Download UNIT TITLE _II_Data and Statistics

UNIT TITLE _Data Analysis and Probability Units 8 & 9
DESIGNED BY Linda Fusco revised by Lesa Curtis and Sharon Ciccone
DATE 8/26/10
Stage 1 – Desired Results
Purpose: Students will learn how to differentiate between the possibility and the probability of
an event happening; and find theoretical and experimental probabilities to make decisions or
predictions about future events. Collect and analyze data to make informed decisions using
statistical results regarding an inquiry; then communicating the findings using graphs, tables,
and the language of statistics.
Example: Students are asked to evaluate the following survey question found on a website:
Would you rather go on beach vacation, a city vacation, or stay at home? The results for the
question are 445 participants responded; 212 preferred the beach; 120 preferred the city; and,
113 preferred to stay at home. Students must include answers to the following questions in
their evaluations: (1) Is the survey question biased? Why or why not? (2) Based on the survey
what are the odds a person will want to stay home on vacation? (3) Based on the survey what
is the probability that a person will travel for vacation? (Randall et. al., 2011, p. 773).
Example: Based on the data analysis, who should have won the 2007 Super Bowl?
New York State Learning Standards:
NYSED MST Standards Content Standard 3
Students will:
- understand the concepts of and become proficient with the skills of mathematics
- communicate and reason mathematically;
- become solvers by using appropriate tools and strategies;
through the integrated study of number sense and operations, algebra, geometry,
measurement, and statistics and probability.
Algebra Strand
A.S.1. Categorize data as qualitative and quantitative
A.S.2 Determine whether data to be analyzed is univariate or bivariate.
A.S.3 Determine when collected data or display of data may be biased.
A.S.4 Compare and contrast the appropriateness of different measures of central tendency for a
given set of data.
A.S.5 Construct histogram, cumulative frequency histogram, and a box-and-whisker plot, given a
set of data
A.S.6 Understand how the five statistical summary ( minimum, maximum, and three quartile) is
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used to construct a box-and- whisker plot.
A.S.7 Create a scatter plot of bivariate data
A.S.8 Construct a reasonable line of best fit for a scatter plot and determine the questions of that
line
A.S.9 Analyze and interpret a frequency distribution table or histogram, a cumulative frequency
distribution table or histogram, or a box –and-whisker plot
A.S.10 Evaluate published reports and graphs that are based on data by considering:
experimental design, appropriateness of the data analysis, and the soundness of the conclusions
A.S.11 Find the percentile rank of an item in a data set and identify the point values for the first,
second, and third quartiles
A.S.12 Identify the relationship between independent and dependent variables from a scatter plot
( positive, negative, or none)
A.S.13 Understand the difference between correlation and causation
A.S.14 Identify variables that have a correlation but not a causation
A.S.15 Identify and describe sources of bias and its effect, drawing conclusions form data
A.S.16 Recognize how linear transformations of one-variable data affect the data’s mean,
median, mode, and range.
A.S.17 Use best fit line to make a prediction involving interpolation or extrapolation
A.S.18 Know the definition of conditional probability and use it to solve for probabilities in
finite sample spaces
A.S. 19 Determine the number of elements in a sample space and the number of favorable events
A.S. 20 Calculate the probability of an event and its compliment
A.S. 21 Determine empirical probabilities based on specific sample data
A.S.22 Determine , based on calculated probability of a set of events if:
-some or all are equally likely to occur
-one is more likely to occur than another
-whether or not an event is certain to happen or not to happen
A.S. 23 Calculate the probability of :
-a series of independent events
- a series of dependent events
-two mutually exclusive events
- two events that are not mutually exclusive
A.N.7. Determining number of possible events using counting techniques or the Fundamental
Principle of Counting
A.N.8. Determine the number of possible arrangements (permutations) of a list of items.
A.A.29 Use Set-builder notation and/or interval notations to illustrate the elements of a set, given
the elements in roster form.
A.A.30 Find the compliment of a subset of a given set.
A.A.31 Find the intersection of sets (no more than three sets) and/or the union of sets (no more
than three sets)
Resources:
Integrated Algebra ( Teachers Edition VOl 1 and VOl 2) R. Charles, B. Hall, D. Kennedy, A.
Bellman, S. Chavis Bragg, W. Handlin, S. Murphy, G. Wiggins Prentice Hall (2011), Vol 1ISBN 13: 978-0-13-251809-3, Vol 2 ISBN -13: 978-0-13-251810-9-
(1)
Chapter 2 Functions and “New York Math A” , Bellman, Bragg, Chapin,
Gardella, Hall, Handlin, Manfre, Prentice Hall , 2001 their Graphs- ( Analyzing
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(2)
Data Using Scatterplots, Relating Graphs to Events, Linking Graphs to Tables,
Functions, Writing A Function Rule, The Three Views of a Functions[rule, able,
graph], The Probability Formula)
“ 4MAT Algebra 4 Algebra” Arelien. Hodenfiled. About Learning Inc, , 2007
Chapter 1 “ In what ways can a person “communicate” more effectively using
Algebra? Chapter 2 Luck “ In what ways does chance show up in the real
world? , Chapter 4 Relationships linear
Enduring Understanding
Big Idea: Real world problem solving
investigations involve surveys that ask unbiased
questions, use valid sampling techniques to
collect , analyze, and represent the results
leading to inferences and predictions about the
probability of an event occurring in larger
populations. (i.e. data representation,
collection, and analysis; and probability)
Using a function derived from a set of data, a
researcher can summarize findings and make
predictions.
Students will understand that……
Data representation, collection, and analysisProbability-Counting methods can be used to find the
number of possible ways to choose objects with
and without regard to order
-Permutation and combination notation can be
used to represent real-world situations,
-The probability of an event, P(event) tells how
likely it is that the event will occur.
- Probabilities can be found by reasoning
mathematically or by using experimental data.
- The Probability of a compound event can
sometimes be found from expressions of the
probabilities of simpler events.
- Different methods must be used for finding
the probability of two dependent events
compared to finding the probability of two
independent events.
- A trend line determined by a scatter plot
represents an approximation.
- Mean, median, and mode are three
measures of central tendency that
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Essential Questions
(1) How can collecting and analyzing data
help you make decisions or predictions?
(2) How can you make and interpret
different representations of data?
(3)
How is probability related to real
world events?
(4) How are statistics used to make
convincing arguments?
(5) How can data trends and correlations
to describe the source of a
disease? ( Prentice Hall TE- p 61)
(6) The Chicago Bears won the coin flip
for the Superbowl XLI making it
the tenth consecutive year in
which the NFC team won the coin
flip. Ray “Dooms” Sullivan, a park
ranger from Virginia, was struck
by lightning seven times. Both of
these events defy the laws of
probability. In what ways does
chance show up in the real world?
4MAT Chapter 2
-
represent three different aspects of the
data.
Interpretations of statistical graphs will
vary and lead to further questions about
the inquiry.
Bias is evident in every quantitative
inquiry.
Knowledge
Skills
Students will know……
Students will be able to……
(a) Categorize data
(b) Compute measures of Central tendency
(c) Display data as histograms, and box
Definition of probability, events ( independent,
dependent, mutually exclusive, overlapping,
compound), outcomes, n factorial,
permutations, combinations, sample space,
odds
- Methods used to develop theoretical
probabilities.
-Formulas used- P (A), P (B), P(A or B), P (A
and B),
- The sum of the probabilities of an event and
its complement is 1.
- measures of central tendency
- the difference between qualitative and
quantitative data
- the proper use of the stem and leaf plot, boxwhiskers plot, histograms, and frequency
histograms
- line of best fit
-linear regressions
-quartiles and percentiles
- five number summary
-analyzing data
-correlation
-causation
and whisker plots
(d) Calculate percentiles and explain how
the percentiles relate to the findings.
(e) Box and whisker plots
(f) Identify and explore biased data
(g) Identify if the linear and non linear
regression
(h) Exploring several ways to draw a trend
line so that as many data points are
above the line as below the line
(i) Using the graphing calculator to draw
and analyze statistical graphs
(histograms, linear regression, scatter
plots, box and whisker plots)
Scatter plots
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Stage 2 – Assessment Evidence
Pre-Assessments :
Hook.
“David Blackwell, Scholar of Probability, Dies as 91”, New York Times , July 17, 2010. Have
students review the video clips of David Blackwell’s journey to becoming a scholar of
probability. Ask students to select what facet of Dr. Blackwell’s life interested them.
http://www.visionaryproject.com/blackwelldavid/ Find other practicing mathematicians and ask
the students to review their lives (such as Evelyn Granville
http://www.visionaryproject.org/granvilleevelyn/)
Ask students to look and listen for messages that include correlations (e.g. “Since I was elected,
crime has dropped 12 percent.” PH Math A TE p 60)
Critical Foundations- Create pre-assessments for the following math skills:
(1) Adding and subtracting fractions.
(2) Multiplying and dividing real numbers.
(3) Distributive Property
(4) Unions and Intersections of Sets
(5) Scatter Plots
See the “Get Ready!” diagnostic assessment ( Randall et al. 2011 p. 711)
Pre- Reading Vocabulary
“Outcomes”- Give students an event ( hockey game) and ask students to name the possible
outcomes. Repeat the diagnostic task for other events (i.e. flips a coin, toss a die, horse race).
“Median”- Have student draw their understanding of a highway median.
“Combinations” As students how they would make the color green from two different paintsask them if it makes a difference if they use each paint in a different sequence.
Assessments: Mid chapter quizzes (Charles, 2011, p 749): Castlelearning;
Informal/ ongoing assessments
Analyzing 2007
AFL and NFL championships
( teacher made TINSPIRE file)
Application to NYS Regents
Performance Tasks:
Data and Analysis:
A.S.1: Categorize the following as quantitative or qualitative: hair color, time, attitude, age,
weight, height, pretty, eyes, DVDs, price of games
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A.S.2: Compare the following. Which is univariate and which is bivariate? Atomic weights on
a periodic table. The edge lengths and volumes of cubes.
A.S.3: To find the average height of high school boys would you only measure basket ball
teams? Every 3rd boy who enters the cafeteria?
A.S. 4: If you want to determine if there are more young people or more old people to know
who is paying into social security and who is receiving it. Which measure of central tendency
would you use? (Mean, median, or mode)
A.S.5: Help your teacher construct a histogram, and cumulative frequency histogram for the
grades in his/her class for the first quarter. Create a box and whisker plot for this same data.
A.S.6: Be able to explain how min, max and quartiles are used in the above box-and-whisker
construction.
A.S.7: Make a scatter plot comparing altitude and temperature (PH TE Int. Alg.2010, p.334).
A.S.8: Draw a line of best fit on the graph of the above data. Determine the equation of this
line.
A.S.9: Interpret the graphs and tables on pp 599-600 (PH NY Math A, 2001) What can we
discover from histograms.
A.S.10:
A.S.11:
A.S.12: Determine the correlation of the scatter plots (PH TE Int. Alg.,2010, p333)
A.S.13: Consider the number of letter in the name of a city and the population of that city.
Should there be any correlation? Why or why not? What is causation based on? Does the
number of letters in a city’s name cause an increase or decrease in its population?
A.S.14:
A.S.15: Does the following question contain a bias? Where would you rather vacation? The
beach? The city? At home? (Should the economics of your population be considered? Perhaps
$2000.00 should be given to any of the three choice)s and then which would the subject prefer.
A.S.16: Discover what happens to a set of data’s measures of central tendency, say six
people’s number of minutes they work out each week, when they each increase their workout
by 5 minutes, then 10 minutes. What would happen to these measures of central tendency if
each workout were increased three times?
A.S.17: PH TE Int. Alg. 2010, p. 335. Using body weights of pandas to interpolate the weight
of a 7 month old panda. Then Decide if the chart, which goes to 9 months, may be used to
extrapolate the weight of a 3 year old panda.
TI NSPIRE activities from Math NSPIRE.com
Teacher made worksheets
Permutations Combinations:
A.N.7. Determining number of possible events using counting techniques or the Fundamental
Principle of Counting
A.N.8. Determine the number of possible arrangements (permutations) of a list of items.
Performance Tasks for PI’s A.N.7 and A.N.8) Give the students sets of manipulatives (20
items: four letter tiles, counters, unique items). Ask the students to use the manipulatives to
create a permutation and combination problems using the same n and r values. The students will
then have to explain why the size of the outcomes for each problem differs. What would happen
to the number of outcomes if n and r are both increased by 1? Have students do exercise
individually and then work with a partner. Provide a grading rubric.
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Probability
A.S.18 Find the probability for the recent sale of a car that the car was domestic and cost less
than $16,000.00.
Know the definition of conditional probability and use it to solve for probabilities in finite
sample spaces
A.S. 19 Determine the number of elements in a sample space and the number of favorable events
A.S. 20 Determine the compliment of the probability of drawing a black jack out of a standard
deck of cards.
Calculate the probability of an event and its compliment.
A.S. 21 Have every person in the class toss a fair coin 20 times. Ask students what they believe
the results will be. Record the data both for heads and for tails. Ask what they believe will
happen if everyone tosses their coins another 30 times. Then have each person toss the coin
another 30 times and add these results onto the initial results and have students note what
happens to the totals. Have them predict what will happen if everyone records their results
another 50 times each and adds that information into the totals. Discuss why multiple entries
are important when drawing conclusions.
Determine empirical probabilities based on specific sample data.
Performance Tasks for PI’s A.S. 19-21. Coin Toss. Have each student determine the possible
outcomes for a coin toss, define the sample space, and predict the outcome when the coin is
tossed 20 times. Ask the students to find the theoretical probability of the number of outcomes
before flipping the coins.
Lottery – Have students calculate the odds of winning the state lotteries- Lotto, Mega,
Sweet Million, Pick Five, etc. Which has better odds for the money that is spent on the ticket?
Simulation – On a graphing calculator generate random integers from 10-99. See Activity 1
on Integrated Algebra ( Charles, 2011) p 763.
A.S.22 Discuss with students if they believe there is a 50/50 chance of a pregnant woman
giving birth to a boy or girl? How about eye color (blue, green, brown, grey, hazel, black), are
they equally likely to occur? What factors can influence the outcome of an event? Are there
certainties or impossibilities in life? What is the probability that you will wake up tomorrow 5
years younger than you are today? What is the probability that the sun will arise in the east
and set in the west?
Determine , based on calculated probability of a set of events if:
-some or all are equally likely to occur
-one is more likely to occur than another
-whether or not an event is certain to happen or not to happen
A.S. 23 PH TE Int. Alg. 2010, p. 778 or try the following website: www.cut-theknot.org/Probability/PokerSampleSpaces , discuss series of independent events, dependent
events, and events that are both mutually exclusive and not.
Calculate the probability of :
-a series of independent events
- a series of dependent events
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Note: Include in Unit -
Stage 3 – Learning Plan
Misleading Graphs and Statistics (Charles, 2011, p. 748) Purpose of the learning experience is
to examine graphs that are well-suited to the data and to recognize when other types of graphs
are a poor choice or drawn so as to be misleading.
Students will compare two presentations of the same data and analyze which of the two
presentations may be misleading.
Simulations- Simulations are models of real life situations such as using a calculator generated
list of numbers to represent a real – life situations like blood types.
Student Interest- Ask students to choose a subject they would like to create a survey for. It
should be a topic that is of interest to the student. Have the students choose the topic in the
early on in the presentations of probability so there is time to develop the survey questions.
Lesson Scope and Sequence:
Lesson 1
Organizing Data- creating tables and spreadsheets
TI Activities Exchange You Are What You Eat!
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=9478
HowRandom!
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=8555
Lesson 2 Frequency & Histograms
TI Activities Exchange Box Plots & Histograms
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=9843
Yankees VS. Mets
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=10997
Lesson 3 Measures of Central Tendency and Dispersions TI Activities Exchange Intro to the
Central Limit Theorem
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=9892
Probability Distributions
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=8972
Lesson 4 Box and Whisker Plots TI Activities Exchange Box Plot Comparison
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=11080
Lesson 5 Samples and Surveys
Lesson 6 Permutations and Combinations TI Activities Exchange Too Many Choices!
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=11763
TI Activities Exchange Perms and Combs?
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=12645
TI Activities Exchange Combinations
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=8433
TI Activities Exchange What’s Your Combination?
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=9839
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TI Activities Exchange Permutations
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=8432
Lesson 7 Theoretical and Experimental Probability TI Activities Exchange Law of Large Numbers
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=13578
TI Activities Exchange Application of Power http://education.ti.com/educationportal/
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=11692activity
exchange/Activity.do?cid=US&aId=13678
TI Activities Exchange Geometric Dartboards
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=8269
TI Activities Exchange Independence Is the Word
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=11692
Lesson 8 Probability of Compound Events TI Activities Exchange Compound Events
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=10136
Lesson 9 Linear Regression see Unit 7 Functions
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