Probability and Statistics

Integrated Math 1 Probability and Statistics Farmington Public Schools Grade 9 Mathematics Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 1

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.... page 12-57 Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 2

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. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 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 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. (12 th ) estimate probabilities and predict outcomes d. (12 th ) select appropriate measures of central tendency g. (12 th ) use scatter plots and curve fitting techniques to interpolate and extrapolate from data i. (12 th ) use simulations to estimate probability c. (8 th ) describe the shape of the data using range, outliers, and measure of central tendency d. (8 th ) select and construct appropriate graphs and measures of central tendency for sets of data Unit Outcome Statement Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 4

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? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 5

Knowledge and Skills Knowledge 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 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 Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 6

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. (12 th ) select appropriate measures of central tendency g. (12 th ) use a scatter plots and curve fitting techniques to interpolate and extrapolate from data i. (12 th ) use simulations to estimate probability c. (8 th ) describe the shape of the data using range, outliers, and measure of central tendency d. (8 th ) select and construct appropriate graphs and measures of central tendency for sets of data Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 7

Tests, Quizzes, and Other Quick and Ongoing Checks for Understanding Quick Quiz on stem and leaf plots (7d 8 th 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 8 th grade) Mean, Median, Mode progress check (7c 8 th grade) Teacher made progress check on choosing the appropriate measure of central tendency (7d 12 th grade) Unit Test - Statistics (7d 12 th grade, 7c, 7d 8 th 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 12 th grade) Probability progress check including sample space and probabilities (7a 12 th grade) Teacher made progress check on Venn diagrams (7a 12 th grade) Teacher made progress check on Fundamental Counting Principle (7a 12 th grade) Test on Probability (7a, 7i, 12 th 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 12 th grade, 7c, 7d 8 th grade) Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 8

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 Checking for Understanding Day 1-5 Picturing data Given a data set, how can you display it? EQ How can data be described and interpreted? 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 U1L2I1 Shapes of distributions # 1-4 pg 14-17 U1L2I2 Producing plots with technology # 1, 3-5 pg 19-22 National/American League worksheet OYO pg 23 MORE pg 24-28 M1 O2 O5 Quiz - Stem and Leaf Hook: What does a picture of data tell us about the population? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 9

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? Days 13-17 Simulations How do you use a simulation to estimate probability? 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? Probability Investigative Spinner Task 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. Probability problem solving worksheet Do simulation results always equal theoretical probabilities? EQ How reliable or useful are the statistical outcomes in decision-making? U7L1I1 How many children? #1-7 pg 484-8 Hook: Is it Fair? (spinners) Why do we use simulations? Day 18-25: Picturing Probability How are diagrams used to Create a sample space for determine probability? rolling two dice including the sum of the two dice. Worksheet OYO pg 490 MORE pg 491-5 M1, M2, M3, O2, O5 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 How can you calculate the size of the sample space without a diagram? Use a tree diagram to model random events. Worksheet Counting Theory worksheet student create a sample space for an event and develop an arithmetic way of answering the question. Complete worksheet Progress check Tree/sample space Complete worksheet for homework Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 10

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? Venn diagram worksheet CAPT question bank for probability and statistics Test Probability Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 11

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 Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 12

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 Mode Integrated Math I Mean, Median & Mode Progress Check Name Find the mean, median and mode for the following set of data: 5 13 9 14 4 11 7 15 19 13 Mean Median Mode Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 13

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 Q 2 Median Q 3 Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 14

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 Q 2 Median Q 3 Maximum d) What is the range of the data? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 15

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? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 16

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. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 17

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 Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 18

Project Rubric Mathematical accuracy computations Appropriate graph Clear labels/scales Communication accuracy-vocabulary Communication conclusion clarity Meets expectations Partially correct Incorrect All mathematics is Some computational Missing computations or shown and correct. errors present or not incorrect mathematics Graphs used are appropriate for the data and correctly drawn All labels and scales are accurate and on the graph Mathematical terminology is used correctly throughout the paper Clear and accurate conclusions are given based on the calculations and graphs 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 Mathematical terminology is generally used correctly however there is consistent incorrect use of some concept. Conclusions are unclear for the graphs or calculations No graph included or an inappropriate graph was used incorrectly Labels and scales are missing Mathematical terminology is used incorrectly or not present. 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. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 19

QuickTime and a TIFF (LZW) decompressor are needed to see this picture. QuickTime and a TIFF (LZW) decompressor are needed to see this picture. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 20

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 DATA 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. TTH TTTTTTH TH TH TH TTTH TH TTTH TTTTH TTTTTH TH H TTH H H TTTTH a. Complete the frequency table for the simulation: Developed a cold Frequency On day number 1 2 3 4 5 6 7 b. Construct a histogram for the frequency table. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 21

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. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 22

Integrated Math 1 TEST Probability Name: Date: 1. 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? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 23

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) What is the probability that the sum will be greater than 9? 4. 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)? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 24

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? 6. If you have some cards numbered 1 through 5. You draw one and then another after returning the first 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) What is the probability of A and E not riding together? 8. 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. Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 25

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? Beben, Lepi DRAFT: 06/30/2006 Farmington Public Schools 26