Post on 12-May-2018
UNIT PLAN: MEASURES OF CENTRAL TENDENCY AND SPREAD
Photo courtesy of http://www.flickr.com/photos/49508892@N08/4534462270/
Kaci Cohn Probability and Statistics
Mrs. Cathy Plowden, Cooperating Teacher
Mrs. Jennifer Cribbs, Professor
Unit Plan: Measures of Central Tendency and spread
Page 1
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
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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).
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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Lesson Two Title of Lesson: “Living” Box Plot
Subject: Probability and Statistics
Grade level: 12
Teacher: Kaci Cohn (Cooperating teacher: Mrs. Cathy Plowden)
Objective(s): (APS 4)
Students understand the meanings of each element of a box plot
Students solidify understanding of distributions including shape, center and spread
SCSDE Curriculum Standard(s) Addressed: (APS 4, 6)
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-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.
DA-4.9 Explain the significance of the shape of a distribution.
NCTM National Curriculum Standard(s) Addressed:
Understand histograms, parallel box plots, and scatterplots and use them to display 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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Materials/Preparation: (APS 6)
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.
Adaptations: (APS 6, 7)
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.
Follow-up Lessons/Activities: (APS7)
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.
Unit Plan: Measures of Central Tendency and spread
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Lesson Three Title of Lesson: Matching Statistics to Plots
Subject: Probability and Statistics
Grade level: 12
Teacher: Kaci Cohn (Cooperating teacher: Mrs. Cathy Plowden)
Objective(s): (APS 4)
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
SCSDE Curriculum Standard(s) Addressed: (APS 4, 6)
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.
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-4.9 Explain the significance of the shape of a distribution.
NCTM National Curriculum Standard(s) Addressed:
Understand histograms, parallel box plots, and scatterplots and use them to display data
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.
Unit Plan: Measures of Central Tendency and spread
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Materials/Preparation: (APS 6)
o Pencil to fill out guided notes (attached in Appendix)
o SMARTBoard
Procedures: (APS 4, 5, 6, 7, 8, 9)
o Introductory Activity
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.
Adaptations: (APS 6, 7)
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.
Unit Plan: Measures of Central Tendency and spread
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Follow-up Lessons/Activities: (APS7)
o Students will be given a handout to complete at home to reinforce the class activity (attached in Appendix).
Unit Plan: Measures of Central Tendency and spread
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Lesson Four Title of Lesson: How Far Are You From the Mean?
Subject: Probability and Statistics
Grade level: 12
Teacher: Kaci Cohn (Cooperating teacher: Mrs. Cathy Plowden)
Objective(s): (APS 4)
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
SCSDE Curriculum Standard(s) Addressed: (APS 4, 6)
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.
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-4.9 Explain the significance of the shape of a distribution.
NCTM National Curriculum Standard(s) Addressed:
Analyze precision, accuracy, and approximate error in measurement situations
Prerequisites:
Students need to know how to find the mean and median of a set of numbers and how to make a dot plot
Unit Plan: Measures of Central Tendency and spread
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Materials/Preparation: (APS 6)
o Paper and pencil to record data, create plots and analyze them
o Rulers
Procedures: (APS 4, 5, 6, 7, 8, 9)
o Introductory Activity
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.
Unit Plan: Measures of Central Tendency and spread
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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.
Adaptations: (APS 6, 7)
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.
Follow-up Lessons/Activities: (APS7)
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).
Unit Plan: Measures of Central Tendency and spread
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Lesson Five Title of Lesson: Variation in Measurement
Subject: Probability and Statistics
Grade level: 12
Teacher: Kaci Cohn (Cooperating teacher: Mrs. Cathy Plowden)
Objective(s): (APS 4)
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
SCSDE Curriculum Standard(s) Addressed: (APS 4, 6)
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.
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.
Unit Plan: Measures of Central Tendency and spread
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DA-4.9 Explain the significance of the shape of a distribution.
NCTM National Curriculum Standard(s) Addressed:
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;
Analyze precision, accuracy, and approximate error in measurement situations
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)
Materials/Preparation: (APS 6)
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 Introductory Activity
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
Unit Plan: Measures of Central Tendency and spread
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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?
Unit Plan: Measures of Central Tendency and spread
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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
Adaptations: (APS 6, 7)
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
Follow-up Lessons/Activities: (APS7)
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).
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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.
Unit Plan: Measures of Central Tendency and spread
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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?
Unit Plan: Measures of Central Tendency and spread
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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?
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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?
Unit Plan: Measures of Central Tendency and spread
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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?
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?
f. Does the standard deviation appear to be a reasonable measure of variability for these data?
Unit Plan: Measures of Central Tendency and spread
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4. Circumference of head
a. How big is the circumference of your head to the closest millimeter?
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?
f. Does the standard deviation appear to be a reasonable measure of variability for these data?
Unit Plan: Measures of Central Tendency and spread
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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?
Unit Plan: Measures of Central Tendency and spread
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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
Unit Plan: Measures of Central Tendency and spread
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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?