinations. I think our lesson went really well. I think

we split the amount of time that each of us was directing the class pretty well, although I did notice that I took

the reins a couple times when the section of the lesson was in her turf. That is definitely something I need to

work on when I’m collaborating with other teachers. We had a lot of control over the class, despite them

being our peers, and I think this is because we had an interesting and engaging lesson. While our lesson

could have gotten a little out of control because of the m&m’s, we managed to keep the class quiet during

class discussions and on task during group discussions.

It’s always more difficult to teach to our peers because they feel more comfortable with us and therefore

don’t see us as nearly as strong of an authority figure as a student finds a teacher, but I think we stayed at a

perfect comfort level. We laughed and joked when appropriate yet stayed concentrating on the material

and task at hand.


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Lesson Two Title of Lesson: “Living” Box Plot


Grade level: 12



Students understand the meanings of each element of a box plot

Students solidify understanding of distributions including shape, center and spread



DA-1.6 Communicate a knowledge of data analysis and probability by using mathematical terminology appropriately

DA-3.2 Organize and interpret data by using pictographs, bar graphs, pie charts, dot plots, histograms, time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots.

DA-4.4 Use procedures and/or technology to find measures of central tendency (mean, median, and mode) for given data.

DA-4.6 Use procedures and/or technology to find measures of spread (range, variance, standard deviation, and interquartile range) and outliers for given data.

DA-4.7 Use procedures and/or technology to find measures of position (including median, quartiles, percentiles, and standard scores) for given data.




For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics

Prerequisites:

Students need to know how to find quartiles and how to construct a box plot


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o Paper and pencil to record data, create plots and analyze them

o For teacher: Pieces of paper to lay on the ground as marks on the axis

Procedures: (APS 4, 5, 6, 7, 8, 9)

o Introductory Activity

o Discussion aiming to review which plot a box plot is and how to find each different element

o Main Activity-

o Collect data

Record the time each student went to sleep last night on a table projected onto the SMARTBoard

o Create the box plot

Using the data, the teacher lays evenly spaced marks on the floor to represent the hours encompassing the earliest and latest times, to create an axis on the floor.

Find out who went to sleep the earliest and latest and have them stand on their respective tick marks.

The rest of the students position themselves at appropriate marks along the axis, according to when they went to sleep. Make sure that students realize the importance of accuracy (9:00p versus 9:30p are two different locations). If multiple students went to sleep at the same time, have them line up behind each other.

Determine which student(s) represents the median and have them step forward.

Do this by counting off starting at each end (1, 2, 3, 4, 5, 4, 3, 2, 1) until the counting meets.

Determine which student(s) represent the first quartile and have them step forward.

Do this by counting off by counting up from the earliest time and down from the person just below the median. Determine where the counting meets (1, 2, 3, 4, 5, 4, 3, 2, 1) and that person (the median of this group) is the value of the first quartile.

Determine which student(s) represent the third quartile and have them step forward.

Do this the same way as the first quartile except using the latest time as an endpoint instead of the earliest.


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Compute the interquartile range.

Do this by pointing out that the IQR is the distance from the Q1 person to the Q3 person.

Determine the outliers

Recall the formula for calculating outliers (Q1-(1.5*IQR) and (1.5*IQR)+Q3). As a class, determine what the outlying values are, and have any student outside of these bounds step forward and turn sideways so that it’s clearly visible where/who the outliers are. Reinforce that the ―whiskers‖ of the box plot extend to the largest/smallest value that is not an outlier.

o Discuss what the box plot tells us about this particular distribution

o Wrap-Up

o Discuss what would happen to the box plot if certain factors were changed

Ask, ―If everyone went to bed an hour earlier tonight, how would that affect the shape of the distribution? The value of the median?‖ Students should see that changing each value by a constant does the same to the median and has no effect on the shape of the distribution,

Have the median sit down. Discuss how this affects the box plot and how the other parts of the box plot are affected by having one less person in the sample.

If the data is skewed: Have students estimate the mean of the distribution and then compute it. Identify where the mean lies on the living box plot and discuss why it’s not the same as the median. Also discuss how, without calculating, which side of the median is on.

Assessment: (APS 3)

o Teacher will assess students knowledge and understanding of each element of the box plot through discussions of the different elements

o The handout given out at the end of class will be collected the next class for a grade. This will help the teacher identify more specifically whether the students understand and which aspects are causing trouble.


o This activity is very hands-on, with both the students and the teacher, therefore the teacher is able to adjust or review concepts if necessary.


o Students will be given a handout to complete at home to reinforce creating a box plot on paper (attached in Appendix).


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REFLECTION—INDIVIDUALLY TAUGHT LESSON TO PEERS

This lesson did not last nearly as long as I planned for it to. With high school students, the discussion of each

element of a box plot, how to calculate it, and what it means would have taken a lot longer. Because we’re

all seniors in college, however, the discussion of each element was under 30 seconds and therefore shrunk my

lesson down. It was a little bit hectic because there wasn’t very much space in our classroom because I didn’t

want to completely rearrange it, therefore everyone was squeezing into a very small area of space.

On the more positive side, the feedback I got from the lesson was mostly positive. They were all excited

about the lesson and the activity, especially since we used our own personal data and were creating a box-

plot in a nonconventional way.

If I were to actually teach this lesson in the classroom, we would collect the bedtime data while students were

still sitting down, this way we could look at it and estimate what the box-plot may look like. I would also

make sure there was plenty of room (maybe go into the hallway if necessary/allowed) and that the discussion

of each element of a box-plot more detailed.


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Lesson Three Title of Lesson: Matching Statistics to Plots


Grade level: 12



Students understand how summary statistics (mean, median, standard deviation) are related to graphs of data

Students explore how box plots are related to histograms and can sketch one after seeing the other

Students can recognize when and how the mean of a distribution differs from the median


DA-1.6 Communicate a knowledge of data analysis and probability by using mathematical terminology appropriately


DA-3.2 Organize and interpret data by using pictographs, bar graphs, pie charts, dot plots, histograms, time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots.




For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics

Prerequisites:

Students need to be familiar with box plots and histograms as well as the concepts of mean, median, and standard deviation.


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o Pencil to fill out guided notes (attached in Appendix)

o SMARTBoard

Procedures: (APS 4, 5, 6, 7, 8, 9)


o Have students create histograms of previously collected data (class heights in inches, how old do you think the professor is) to review the basic histogram construction.

o Main Activity- Students will proceed using guided notes (attached)

o Students will begin by looking at question 1 on the guided notes and matching the histograms to the appropriate variable.

o In partners, discuss their responses including which histogram matched to which variable and why.

o As a class, discuss the correct answers and the reasons behind them.

o Complete question 2 similarly to question 1 (when finished, discuss in pairs and then as a group).

o Wrap-Up

o In pairs of pairs, students discuss the following question: ―What features of a distribution determine whether the mean and the median will be similar? When does the mean exceed the median?‖ Then discuss this question as a class.

o Again, in pairs of pairs, discuss the following question: ―What features of a distribution influence how large the standard deviation is?‖ Discuss as a class.

Assessment: (APS 3)

o Teacher will circulate the room while students are matching the plots as well as during group discussion, available to redirect off-task students, students who are on the incorrect line of thinking, and to answer any questions/clear up confusion.

o The handout given out as homework will be collected for a grade.


o This lesson is not that difficult of a lesson therefore it shouldn’t need too many adaptations, but it can be adapted through the discussions, in order to focus more intensely on confusing topics.


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o Students will be given a handout to complete at home to reinforce the class activity (attached in Appendix).


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Lesson Four Title of Lesson: How Far Are You From the Mean?


Grade level: 12



Students learn how to measure variability in data

Students will be introduced to the concept of standard deviation inexplicitly

Students learn how the measure of variability reflects the actual data sets




terminology appropriately


DA-3.2 Organize and interpret data by using pictographs, bar graphs, pie charts, dot plots,

histograms, time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots.








Prerequisites:

Students need to know how to find the mean and median of a set of numbers and how to make a dot plot


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o Paper and pencil to record data, create plots and analyze them

o Rulers

Procedures: (APS 4, 5, 6, 7, 8, 9)


o Discussion aiming to review which plot a dot plot is and how to find the mean and median.

o Main Activity- Students should be grouped into groups of 4

o Collect data

In groups, each student measures his/her ―V-Span‖

With the palm of the writing hand on a flat surface, make a ―V‖ between the

pointer and middle finders. Measure the distance from the outside of the

index finger to the outside of the middle finger when you spread them as far

as possible. Measure to the nearest tenth of a centimeter.

Compute the group’s median ―V-Span‖.

Make a dot plot of the measurements. Differentiate between the dots either by

writing names/initials above the dot, color coding, or different symbols. Mark the

median with an arrow below the number line.

o Discussion: In groups, discuss the different sources of variability in the measurements. Then,

bring the groups together as a class to discuss each group’s main thought.

o Describing the spread of the measurements in each group

Have students calculate the absolute value of the difference between their personal

V-Span and the median of the group.

Make a second dot plot of these differences (individual differences from the median).

Use the same labeling system as on the previous plot.

Discuss in groups how to get the second dot plot (absolute differences) from

the first dot plot (actual measurements), without actually computing any

differences. [Fold the dot plot at the median]

Using the idea of the differences from the median, calculate a number that gives a

―typical‖ distance from the median. Represent it graphically on both plots.

Discuss the different measures of spread that the groups come up with.


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Compare their advantages and disadvantages.

Discuss whether they’re ―reasonable‖ measures of spread.

Many groups will focus on the mean absolute difference from the median because

as the teacher, we have directed them towards this.

Record each group’s mean absolute difference from the median on the board and as

a class, construct a dot plot from this data.

What can we tell about a group based on its mean absolute difference from

the median? How do you explain the variation in the variations?

o Wrap-Up

o What sorts of groups tended to have small mean absolute differences from the median?

What sorts of groups large ones? Explain why this occurs.

o Why might it be important to have a quantitative measure of spread, instead of one with

actual measurement values on it?

Assessment: (APS 3)

o Teacher will assess students knowledge and understanding of each element of the box plot

through discussions of the different elements

o The handout given out at the end of class will be collected the next class for a grade. This will

help the teacher identify more specifically whether the students understand and which aspects

are causing trouble.


o Students are put in groups so that if there is a student who is struggling, the other group

members (or teacher who’s circulating the room) can assist them.


o Students will be given a handout to complete at home to first solidify the mean absolute differences

from the median and then extend the lesson to exploring the mean differences from the mean

(attached in Appendix).


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Lesson Five Title of Lesson: Variation in Measurement


Grade level: 12



Students practice describing sets of measurements in the face of variability

Students observe the data in multiple forms: graphically (students can see the distribution’s

symmetry or skewedness), numerically (students can compute measures of central tendency and

spread)

Students understand how to compute measures of center and spread, including understanding the

term outliers by using graphs and calculations and are able to discuss and reason behind their

computations

Students understand the difficulty in measuring accurately

Students explore the reasons behind variability and potential causes for it




terminology appropriately.


DA-3.2 Organize and interpret data by using pictographs, bar graphs, pie charts, dot plots,

histograms, time-series plots, stem-and-leaf plots, box-and-whiskers plots, and scatterplots.





DA-4.7 Use procedures and/or technology to find measures of position (including median, quartiles,

percentiles, and standard scores) for given data.


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For univariate measurement data, be able to display the distribution, describe its shape, and select

and calculate summary statistics

Understand histograms, parallel box plots, and scatterplots and use them to display data;


Prerequisites:

Students need to be familiar with plotting data (including box plots with outliers) and calculating numerical

summaries of data (mean, median, interquartile range, and standard deviation)


o Coins

o Measuring tape

o Tennis ball

o Ruler

o Paper (Guided Notes; attached in Appendix) and pencil to create plots

o Water dropper

o Cup of water

Procedures: (APS 4, 5, 6, 7, 8, 9)


o Students look ―out the window‖ (a picture projected onto the SMARTBoard) and describe

instances of variability in what they see. What do you think are the causes of the variability?

Is the variability good or not so good?

o Main Activity- Use guided notes to record data and create graphs (attached)

o Collect personal data for the following things:

Count the dollar value of the change the student currently has with them


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Using the given measuring device, measure the diameter of a tennis ball

Using the measuring tape, measure the circumference of your head to the nearest

millimeter

o Record the data as a class

o Analyze the data

For each step, create a plot of the measurements that shows the shape of the

distribution. Describe the shape.

Construct a box plot for the set of measurements. Describe any interesting features of

the box plot. Are there any outliers shown by the IQR rule?

Find the mean of the measurements and compare it with the median. Explain why

they’re the same/different.

Find the standard deviation of the measurements. Identify any observations that are

more than 2 standard deviations away from the mean. Are these ―unusual‖

observations the same ones we identified in the box plot? Does the standard deviation

seem to be a reasonable measure of variability?

o How many drops of water will fit on a penny?

Estimate how many drops fit first and record this number.

Using the penny, eyedropper and cup of water, place as many drops of water as

possible on the penny. Count the number of drops carefully and record this number

Record the data as a class.

Use graphical and numerical methods to analyze the data (Summarize the distribution

and find any variation within the data)

How could you reduce the variability?

o Wrap-Up

o Discuss the two reasons for variability

Measurement device/system (using a balance to measure the weight of an object

Difference in actual objects (number of chairs in a classroom: classrooms have different

number of chairs but they can be accurately counted within each room)

o Determine what the reason for the variability in each step is (How could you reduce the

amount of variability?


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Assessment: (APS 3)

o Throughout the lesson both my cooperating teacher and I circulate the classroom, answering

any questions and confusion; when there is a common question, address the answer to the

whole group

o As a class, we discuss questions 5 and 6 on the worksheet, first exploring the different types of

variation and potential causes for each

o The guided notes/worksheet is collected for a grade, graded with accuracy according to the

specific class data

o Students must be able to create the box plots and perform the calculations successfully in

order to demonstrate mastery

o There are two worksheets to gather a sense of understanding from each student (worksheets

attached):

Students assess their comfort level with specific vocabulary (variability, outliers,

standard deviation, IQR, and more) before the lesson, and then fill out the exact same

survey (printed on the back of the first) to assess their comfort level after the lesson

Students decide whether certain situations are better to have a high or low variability

and asked why, reinforcing the different typed of variability and why we would want

it either high or low


o Students will work in groups to help each other when they’re struggling

o The circumference of the head measurement can be left out for students who are taking longer

to complete the graphs and calculations

o Calculators can be used to graph the box plot first if the graphing is taking a student a long

time to do on their own, once they create one by themselves without it


o This lesson is the final one in their unit, right before their test. Students will be given a handout to

complete if they want to continue practicing at home before their test (attached).


Page 33

Appendix TO MAKE THE VOCABULARY BOOK: http://www.youtube.com/watch?v=mmPZlFBR6I8&feature=related

This video shows the step by step instructions on how to make the small book for vocabulary out of one sheet

of paper. The students have made these books before, but I will be walking them through the steps (not the

video, simply attached as reference for clarification).

EXIT SLIP

Students will take out a sheet of paper and anonymously answer the following:

What are the main concepts we discussed this week?

How could these be used outside of school?

I understand this week’s material… [Students will describe how well they understand the week’s

material on a scale of 0-10, 0 being they have absolutely no clue what we’ve even been

learning, 5 being they’re on the way to understanding but there are still a lot of roadblocks,

and 10 being there is no confusion, it all makes perfect sense]

I’m still confused about/don’t understand…

I liked/disliked ____ that we did this week.

By keeping these slips anonymous, students will feel more comfortable being honest.


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LESSON TWO

The following is the worksheet given out at the end of class for extra

practice in making a box plot, to be completed for homework.

Following are the violent crime rates as of 1999, of 23 of the largest cities in the United States (in incidents

per year per 100,000 population; data from the FBI’s Uniform Crime Reporting System, found at

http://www.fbi.gov/ucr/99cius.htm)

City Violent

Crime Rate

City Violent

Crime Rate

Austin 529 Milwaukee 1043

Boston 1302 Minneapolis-St. Paul 1161

Columbus 855 Nashville 1607

Dallas 1414 New York 1063

Detroit 2254 Philadelphia 1604

El Paso 686 Phoenix 832

Honolulu County 254 San Antonio 561

Houston 1187 San Diego 598

Indianapolis 1016 San Francisco 866

Jacksonville 1034 San Jose 581

Las Vegas 665 Seattle 767

Los Angeles 1283


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1. Construct a box plot of the data.

2. Describe the distribution of the box plot.

3. Based on these data, how large or how small would a crime rate have to be to be an outlier?

(Remember: Q1-(1.5*IQR), (1.5*IQR)+Q3)

4. Portland, Oregon had a rate of 1236. Add this city to the 23 cities listed in table 1 and construct a

box plot of the 24 data values.


Page 36

LESSON THREE

The following is the set of guided notes that students are to fill out

during class.

1. Consider the following group of histograms and summary statistics. Each of the variables (1-6)

corresponds to one of the histograms.


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Write the letter of the histogram next to the appropriate variable number in Table 1 and explain how

you made your choices.

Variable Mean Median Standard

Deviation

Table

Letter Reasoning

1 60 50 10

2 50 50 15

3 53 50 10

4 53 50 20

5 47 50 10

6 50 50 5


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2. Consider the following group of histograms and box plots.

Each box plot corresponds to one of your histograms. Match the box plots to the histograms and

explain how you made your choices.


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This document is the document handed out at the end of class, to be

competed as further practice for homework.

1. Estimate the mean, median, and standard deviation of each of the distributions graphed below.

2. Sketch a histogram of a variable for which the mean is greater than the median.


Page 40

LESSON FOUR

The following is the worksheet handed out at the end of class as

homework to reinforce concepts learned that day.

1. Calculate the mean absolute difference from the median (MAD) of these numbers:

{1, 2, 3, 3, 3, 4, 6, 8, 10, 14, 20}.

2. Mavis likes this measure of spread: just take the maximum value and subtract the minimum value.

The result represents the entire range and shows how spread out the data are. To get a typical

distance, divide that by two. What are the advantages and disadvantages of Mavis’s method?

3. Invent two sets of numbers that have the same median and the same range but different values for

the MAD. Describe what the difference in the MADs tells you about differences between the two

sets of numbers.

4. We could calculate the median V-Span of the entire class, calculate everyone’s distance from that median, and then figure out the MAD of the V-Spans for the whole class. But suppose we lost the individual V-Spans and only had the MADs for the groups. Could we still calculate the MAD for the whole class? If so, how? If not, why not? Also if not, what could we tell about the whole class MAD from the group MADs?


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5. Using the data set from the “V-Span” activity in class, compute the mean difference from the mean (MDFM) without taking the absolute value first. What do you get? What could you use this measure for?

6. Why is the MDFM always the same value?


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LESSON FIVE

This is the survey taken both before and after class by e ach student to

check student knowledge.

Evaluation Scale: (5) Perfectly (4) Pretty Well (3) I have a general idea (2) I vaguely remember talking about it (1) I have no clue

Before Today’s Activity…

I understand what the following terms mean: Variability 5 4 3 2 1

Outliers 5 4 3 2 1 Standard deviation 5 4 3 2 1 IQR 5 4 3 2 1

I understand how to calculate/find/create the following things: Mean 5 4 3 2 1 Median 5 4 3 2 1 IQR 5 4 3 2 1 Standard deviation 5 4 3 2 1 Outliers 5 4 3 2 1 Box plot 5 4 3 2 1


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Evaluation Scale: (5) Perfectly (4) Pretty Well (3) I have a general idea (2) I vaguely remember talking about it (1) I have no clue

After Today’s Activity…

I understand what the following terms mean: Variability 5 4 3 2 1

Outliers 5 4 3 2 1 Standard deviation 5 4 3 2 1 IQR 5 4 3 2 1

I understand how to calculate/find/create the following things: Mean 5 4 3 2 1 Median 5 4 3 2 1 IQR 5 4 3 2 1 Standard deviation 5 4 3 2 1 Outliers 5 4 3 2 1 Box plot 5 4 3 2 1


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These are the guided notes that will help students step -by-step through

the lesson.

1. Out the “Window”

a. What instances of variation do you see out the window?

Is this a good example of variability or no? Why or why not?

2. Pocket Change

a. How much change in coins do you have with you?

b. Have one representative from your group record all three of your findings on your team’s side.

c. Construct a plot of these measurements that show the shape of the distribution.

d. Construct a box plot for the set of measurements. Define n, the mean, and median.

e. Find the standard deviation of the measurements. Identify any observations that are more

than 2 standard deviations away from the mean. Are these the same “unusual” observations

that were identified by the box plot?


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f. Does the standard deviation appear to be a reasonable measure of variability for these data?

3. Tennis Ball

a. How big is the diameter of the tennis ball to the nearest millimeter?









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4. Circumference of head

a. How big is the circumference of your head to the closest millimeter?









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5. What was the main source of variation for each experiment?

a. Pocket Change-

b. Tennis Ball-

c. Circumference of Head-

6. How could you reduce the variation in each of these activities if you were to repeat them?

7. Water Droplets on a Penny

a. Guess how many drops will fit on a penny.

b. Using the penny, eyedropper, and cup of water provided, determine how many drops of water

can fit on the surface of the penny. Remember to count while carefully placing the drops.

c. Send a group representative to the board to record your group’s findings.

d. Create 2 plots to represent the class data. Analyze the data similarly to the last three

experiments (refer to steps d, e, and f of each experiment).

e. What is the source(s) of variation in this experiment, if any?


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This is the worksheet given out after working through the lesson and

discussing variability and the different types and causes for it, that will

help check student understanding of the term variation.

Consider each of the following variables. Do you think it would be better for the variability to be high or

low? Explain your decision.

1. Age of trees in a national forest

2. Diameter of new tires coming off one production line

3. Scores on an aptitude test given to a large number of job applicants

4. Daily rainfall


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This is the extra practice given out as homework to reinforce the activity

that we did in class.

1. Measure the area of your desktop with a meter stick to the closest square centimeter. Do this five

separate times. (If only have a ruler, find a smaller rectangular object to measure.)

a. Comment on the variability among your five measurements.

b. Comment on the sources of variability for the process of measuring the area of a desktop.

c. How would you combine the five trials into a single measure of area to report to the rest of the class? (Hint: Think about the different measures of central tendency.)

2. To determine how much sleep students get on a typical night, an instructor asked the class to report

how many hours they slept last night. The data are shown in the following figure.

Figure 1

a. Describe two different sources of variation in these data.

b. How would you suggest the measurements be made if the goal were to find how much sleep

students get on a typical night?

Unit Plan: Measures of Central Tendency and...

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