Download Probability and Statistics - Farmington Public Schools

Integrated Math 1
Probability and Statistics
Farmington Public Schools
Grade 9
Mathematics
Beben, Lepi
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Table of Contents
Unit Summary
………………….….…………..page 3
Stage One: Standards
Stage One identifies the desired results of the unit including the broad understandings, the unit
outcome statement and essential questions that focus the unit, and the necessary knowledge and
skills.
The Understanding by Design Handbook, 1999
…………………………….... page 4-6
Stage Two: Assessment Package
Stage Two determines the acceptable evidence that students have acquired the understandings, knowledge
and skills identified in Stage One.
……………………………… page 7-8
Stage Three: Curriculum and Instruction
Stage Three helps teachers plan learning experiences and instruction that aligns with Stage One and
enables students to be successful in Stage two. Planning and lesson options are given, however teachers are
encouraged to customize this stage to their own students, maintaining alignment with Stages One and Two.
………………..……………… page 9-11
Appendices
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….....………………………. page 12-57
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Unit Summary
This unit on Statistics and Probability is taught in Integrated Math 1
after a unit on exponential patterns and before a unit on networks. This
unit will take approximately 5-6 weeks of class time.
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Stage One: Standards
Stage One identifies the desired results of the unit including the broad understandings, the unit outcome
statement and essential questions that focus the unit, and the necessary knowledge and skills.
The Understanding by Design Handbook, 1999
Essential Understandings and Content Standards
#7 Students will understand that people use basis concepts of probability
and statistics to meaningfully collect, organize, display and analyze data to
simulate events.
Students will:
a. (12th) estimate probabilities and predict outcomes
d. (12th) select appropriate measures of central tendency
g. (12th) use scatter plots and curve fitting techniques to interpolate
and extrapolate from data
i. (12th) use simulations to estimate probability
c. (8th) describe the shape of the data using range, outliers, and
measure of central tendency
d. (8th) select and construct appropriate graphs and measures of
central tendency for sets of data
Unit Outcome Statement
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Consistently aligning all instruction with this statement will maintain focus in this unit.
Students will use statistics and probability concepts to make decisions about
real world situations using a sample. As a result of the statistics component
of this unit students will organize, display, and describe data using
appropriate statistical and graphical methods. As a result of the probability
component of this unit students will understand and apply the basic concepts
of probability using sample spaces, Venn diagrams and tree diagrams as well
as perform simulations.
Essential Questions
These questions help to focus the unit and guide inquiry.
•
•
•
How reliable or useful are the statistical outcomes in decision making?
How can data be described and interpreted?
How can estimations of likelihood be made for random events?
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Knowledge and Skills
The Knowledge and Skills section includes the key facts, concepts, principles, skills, and processes called for by
the content standards and needed by students to reach desired understandings.
The Understanding by Design Handbook, 1999
Knowledge
♦
♦
♦
♦
♦
♦
♦
♦
Distinguish among mean, median and mode
Define simple probability
Describe a random event
Understand the Law of Large Numbers and its relationship to probability
Decide when to use a sample space, tree diagram, or Venn diagram
How to construct and perform a simulation
Determine the 5-number summary for data
Identify the different graphical displays for data
o Stem and leaf
o Histogram
o Box plot
o Scatter plot
o Time plot
Skills/Processes
♦ Apply the Fundamental Counting Principle
♦ Calculate probabilities of an event by using a sample space, trees or Venn
diagram
♦ Calculate the 5-number summary
♦ Identify the shape of a distribution (skewed, symmetric, center)
♦ Perform an appropriate simulation for an event
♦ Construct the appropriate graphical displays for data
o Stem and leaf
o Histogram
o Box plot
o Scatter plot
o Time plot
Thinking Skills
♦
♦
♦
♦
♦
Recognizing intended meaning
Making inferences
Identifying appropriate evidence
Recognizing fact vs. opinion
Sorting/categorizing
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Stage Two: Assessment Package
Stage Two determines the acceptable evidence that students have acquired the understandings, knowledge and
skills identified in Stage One.
Authentic Performance Task: Analyzing spending trends using Statistics
Goal: Students will analyze the spending habits of families in Farmington using the
grocery receipts. They will report their findings in a newspaper article.
using the data, graphs, measures of central tendency and probability values for
specific types of food items.
Role: You are a reporter for the Farmington Times Weekly newspaper. You have
been assigned the task of writing an article for the newspaper about the spending
habits of families in Farmington for publication in the local paper.
Audience: The editor of the newspaper and the readers of the newspaper article
Situation: With the emphasis on eating healthy and the new food pyramid, this
article will be to inform readers on the trends you see and make inferences about the
nutritional level of the diet of the families.
Product/Performance/Purpose: Using the receipts collected you will write an article
for the newspaper that will include the Who, What, When, Where and How of the
investigation. Within the article you will include:
• A categorical table of data including: dairy, produce, meat, grain, other foods,
and non-food purchases.
• Appropriate graphic for the data
• Measures of central tendency and range for total spent at the store as well as
for each category in the data table
• A time plot of (time shopped, $ spent) along with a linear regression analysis
• Comparison of food purchased to the food pyramid (attached)
• Percent saved using coupons
Standards & criteria for success: You will be assessed on:
o Mathematical accuracy including work
o Appropriate graphic with clear labels
o Proper use of mathematical terminology
o Clear conclusions drawn from the data analysis
o Neatness
Content standards addressed:
d. (12th) select appropriate measures of central tendency
g. (12th) use a scatter plots and curve fitting techniques to interpolate and
extrapolate from data
i. (12th) use simulations to estimate probability
c. (8th) describe the shape of the data using range, outliers, and measure of
central tendency
d. (8th) select and construct appropriate graphs and measures of central
tendency for sets of data
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Tests, Quizzes, and Other Quick and Ongoing Checks for Understanding
Quick Quiz on stem and leaf plots (7d 8th grade)
Baseball Data stem and leaf assessment, students make a stem and leaf for both the
American and National League homerun data and compare and contrast their
findings. (7 c, 7d 8th grade)
Mean, Median, Mode progress check (7c 8th grade)
Teacher made progress check on choosing the appropriate measure of central
tendency (7d 12th grade)
Unit Test - Statistics (7d 12th grade, 7c, 7d 8th grade)
The focus on the test was on measures of central tendency, constructing and
interpreting graphs, the 5-number summary IQR.
Quiz on Simulations for a random event (7i 12th grade)
Probability progress check including sample space and probabilities (7a 12th grade)
Teacher made progress check on Venn diagrams (7a 12th grade)
Teacher made progress check on Fundamental Counting Principle (7a 12th grade)
Test on Probability (7a, 7i, 12th grade)
The focus of the test is on creating sample spaces, using tree and Venn diagrams for
probabilities and using the Fundamental Counting Principle to solve real world
problems
Projects, Reports, Etc.
Students will find two articles from the newspaper or magazine that incorporates
the use of statistics and /or probability. They will then write an analysis of the
appropriate use of statistics and/or probability in the decision making process of the
article. (7d 12th grade, 7c, 7d 8th grade)
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Stage Three: Learning Experiences and Instruction
Stage Three helps teachers plan learning experiences and instruction that align with Stage One and enables
students to be successful in Stage Two.
Learning Experiences and Instruction
The learning experiences and instruction described in this section provide
teachers with one option for meeting the standards listed in Stage One. Teachers
are encouraged to design their own learning experiences and instruction,
tailored to the needs of their particular students
Middle Grades and High School
Guiding Questions
Instructional Strategies
Day 1-5 Picturing data
Given a data set, how can
you display it?
EQ How can data be
described and interpreted?
Checking for
Understanding
Think about this pg 2, given
a graph or table discuss with
class how it can be used to
make decisions.
NOTE- U#L#I#=Unit,
Lesson. Investigation in text.
These are student-centered
investigation for groups of
two to four students.
Students instigate a
situation and develop a
problem solving strategy for
the concept in the title of the
investigation. Parts of the
investigations include a
student self-assessment of
the concept. After studentcentered discussion, the
teacher will bring the group
to an understanding of the
concept.
Lesson 2 pg 10 #5 checkpoint
pg 11
U1L1I2 Describing patterns
in data #2, 4 a-c pg. 9-10
National/American League
worksheet
U1L2I1 Shapes of
distributions # 1-4 pg 14-17
OYO pg 23
MORE pg 24-28 M1 O2 O5
U1L2I2 Producing plots with
technology # 1, 3-5 pg 19-22
Quiz - Stem and Leaf
Hook: What does a picture of
data tell us about the
population?
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Day 6-12 Measuring Data
Given a set of data, how can
you describe the center,
shape and spread?
EQ How can data be
described and interpreted?
U1L2I3 Measures of center
#1, 2, 4a,b,g, 6 pg 31-6
U1L3I1 The five number
summary#1-5 pg 48-9
U1L3I2 Picturing variability
#4 pg 53 one by hand, one on
the calculator
Hook: What does 'Average"
mean? What is the best way
to describe the center and
spread of a data set?
Lesson 3 pg 35-6 #5
Worksheet on Measures of
central tendency
MORE pg 38-44 M2, O1, O3,
R2
Progress check - mean
median mode
Lesson 1 pg 50 #7-9
Lesson 2 pg 54 #5
Worksheet - statistics
Worksheet Review of
histograms, 5# summary and
box plots
Authentic Assessment
Test Statistics
Dietary Change and
Cholesterol. Could be used
for review for test.
Days 13-17 Simulations
How do you use a simulation
to estimate probability?
Probability Investigative
Spinner Task
Probability problem solving
worksheet
Do simulation results always
equal theoretical
probabilities?
U7L1I1 How many children?
#1-7 pg 484-8
OYO pg 490
MORE pg 491- 5 M1, M2,
M3, O2, O5
EQ How reliable or useful
are the statistical outcomes
in decision-making?
Hook: Is it Fair? (spinners)
Why do we use simulations?
Day 18- 25: Picturing Probability
Create a sample space for
How are diagrams used to
rolling two dice including the
determine probability?
sum of the two dice.
Worksheet
How can you calculate the
size of the sample space
without a diagram?
Beben, Lepi
Quiz
Teacher created-Students
research project of finding
newspaper or magazine
articles and analysis of the
appropriateness of the use of
statistics or probability.
Calculate probabilities from
this sample space
Use a tree diagram to model
random events. Worksheet
Complete worksheet
Counting Theory worksheet
student create a sample
space for an event and
develop an arithmetic way of
answering the question.
Complete worksheet for
homework
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Progress check Tree/sample
space
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EQ How can estimations of
likelihood be made for
random events?
Abby's Kennel Investigation
to organize data using a
Venn diagram.
Hook: What is the
probability of rolling a sum
of seven on a pair of dice?
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Venn diagram worksheet
CAPT question bank for
probability and statistics
Test Probability
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Appendices
Complete set of Essential understandings for your discipline
Any student work sheets
List of resources including texts, videos, field trips, web sites, etc.
Navigating through Probability in grades 9-12, NCTM
Navigating through Data Analysis in grades 9-12, NCTM
Contemporary Mathematics In Context, Course 1 A and B
Mathematics binder for Integrated Math 1 semester 1 and 2
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Integrated Math I
Mean, Median & Mode Progress Check
Name_______________________________
Find the mean, median and mode for the following set of data:
5 13 9 12 4 11 5 15 19 8
Mean__________
Median___________
Integrated Math I
Mean, Median & Mode Progress Check
Mode__________
Name_______________________________
Find the mean, median and mode for the following set of data:
5 13 9 14 4 11 7 15 19 13
Mean__________
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Median___________
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Mode__________
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Integrated Math I
Unit 1Test
Name _________________________
Date __________________________
1. A mathematics teacher, Mrs. Valdez, wanted to test how the use of a
calculator might affect her students’ scores on a test. On the day of
the test, the students who were selected to use calculators received
the following scores: 100, 55, 78, 66, 93, 73, 73, 65, 83, 72, and 57.
a) Make a stem and leaf plot.
b) Find the mean, median and mode (measures of center).
Mean __________
Median _____________
Mode _____________
c) If Mrs. Valdez has a student who was absent and takes a make up test, which
measure of center mentioned above will be the most affected if the student gets a
30? Why
2. On a test of eye-hand coordination, the number of seconds required to finish a task
involving fitting various shapes into corresponding holes is recorded. A group of 15
students in the second grade had the following number of seconds:
Coordination Scores in Seconds
83 76 90 58 66 44 86 66 61 59 50 53 61 64 73
a) Find the mean
Mean _____________
b) Find the five number summary
Minimum _____________
Q2 _____________
Median _____________
Q3 _____________
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Maximum _____________
c) Find the IQR (Interquartile Range). Explain what the IQR means in terms of the
eye-hand coordination.
IQR ____________
Explanation:
3.
(insert a box plot for this question)
Using the box plot shown:
a) What does the “n = 20” mean?
b) How many scores are greater than 20?
c) What is the five number summary?
Minimum _____________
Q2 _____________
Median _____________
Q3 _____________
Maximum _____________
d) What is the range of the data?
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4. Mr. Thomas has three Integrated Math 1 classes. Last week he gave a quiz with a
maximum of 10 points. The grades for each class are given in the table:
Per 1 8 7 9 8 6 7 8 7 6 6 8 8 6 6 7 6 4 9 7 7
Per 2 10 8 10 9 6 10 5 7 5 8 9 9 5 6 6 6 8 8 8 8
Per 3 8 9 9 8 5 9 9 8 8 7 8 9 9 7 6 5 2 9 7 9
a) Use the table to create a box plot for each of the classes and copy them below.
1
2
3
4
5
6
7
8
9
10
b) Which class has the largest interquartile range (IQR)?
c) Compare each class using the box plots. Are they similar? Different?
d) Which class would you say has the better scores? How did the box plots help
with your decision?
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5. Based on the description below, construct a histogram:
The minimum score is between 20 and 29.
The maximum score is between 80 and 89.
Most of the scores were in the 40’s.
No one received quiz scores in the 50’s.
The graph is skewed to the right.
6. The histogram below shows the performance of a social studies class at Farmington
High School on a recent quiz. The quiz scores were grouped into intervals of size 5. A
score on the edge of a bar is counted in the bar on the right.
Describe the data shown in the histogram by using what you have learned in class and by
using your handout (Master 9a/9b). Be specific and thorough.
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Integrated Math 1
Stats/Prob Authentic Assessment
Name ______________________
Date _____________
You are a reporter for the Farmington Times Weekly newspaper. You have been
assigned the task of writing an article for the newspaper about the spending habits
of families in Farmington for publication in the local paper. Your emphasis should be
on eating healthy and the new food pyramid. The article will be to inform readers on the trends
you see and make inferences about the nutritional level of the diet of the families.
Preliminary to writing the article you need to collect between 10 and 15 grocery store receipts for
family shopping trips. Before you write your article you will need to analyze the data on these
receipts and study the food pyramid provided. You should then check with the editor (teacher) to
make sure the data is usable for the article.
Using the analysis of the data collected you will write the article for the newspaper. The article
should include information about the data (Who, What, When, Where and How). Within the
article you will include:
• A categorical table of data including: dairy, produce, meat, grain, other foods, and nonfood purchases.
• Appropriate graphic for the data (at least one graph)
• An appropriate measure of central tendency and range for total spent at the store as well
as for each category in the data table
• A time plot of (time shopped, $ spent) along with a linear regression analysis (best fit)
• Comparison of distribution of food purchased in relationship to the guidelines in the
food pyramid (attached)
• Percent saved using coupons (if applicable)
Due date _______________________
Grade will be based on:
o
o
o
o
o
o
Mathematical accuracy including work
Appropriate graphic with clear labels
Proper use of mathematical terminology
Clear conclusions drawn from the data analysis
How well you communicated the outcome of your work
Neatness
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Project Rubric
Mathematical accuracy
computations
Meets expectations
All mathematics is
shown and correct.
Appropriate graph
Graphs used are
appropriate for the data
and correctly drawn
Clear labels/scales
All labels and scales are
accurate and on the
graph
Mathematical
terminology is used
correctly throughout the
paper
Communication
accuracy-vocabulary
Communication
conclusion clarity
Clear and accurate
conclusions are given
based on the
calculations and graphs
Partially correct
Some computational
errors present or not
included.
Graphs used is not
appropriate but correctly
drawn, or is a correct
graph but
inappropriately drawn
Some labels are missing
or scale is incorrect
Incorrect
Missing computations or
incorrect mathematics
Mathematical
terminology is generally
used correctly however
there is consistent
incorrect use of some
concept.
Conclusions are unclear
for the graphs or
calculations
Mathematical
terminology is used
incorrectly or not
present.
No graph included or an
inappropriate graph was
used incorrectly
Labels and scales are
missing
Conclusion are missing
or inappropriate for the
graph or calculations
Overall neatness = 5 points
The IM1 team will develop scoring of this rubric when the assignment is given.
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QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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Integrated Math 1
Unit 7 lesson 1 quiz A
Name ___________________
1.
Suppose a population control plan for Transville allows parents to have at most four children
each, and must stop having children when they have two girls. Explain how to use a coin to
simulate the experiment of having children until you have either two girls or four children.
2.
Ross was exposed to a cold virus and now has a 50% chance of developing a cold on each of the
next 7 days. A simulation was run in which a fair coin was tossed.
H = develops a cold
T = does not develop a cold
Each coin toss represents one day. So for example, TTH means that Ross developed a cold on the
third day, and TTTTTTT mean no cold for 7 days.
DATA
TTH
TTTTH
TTTTTTH
TTTTTH
TH
TH
TH
H
TH
TTH
TTTH
H
TH
H
TTTH
TTTTH
a. Complete the frequency table for the simulation:
Developed a cold
On day number
1
2
3
4
5
6
7
Frequency
________
________
________
________
________
________
________
b. Construct a histogram for the frequency table.
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c. Using the frequency table or histogram, estimate the average number of days it took Ross to develop a
cold.
d. Using the frequency table or histogram, estimate the percent of times that Ross will develop a cold
before the third day.
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Integrated Math 1
TEST – Probability
1.
Name:________________________
Date: ________________________
A bowl has several jelly beans in it, 4 red, 3 black, 2 green, 2 purple, and 1 orange.
If you choose one jelly bean,
a) What is the probability you will get a red one?
b) What is the probability you will get a purple or an orange one?
2.
A bag holds blocks numbered 1 to 5. You draw one block out of the bag and then without
replacing it you draw a second block.
a) Produce a sample space for this situation.
b) What is the probability you will draw the same number twice?
c) What is the probability the second number will be odd?
d) If the first number is even, what is the probability the second number is odd?
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3.
Make a table for the arithmetic sums of rolling two dice.
a)
What is the probability you will get a sum of 5?
b) What is the probability you will get an even number for a sum?
c)
4.
What is the probability that the sum will be greater than 9?
Make a tree that represents the tossing of a fair coin 3 times. List the sample space.
a) What is the size of the sample space?
b) What is the probability of getting 3 heads (in a row)?
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c) What is the probability of getting 2 heads and 1 tail (in any order)?
5.
Jenna has in her suitcase 4shirts and 3 pairs of jeans along with 2 pairs of shoes.
How many possible outfits does she have?
If you have some cards numbered 1 through 5. You draw one and then another after returning the first
6.
one.
a)
What is the probability of choosing a 9?
b) What is the probability of choosing 2 even cards?
7.
Six friends (let’s call them A, B, C, D, E, and F) are going in pairs on a roller coaster.
a) What is the sample space for pairs of riders? (AB is the same as BA)
b) What is the probability of A and E riding together?
c)
8.
Beben, Lepi
What is the probability of A and E not riding together?
Suppose a new population growth plan for China is proposed. Parents will be allowed to have at
most 3 children and must stop having children as soon as a boy is born. Explain how to simulate the
experiment of having children until you either have one boy or three children.
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9.
There are 49 people who own pets. 15 people own only dogs, 5 own cats and dogs only, and 3 own
cats, dogs and birds, and 4 own only a bird.
a. Draw a Venn diagram.
b.
How many people own only cats?
c.
How many pets are there?
In the game of “Shoot!,” three friends put out one or two fingers to find the sum.
If the sum of the three is odd, Mike gets 3 points.
If there are at least 2 numbers the same, Sue gets 2 points.
If all numbers are the same, Alice gets 14 points.
What is the probability of each of them winning?
Is this a fair game? Why or why not?
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