Teacher s guide Table of Contents Volume I Teacher s Guide Introduction.................................................................................. TG1 Pacing Guide.................................................................................. TG9 Standards Correlations........................................................................ TG29 Graphic Organizers........................................................................... TG31 z-scores Table................................................................................ TG55 Unit 1: Data Analysis............................................................... 1 Lesson 2: Determining Margins of Error and Confidence Intervals..................................... 36 Lesson 3: Making Inferences Using Margins of Error and Confidence Intervals........................... 65 Answer Key..................................................................................... 93 Station Activities: Distributions and Estimating with Confidence...................................... 99 Unit 2: Sequences and Series Lesson 1: Sequences............................................................................ 113 Lesson 2: Series................................................................................ 156 Answer Key.................................................................................... 195 Station Activities Set 1: Sequences and Their Limits............................................................ 199 Set 2: Sequences and Series.................................................................. 209 Unit 3: Rational Functions Lesson 1: Characteristics of Rational Functions..................................................... 219 Lesson 2: Inverses.............................................................................. 312 Lesson 3: Solving............................................................................... 340 Answer Key.................................................................................... 429 Station Activities: Rational Functions............................................................. 441 Unit 4: Introduction to Trigonometry Lesson 1: Angles in Degrees and Radians........................................................... 451 Lesson 2: Trigonometric Functions................................................................ 475 Lesson 3: The Unit Circle........................................................................ 509 Answer Key.................................................................................... 543 Station Activities: Trigonometric Functions........................................................ 547 067065 iii
TEACHER S GUIDE Table of Contents Volume II Unit 5: Graphs and Inverses of Trigonometric Functions Lesson 1: Trigonometric Functions............................................................... 565 Lesson 2: Graphs of Transformations and Periodic Phenomena....................................... 595 Lesson 3: Inverse Trigonometric Functions........................................................ 629 Answer Key.................................................................................... 657 Station Activities: Graphs and Inverses of Trigonometric Functions................................... 663 Unit 6: Trigonometric Identities Lesson 1: Identities............................................................................. 681 Lesson 2: Trigonometric Sum and Difference Formulae.............................................. 701 Lesson 3: Trigonometric Double Angle Formulae................................................... 724 Answer Key.................................................................................... 745 Station Activities: Trigonometric Identities........................................................ 749 Unit 7: Extended Trigonometry Lesson 1: Solving Trigonometric Equations......................................................... 761 Lesson 2: Applications of Trigonometric Equations................................................. 810 Lesson 3: Vectors............................................................................... 857 Lesson 4: Complex Numbers and Trigonometry.................................................... 924 Lesson 5: Parametric Forms...................................................................... 977 Lesson 6: Polar Equations...................................................................... 1019 Answer Key................................................................................... 1069 Station Activities: Exploring Trigonometry....................................................... 1085 Unit 8: Investigations of Functions Lesson 1: Comparing and Contrasting Functions.................................................. 1095 Lesson 2: Transforming Functions............................................................... 1126 Lesson 3: Operating with Functions and the Resulting Function..................................... 1147 Answer Key................................................................................... 1167 Station Activities: Investigations of Functions..................................................... 1173 iv
Teacher s Guide Introduction The Georgia Math 3 Teacher Resource Binder is a complete set of materials developed to support achievement of Georgia s Performance Standards, as well as to prepare students for the EOCT and further studies in mathematics. Topics are built around accessible core curricula, ensuring that the Georgia Math 3 Teacher Resource Binder is useful for striving students and diverse classrooms. This program realizes the benefits of exploratory and investigative learning and employs a variety of instructional models to meet the learning needs of a range of students. The Georgia Math 3 Teacher Resource Binder includes components that support inquirybased learning, instruct as needed, provide practice, and assess students skills. al tools and strategies are embedded throughout. The scope and sequence address topics included in both the Math 3 Georgia Performance Standards and the National Council of Teachers of Mathematics Standards. The binders include: More than 150 hours of lessons, with warm-ups and engagement activities Essential Questions for each instructional topic Vocabulary Tasks Guided Practice of skills necessary for completing learning tasks Step-by-step graphing calculator instructions for the TI-83/84 and the TI-Nspire Hands-on station activities to promote problem-solving skills Purpose of Materials The Georgia Math 3 Teacher Resource Binder has been organized to coordinate with the Math 3 Georgia Performance Standards. Each lesson includes activities that offer opportunities for exploration and investigation. These activities incorporate concept and skill development and guided practice, then move on to the application of new skills and concepts in problem-solving situations. TG1
Teacher s guide Introduction This program includes all the topics addressed in the Georgia Performance Standards for Math 3 and the EOCT. These include: Data Analysis Sequences and Series Rational Functions Introduction to Trigonometry Graphs and Inverses of Trigonometric Functions Trigonometric Identities Extended Trigonometry Investigations of Functions Problem solving, reasoning and proof, communication, connections, and representations are infused throughout. Structure of the Binder The materials in the Georgia Math 3 Teacher Resource Binder are completely reproducible. Tabs allow you to access the sections of the binder quickly and easily. The Teacher s Guide is the first section. Written for you, this section helps you to navigate the materials with the pacing guide, offers several graphic organizers and suggested strategies for their use, and shows how the lessons and sub-lessons correlate to the Georgia Performance Standards. The next sections focus on content, knowledge, and application of the eight units in the Georgia Math 3 curriculum: Data Analysis; Sequences and Series; Rational Functions; Introduction to Trigonometry; Graphs and Inverses of Trigonometric Functions; Trigonometric Identities; Extended Trigonometry; and Investigations of Functions. The units in the Georgia Math 3 Teacher Resource Binder can be implemented as prescribed in the pacing guide, yet the design is flexible so that you can mix and match activities as the needs of your students and your instructional style dictate. Each lesson ends with a post-assessment. These allow you to assess students progress as you move from lesson to lesson, enabling you to gauge how well students have understood the material. The Station Activities correspond to the content in the units and provide students with the opportunity to apply concepts and skills, while you have a chance to circulate, observe, speak to individuals and small groups, and informally assess and plan. TG2
Teacher s guide Introduction Structure of Units Nearly all of the instructional units have some common features. Each lesson begins with a pre-assessment, followed by Essential Questions; vocabulary, titled Words to Know; a warm-up; a learning task; and Questions to Explore relating to the task. The materials provide instruction for each question presented in the task s Questions to Explore section, and are followed by guided practice examples that address the skills necessary for completing the learning tasks. The instructional portion of the lesson ends with a closure activity that asks students to reflect on one or more of the essential questions presented at the beginning of the lesson. Lessons are broken down into sub-lessons where appropriate. Each sub-lesson contains Skills Practice and Practice-in-Context activities designed to aid students in transferring their newly learned skills to different situations. 1. Pre-Assessment This can be used to gauge students prior knowledge and to inform instructional planning. 2. Georgia Performance Standards for the Lesson All Georgia Performance Standards that are addressed in the entire lesson are listed. 3. Essential Questions These are intended to guide students thinking as they proceed through the investigations of the learning task. By the end of each lesson, students should be able to answer the questions. 4. Words to Know Vocabulary terms and formulas are provided as background information for instruction or to review key concepts that are addressed in the lesson. 5. Warm-Up Each warm-up takes approximately 5 minutes and addresses either prerequisite and critical-thinking skills or the particular math concepts in the unit. 6. Task Each lesson presents students with a situation and a problem to solve that will be addressed through the investigation of the Questions to Explore. TG3
Teacher s guide Introduction 7. Questions to Explore Each task is followed by a series of investigative questions that you can use in guiding your students through the concepts and skills being taught in the lesson. 8. Georgia Performance Standards for the Sub-Lesson When lessons are broken down into sub-lessons, the specific standard or standards that are addressed are presented at the beginning of the instructional portion of the lesson. 9. Written for you, this section gives some direction for the skills presented in the learning tasks and helps you to guide students toward the learning of these skills. al strategies include investigation, direct instruction, modeling, and discussion. 10. Guided Practice This section provides examples of skills necessary for completing learning tasks. The examples are solved through step-by-step instructions. 11. Recommended Closure Activity Students are given the opportunity to synthesize and reflect on the lesson through a journal entry discussion of one or more of the essential questions. 12. Student Practice Sheets Each sub-lesson includes practice problems to support students achievement of the learning objectives. These sheets are written for the student. They can be used in any combination of teacherled instruction, cooperative learning, or independent application of knowledge. 13. Post-Assessment Each lesson ends with 10 or more multiple-choice questions and one extended-response question that incorporates critical thinking and writing components. This can be used to document the extent to which students grasped the concepts and skills addressed during instruction. TG4
Teacher s guide Introduction 14. Answer Key Answers for all of the Warm-Up activities, Skills Practices, and Practice-in-Context problems from the binder and all of the problems from the student book are provided at the end of each unit. (Student editions include odd answers for the exercises in their book.) 15. Station Activities Most units provide at least one set of hands-on activities that correspond to instructional topics. They can be used to introduce new concepts or to culminate a sequence of instructional experiences. (See Station Activities Introduction on the following page.) 16. Technology Most instructional topics list an EasiTeach RM keyword for computer-based exploration that can be used as available in your school. Additionally, step-by-step instructions for using the TI-83/84 and TI-Nspire are provided whenever graphing calculators are referenced. TG5
Teacher s Guide Station Activities Guide Intoduction Each unit includes a collection of station-based activities to provide students with opportunities to practice and apply the mathematical skills and concepts they are learning. You may use these activities in addition to the instructional lessons, or, especially if the pre-test or other formative assessment suggests it, instead of direct instruction in areas where students have the basic concepts but need practice. The debriefing discussions after each set of activities provide an important opportunity to help students reflect on their experiences and synthesize their thinking. It also provides an additional opportunity for ongoing, informal assessment to guide instructional planning. Implementation Guide The following guidelines will help you prepare for and use the activity sets in this section. Setting Up the Stations Each activity set consists of four stations. Set up each station at a desk, or at several desks pushed together, with enough chairs for a small group of students. Place a card with the number of the station on the desk. Each station should also contain the materials specified in the teacher s notes, and a stack of student activity sheets (one copy per student). Place the required materials (as listed) at each station. When a group of students arrives at a station, each student should take one of the activity sheets to record the group s work. Although students should work together to develop one set of answers for the entire group, each student should record the answers on his or her own activity sheet. This helps keep students engaged in the activity and gives each student a record of the activity for future reference. Forming Groups of Students All activity sets consist of four stations. You might divide the class into four groups by having students count off from 1 to 4. If you have a large class and want to have students working in small groups, you might set up two identical sets of stations, labeled A and B. In this way, the class can be divided into eight groups, with each group of students rotating through the A stations or B stations. TG6
TEACHER S GUIDE Station Activities Guide Assigning Roles to Students Students often work most productively in groups when each student has an assigned role. You may want to assign roles to students when they are assigned to groups and change the roles occasionally. Some possible roles are as follows: Reader reads the steps of the activity aloud Facilitator makes sure that each student in the group has a chance to speak and pose questions; also makes sure that each student agrees on each answer before it is written down Materials Manager handles the materials at the station and makes sure the materials are put back in place at the end of the activity Timekeeper tracks the group s progress to ensure that the activity is completed in the allotted time Spokesperson speaks for the group during the debriefing session after the activities Timing the Activities The activities in this section are designed to take approximately 10 minutes per station. Therefore, you might plan on having groups change stations every 10 minutes, with a two-minute interval for moving from one station to the next. It is helpful to give students a 5-minute warning before it is time to change stations. Since each activity set consists of four stations, the above time frame means that it will take about 50 minutes for groups to work through all stations. Guidelines for Students Before starting the first activity set, you may want to review the following ground rules with students. You might also post the rules in the classroom. All students in a group should agree on each answer before it is written down. If there is a disagreement within the group, discuss it with one another. You can ask your teacher a question only if everyone in the group has the same question. If you finish early, work together to write problems of your own that are similar to the ones on the student activity sheet. Leave the station exactly as you found it. All materials should be in the same place and in the same condition as when you arrived. TG7
TEACHER S GUIDE Station Activities Guide Debriefing the Activities After each group has rotated through every station, bring students together for a brief class discussion. At this time you might have the groups spokespersons pose any questions they had about the activities. Before responding, ask if students in other groups encountered the same difficulty or if they have a response to the question. The class discussion is also a good time to reinforce the essential ideas of the activities. The questions that are provided in the teacher s notes for each activity set can serve as a guide to initiating this type of discussion. You may want to collect the student activity sheets before beginning the class discussion. However, it can be beneficial to collect the sheets afterward so that students can refer to them during the discussion. This also gives students a chance to revisit and refine their work based on the debriefing session. If you run out of time to hold class discussions, you might want to have students journal about their experiences and follow up with a class discussion the next day. TG8
NAME: Unit 1 Data Analysis Pre-Assessment Circle the best answer. Assessment 1. Which frequency distribution that follows is normal? a. c. b. d. 2. A sample set of 50 data points has a distribution that is skewed to the left. The same data is collected from 100 samples, each containing 50 data points. What will be the distribution of the 100 sample means? a. skewed to the left b. skewed to the right c. normal d. not enough information given to determine continued 1
NAME: Unit 1 Data Analysis Assessment 3. The following samples of data are collected: {10, 13, 25, 19, 12}; {17, 20, 18, 11, 12}; {14, 21, 17, 19, 18}; {12, 16, 22, 25, 12}. What are the respective sample means? a. 13, 17, 18, 16 b. 15.8, 15.6, 17.8, 17.4 c. 79, 78, 89, 87 d. 25, 20, 21, 25 4. Which data set is a random set of integers between 1 and 20, inclusive? a. {8.5, 10, 15, 15} b. {1, 7, 9, 13} c. {1, 26, 30, 35} d. {0.1, 1.0, 2.5, 6.7} 5. Which set of data has a normal distribution? a. {0, 0, 0, 1, 1, 2, 6} b. {1, 2, 3, 3, 5, 6, 8} c. {1, 2, 5, 7, 12, 12, 12} d. {1, 2, 3, 3, 3, 4, 5} 2
Unit 1 Data Analysis Lesson 1: The Central Limit Theorem Georgia Performance Standard MA3D1. Using simulation, students will develop the idea of the central limit theorem. Essential Questions 1. When the sample sizes are large (greater than 30 in each sample), what shape is the distribution of multiple sample means? 2. How can sample means be used to estimate the mean of a population? 3. How can analysis of frequency distributions determine if a data set is normal? WORDS TO KNOW Central Limit Theorem dot plot mean median mode normal distribution skewed distribution states that even if a sample of data is not normal, the sample means of at least 30 samples of the same data will be normal a frequency diagram where each data value is given a unique point the mean of a data set can be calculated using: sum of data points x = ; the average value of the data number of data points the point of a data set for which there are an equal number of data points greater than and less than the median; if there are an even number of data points, the median is the average of the two middle data values the most frequently populated data value in a data set data that conforms to a normal distribution, when graphed using a frequency diagram, creates a normal, bell-shaped curve; the mean, median, and mode of data that has a normal distribution are equal data that, when graphed in a frequency diagram, does not have a mode that is in the center of the data values; data values are clustered to the right or left of the frequency distribution 3
NAME: Unit 1 Data Analysis Warm-Up Use a calculator to help answer each question. 1. Find the mean of the following data set: {41, 52, 58, 32, 49} 2. Create a random sample set of 10 integers between 30 and 50, inclusive. 3. Draw the general shape of a normal curve. 4
NAME: Unit 1 Data Analysis Lesson 1.1.1: The Central Limit Theorem Georgia Performance Standard MA3D1. Using simulation, students will develop the idea of the central limit theorem. Task Mrs. Ryan s class decides to explore the average number of children in each student s family. There are 20 students in the class, and they record the total number of children in each student s family in the table that follows: Student Total children in family A 3 B 1 C 2 D 2 E 5 F 1 G 2 H 3 I 1 J 4 K 3 L 2 M 1 N 6 O 2 P 4 Q 3 R 3 S 2 T 4 5
NAME: Unit 1 Data Analysis The class decides to increase the number of samples and to randomly generate 4 more samples of 20 with the total number of children ranging from 1 to 6. The 5 total samples are in the screen shot that follows. 6
NAME: Unit 1 Data Analysis Questions to Explore 1. Plot the first sample, from Mrs. Ryan s class, on a dot plot. What is the shape of the sample? 2. What is the mean of the first sample? 3. Let s break the first sample into four smaller samples. Since the data is not in any particular order, let s set the smaller samples as: 1 5, 6 10, 11 15, 16 20. What is the mean of each smaller sample? 4. Similarly, create four sample groups of each of the next four random samples. What is the mean of each smaller sample? 5. Create a dot plot of all of the 5-set sample means. Compare the distribution of the data points to the distribution of the sample means. 7
NAME: Unit 1 Data Analysis 6. Let s continue to explore sample means with more data. To simulate more data collection, the list that follows is 30 means of random data sets containing 5 items, integers ranging from 1 to 6, inclusive. {3.6, 3.6, 2.2, 2.6, 4.6, 3.4, 3.6, 2.6, 3.8, 2.6, 3.8, 4.8, 3.6, 3, 3.8, 4.2, 4.2, 3.8, 2.2, 2.4, 3.4, 3.4, 3.4, 2.8, 4.2, 3, 3.2, 4.6, 3, 3.6} Each mean in the list above is from a sample set of 5 data points. Plot the means in a dot plot. Compare the shape of the distribution to the previous dot plots. 7. Summarize your observations about the shape of a dot plot containing: 20 data points, 20 sample means (sample size of 5), and 30 sample means (sample size of 5). All data is related to the number of children in a family, based on our original collected data, where the number of children ranged from 1 to 6. 8. Determine the mean of the sample means of the 30 samples with 5 items in each sample. How does this relate to the mean of the population? 8
Unit 1 Data Analysis Warm-Up Debrief Remind students of the concepts of determining the mean, finding a random sample, and drawing a normal distribution. Calculating means and analyzing the distributions of both samples and sample means drive the Task. Introduction During the Task, students will explore a mean of a sample, then determine and analyze the means of multiple samples. The mean is the average value of the data points. The students will informally explore the Central Limit Theorem, which states that even if a sample of data is not normal, the sample means of at least 30 samples of the same data will be normal (Essential Question 1). The distribution will be more normal given larger sample sizes. The theorem also states that the mean of the sample means will be the mean of the entire population (Essential Question 2). The students are given the random samples to work with, to focus their analysis on the distribution of the sample and sample means, as well as ensuring that all students have the same data values. The sample is skewed, which means the data values do not cluster around the median data value. Instead, the data values cluster toward the right or to the left of the distribution. This can be seen when the shape of the distribution is not symmetrical. If the distribution is skewed to the right, the data has a tail to the right. If the distribution is skewed to the left, the distribution has a tail to the left. Additional samples are presented, and students break down these samples to determine sample means. Students use a frequency plot, a dot plot, where each data point (or sample mean) is given an individual point on the plot. A frequency plot allows students to see how the values are distributed. A set of data that has a normal distribution has a mode, median, and mean that are equal. The data values are also symmetric about the median (Essential Question 3). The distribution that follows is normal. Normal distribution 1 2 3 4 5 6 7 8 9 10 11 12 9
Unit 1 Data Analysis A distribution that is skewed to the right has a tail of data that trails to the right. The distribution that follows is skewed to the right. If the data is skewed to the right, the mean will be at least slightly higher than the median, because the tail of higher values increases the average of the data. The median, being the middle-most data value in the set, is not as affected by the higher values of data. The mode will be closer to the value of the median. Distribution with data skewed to the right 1 2 3 4 5 6 7 8 9 10 11 12 10
Unit 1 Data Analysis A distribution that is skewed to the left has a tail of data that trails to the left. The distribution that follows is skewed to the left. If the data is skewed to the left, the mean will be at least slightly lower than the median, because the tail of lower values decreases the average of the data. The median, being the middle-most data value in the set, is not as affected by the lower values of data. The mode will be closer to the value of the median. Distribution with data skewed to the left 1 2 3 4 5 6 7 8 9 10 11 12 11
Unit 1 Data Analysis Question 1 Plot the first sample from Mrs. Ryan s class on a dot plot. What is the shape of the sample? Using a TI-Nspire: Enter the data points into your calculator. 1. Press the Home key. 2. Select the fourth icon: Lists & Spreadsheets. 3. Navigate to the white space below column A. Type the column label s1, and press enter. 4. Enter data. Next, create the dot plot. 5. Press the Home key. 6. Select the fifth icon: Data & Statistics. 7. Use the nav pad to roll over the rectangle at the bottom of the screen, under the x-axis, then press the Click key. 8. Choose s1 from the menu that pops up, and press enter. 9. To move back to the Lists & Spreadsheets application, press ctrl and the left arrow key. To return to the dot plot, press ctrl and the right arrow key. Using a TI-83/84: Enter the data points into your calculator. 1. Press the STAT key. 2. Press 1. 3. Enter the data values into L1. A histogram on the TI-83/84 will provide a better representation of the shape of the sample. 4. Press 2nd Y=. 5. With Plot 1 highlighted, make sure that On is also highlighted. When the other Plots are selected, Off should be highlighted. (continued) 12
Unit 1 Data Analysis 6. Select the histogram option; the third graph to the right of the word Type: 7. Next to Xlist:, type L1. 8. Press WINDOW. 9. Choose an Xmin, Xmax, and Xscl to fit the data values. The Ymin, Ymax, and Yscl should be chosen based on the greatest frequency of any of the data values. 10. Press GRAPH. The sample is skewed to the right. 13
Unit 1 Data Analysis Question 2 What is the mean of the first sample? Using the TI-Nspire: 1. Press the menu key. 2. Choose 4: Statistics, then 1: Stat Calculations. 3. Select 1: One-Variable Statistics, and press enter. 4. Make sure that the X1 list is: s1[], the Frequency List is: 1, and the 1st Result Column is: b[]. Use the tab key to move between fields to make any edits. 5. Tab to the OK button, and press enter. 6. The value x is the mean. Using a TI-83/84: 1. Press the STAT key. Use the right arrow to select CALC. 2. Press 1, or highlight 1: and press Enter, to select 1-Var Stats. 3. L1 will be the default list to calculate the statistics. The value x is the mean. The mean of the first sample is 2.7. 14
Unit 1 Data Analysis Question 3 Let s break the first sample into four smaller samples. Since the data is not in any particular order, let s set the smaller samples as: 1 5, 6 10, 11 15, 16 20. What is the mean of each smaller sample? Students can use pencil and paper to determine the mean of each smaller sample by using: sum of data points x =. The mean of the four smaller samples is: number of data points 13 s1.1: x = = 5 11 s1.2: x = = 5 14 s1.3: x = = 5 16 s1.4: x = = 5 2. 6 2. 2 2. 8 3. 2 15
Unit 1 Data Analysis Question 4 Similarly, create four sample groups of each of the next four random samples. What is the mean of each smaller sample? Students can similarly use pencil and paper to determine the mean of each smaller sample by using the method in Question 3. Breaking the samples into smaller groups as was done with s1, the sample means are: 17 s2.1: x = = 5 15 s2.2: x = = 5 23 s2.3: x = = 5 16 s2.4: x = = 5 19 s3.1: x = = 5 16 s3.2: x = = 5 16 s3.3: x = = 5 12 s3.4: x = = 5 18 3. 4 s4.1: x = = 5 19 3 s4.2: x = = 5 14 4. 6 s4.3: x = = 5 14 3. 2 s4.4: x = = 5 9 3. 8 s5.1: x = = 5 14 3. 2 s5.2: x = = 5 19 3. 2 s5.3: x = = 5 17 2. 4 s5.4: x = = 5 3. 6 3. 8 2. 8 2. 8 1. 8 2. 8 3. 8 3. 4 16
Unit 1 Data Analysis Question 5 Create a dot plot of all of the 5-set sample means. Compare the distribution of the data points to the distribution of the sample means. See Question 1 to input the data into either the TI-Nspire or the TI-83/84 and to create a dot plot. The distribution of the sample means appears slightly skewed, but more normal. 17
Unit 1 Data Analysis Question 6 Let s continue to explore sample means with more data. To simulate more data collection, the list that follows is 30 means of random data sets containing 5 items, integers ranging from 1 to 6, inclusive. {3.6, 3.6, 2.2, 2.6, 4.6, 3.4, 3.6, 2.6, 3.8, 2.6, 3.8, 4.8, 3.6, 3.0, 3.8, 4.2, 4.2, 3.8, 2.2, 2.4, 3.4, 3.4, 3.4, 2.8, 4.2, 3.0, 3.2, 4.6, 3.0, 3.6} Each mean in the list is from a sample set of 5 data points. Plot the means in a dot plot. Compare the shape of the distribution to the previous dot plots. See Question 1 to input the data into either the TI-Nspire or the TI-83/84 and to create a dot plot. The distribution of the data appears more normal than any of the other dot plots. 18
Unit 1 Data Analysis Question 7 Summarize your observations about the shape of a dot plot containing: 20 data points, 20 sample means (sample size of 5), and 30 sample means (sample size of 5). All data is related to the number of children in a household, based on our original collected data, where the number of children ranged from 1 to 6. The dot plot containing 20 data points was slightly skewed to the right. The dot plot with 20 sample means of sample size 5 was more normal, and the dot plot of the 30 sample means of sample size 5 was even more normal. Despite the original data being skewed, the plot of the sample means is normal. The larger the sample size, the more likely the sample means will have a normal distribution. Question 8 Determine the mean of the sample means of the 30 samples with 5 items in each sample. How do you think this relates to the mean of the population? The mean of the 30 sample means is 3.43. As the number of samples increases, the mean of the sample means would approximate the mean of the entire population. If students are struggling with these concepts, you might want to walk them through the Guided Practice that follows. 19
Unit 1 Data Analysis Guided Practice 1.1.1 Students are asked to analyze given random samples. To further enable students to simulate data collection, work through Example 1, which requires students to use technology to generate their own random samples. Students will then examine whether a sample has a normal distribution. The general features of a normal distribution will be reviewed in Example 2. Example 1 Dillon wants to analyze the results of tossing three six-sided dice. He wants to sum the three resulting numbers and display the sums. Simulate 10 rolls of the three six-sided dice and plot the resulting sums in a dot plot. 1. We can use three different lists to track the results from each die. On the TI-Nspire: 1. Press the Home key. 2. Select the fourth icon: Lists & Spreadsheets. 3. Click on the header of column A, the grayed-out cell. 4. Type the expression for generating random integers: randint(1,6,10). This will generate a list of 10 simulated rolls of a six-sided die. Press the book key and type r. Navigate to randint( and press [enter]. Type in [1] [,] [6] [,] [10]. Then, press [enter]. 5. Type the same expression in the header of columns B and C to generate the simulations of the other two dice. On the TI-83/84: 1. Press the MATH key. 2. Use the right arrow to highlight PRB. 3. Press 5, or select 5: randint(. 4. Complete the randint expression with: randint(1,6,10) to generate 10 numbers between 1 and 6 to simulate the tossing of one die 10 times. 5. Re-enter the randint( expression to generate two more sets of 10 random numbers, to simulate tossing the other two dice. 20
Unit 1 Data Analysis Sample set of random integers: A B C 3 2 5 6 2 3 5 2 4 5 3 6 6 5 2 3 1 2 1 4 3 2 5 6 2 5 2 5 1 1 2. Sum each row to determine the sum of the three dice. A B C Sum 3 2 5 10 6 2 3 11 5 2 4 11 5 3 6 14 6 5 2 13 3 1 2 6 1 4 3 8 2 5 6 13 2 5 2 9 5 1 1 7 21
Unit 1 Data Analysis 3. Plot the sums on a frequency plot. Using a TI-Nspire: Enter the data points into your calculator. 1. Press the Home key. 2. Select the fourth icon: Lists & Spreadsheets. 3. Navigate to the white space to below column A. Type the column label s, and press enter. 4. Enter data. Next, create the dot plot. 5. Press the Home key. 6. Select the fifth icon: Data & Statistics. 7. Use the nav pad to roll over the rectangle at the bottom of the screen, under the x-axis, then press the Click key. 8. Choose s from the menu that pops up, and press enter. 9. To move back to the Lists & Spreadsheets application, press ctrl and the left arrow key. To return to the dot plot, press ctrl and the right arrow key. Using a TI-83/84: Enter the data points into your calculator. 1. Press the STAT key. 2. Press 1. 3. Enter the data values into L1. A histogram on the TI-83/84 will provide a better representation of the shape of the sample. 4. Press 2nd Y=. 5. With Plot 1 highlighted, make sure that On is also highlighted. When the other Plots are selected, Off should be highlighted. 6. Select the histogram option; the third graph to the right of the word Type: 7. Next to Xlist:, type L1. 8. Press WINDOW. 9. Choose an Xmin, Xmax, and Xscl to fit the data values. The Ymin, Ymax, and Yscl should be chosen based on the greatest frequency of any of the data values. 10. Press GRAPH. 22
Unit 1 Data Analysis 23
Unit 1 Data Analysis Example 2 Hailey visits multiple grocery stores and collects the data that follows. Store Cost of brand X bread A 2.25 B 3.10 C 2.99 D 2.25 E 3.05 F 3.00 G 2.75 H 2.10 I 3.50 J 2.80 K 3.10 L 2.85 M 2.50 N 2.90 O 2.95 P 3.25 Q 1.50 R 2.40 S 2.99 T 3.60 Does the cost of a loaf of bread represent a normal distribution? 24
Unit 1 Data Analysis 1. Create a dot plot of the data. See Example 1 for step-by-step instructions to create the dot plot with your TI-Nspire or TI-83/84. 2. Analyze the shape of the frequency distribution. If the distribution is normal, the data values should be clustered in the middle of the data values. This data appears to be slightly skewed left. 25
Unit 1 Data Analysis 3. Use measures of mean, median, and mode to further analyze if the data is normal. The median and mean can be determined using a calculator. Using the TI-Nspire: 1. Press the menu key. 2. Choose 4: Statistics, then 1: Stat Calculations. 3. Select 1: One-Variable Statistics, and press enter. 4. Make sure that the X1 list is: s1[], the Frequency List is: 1, and the 1st Result Column is: b[]. Use the tab key to move between fields to make any edits. 5. Tab to the OK button, and press enter. 6. The value x is the mean, and the value MedianX is the median. Using a TI-83/84: 1. Press the STAT key. Use the right arrow to select CALC. 2. Press 1, or highlight 1: and press Enter, to select 1-Var Stats. 3. L1 will be the default list to calculate the statistics. The value x is the mean. Scroll down to see Med=, which is the value of the mean. 26
Unit 1 Data Analysis The median of the data set is 2.925, and the mean is 2.7915. Reorder the list from least to greatest to determine the mode. Store Cost of brand X bread Q 1.50 H 2.10 A 2.25 D 2.25 R 2.40 M 2.50 G 2.75 J 2.80 L 2.85 N 2.90 O 2.95 C 2.99 S 2.99 F 3.00 E 3.05 B 3.10 K 3.10 P 3.25 I 3.50 T 3.60 There are three modes, since three of the values repeat 2 times: 2.25, 2.99, and 3.10. The median is greater than the mean, confirming that the data is skewed to the left. Recommended Closure Activity Select one or more of the Essential Questions for a class discussion or as a journal entry prompt. 27
NAME: Unit 1 Data Analysis Skills Practice 1.1.1: Simulating and Describing Data For problems 1 5, simulate collecting data by finding a random set of integers with the given parameters. Plot the sample set on a dot plot and describe the distribution of the data. 1. Ten integers between 1 and 5, inclusive 2. Twenty integers between 10 and 15, inclusive 3. Twenty integers between 70 and 90, inclusive 4. Ten integers between 100 and 105, inclusive 5. Thirty integers between 20 and 30, inclusive For problems 6 10, simulate collecting data by finding random sets of integers with the given parameters. Plot the sample means on a dot plot and describe the distribution of the data. 6. Ten samples with 5 data points, containing integers from 1 to 4 7. Ten samples with 10 data points, containing integers from 9 to 18 8. Ten samples with 10 data points, containing integers from 40 to 45 9. Six samples with 20 data points, containing integers from 2 to 8 10. Six samples with 20 data points, containing integers from 51 to 59 28
NAME: Unit 1 Data Analysis Practice in Context 1.1.1: Analyzing Data Use the problem statements to answer the questions that follow. All twelve- and thirteen-year-old girls in Joelle s math class record their height in the following table: Student Height (in inches) A 58 B 61 C 62 D 59 E 60 F 62 G 60 H 61 I 60 J 59 K 61 L 63 M 57 N 60 O 61 1. Plot the class data in a dot plot. Describe the distribution based on the shape of the frequency plot and measures of center. continued 29
NAME: Unit 1 Data Analysis 2. Break the sample into 3 smaller samples, each containing 5 data points. Simulate collecting 5 more samples of data, each containing 5 data points, with heights ranging from 57 to 63. Plot the sample means on a dot plot, and describe the distribution of the plot and measures of center. All twelve- and thirteen-year-old boys in Joelle s math class record their height in the following table: Student Height (in inches) A 59 B 56 C 61 D 57 E 58 F 63 G 62 H 60 I 58 J 59 K 60 L 64 M 58 N 57 O 59 continued 30
NAME: Unit 1 Data Analysis 3. Plot the class data in a dot plot. Describe the distribution based on the shape of the frequency plot and measures of center. 4. Break the sample into 3 smaller samples, each containing 5 data points. Simulate collecting 5 more samples of data, each containing 5 data points, with heights ranging from 56 to 64. Plot the sample means on a dot plot, and describe the distribution of the plot and measures of center. 31
NAME: Unit 1 Data Analysis Post-Assessment Circle the best answer. Assessment 1. Ten samples are collected, each containing 30 items. The means of the samples are: 29.03, 29.17, 28.8, 28.63, 29.03, 28.73, 29.03, 28.8, 28.93, 29.93. What is the mean of the sample means? a. 28.98 c. 29.03 b. 29.00 d. 290.08 2. The frequency plot of a data set is shown below. How can the distribution of the data be described? a. normal c. skewed to the left b. skewed to the right d. bimodal 3. A random set of integers is generated to simulate a data collection of integers from 40 to 45. Which set of data that follows could be the generated integers? a. {40, 41, 41, 42.5, 44} b. {41, 43, 45, 45, 46} c. {43, 44, 45, 45, 46} d. {44, 44, 44, 45, 45} continued 32
NAME: Unit 1 Data Analysis Assessment 4. A sample of 100 data points is collected. The data is skewed to the left. Ten more samples of data are collected, each with 100 data points. What is the distribution of all the data points? a. normal b. skewed to the left c. skewed to the right d. distribution is not certainly known 5. Which set of data has a normal distribution? a. {0.4, 0.5, 0.6, 0.6, 0.6, 0.7, 0.8} b. {1, 3, 3, 3, 5, 7, 10} c. {112, 112, 116, 118, 118, 120, 125} d. {1.3, 2.0, 2.5, 2.8, 3.4, 4.1, 5.6} 6. A set of 10 data items is collected and listed below. What are the mean, median, and mode of the data set? {6, 7, 9, 10, 9, 7, 8, 6, 9, 11} a. Mean: 8.2; Median: 8.5; Mode: 9 b. Mean: 8.5; Median: 8.2; Mode: 9 c. Mean: 8.2; Median: 8; Mode: 7 d. Mean: 8.5; Median: 9; Mode: 6 continued 33
NAME: Unit 1 Data Analysis 7. Which dot plot that follows has a normal distribution? Assessment a. c. b. d. 8. A data set of 40 points is skewed to the right. Fifteen more samples of data are collected, each with 40 data points. What will be the distribution of the 16 sample means? a. skewed to the right b. skewed to the left c. normal d. distribution is not certainly known continued 34
NAME: Unit 1 Data Analysis 9. Which data set has a distribution that is skewed to the left? a. {1, 2, 3, 3, 3, 3, 3, 4, 5} b. {14, 15, 15, 15, 16, 16, 17, 18, 21} c. {25, 25, 26, 27, 28, 29, 29, 30, 31} d. {32, 36, 38, 40, 41, 42, 43, 43, 44} Assessment 10. A random set of integers is generated to simulate a data collection of 6 integers from 21 to 28. Which set of data that follows could be the generated integers? a. {27, 28, 28, 28, 28, 28} b. {21, 22, 23, 24, 28, 29} c. {21.0, 22.1, 23.4, 24.1, 25.0, 26.0} d. {25, 25, 28, 28, 28, 29} 11. Simulate collecting data by finding 10 random sets of 10 integers from 13 to 18. Plot the sample means on a dot plot and describe the distribution of the data. 35
Unit 1 data analysis Station Activities: Distributions and Estimating with Confidence Goal: To provide opportunities for students to analyze distributions of simulated data and confidence intervals and to calculate point estimates Georgia Performance Standards MA3D1. MA3D2. MA3D3. Using simulation, students will develop the idea of the central limit theorem. Using student-generated data from random samples of at least 30 members, students will determine the margin of error and confidence interval for a specified level of confidence. Students will use confidence intervals and margins of error to make inferences from data about a population. Technology is used to evaluate confidence intervals, but students will be aware of the ideas involved. The student will: 1. analyze the distribution of simulated data. 2. describe the distribution of sample means of simulated data. 3. analyze confidence intervals. 4. calculate point estimates with different levels of confidence. Student Activities Overview and Answer Key Station 1 Students will work in small groups. Each student will simulate collecting 50 numbers between 1 and 6, and will plot these values on a line graph. The distribution of the plots will vary by student. Students will be asked to describe the line plot s shape and also to determine the mean and standard deviation. Students will then collaborate with their group members. Initially, they will discuss their individual plots, and will then compile their data to create a larger data set. Theoretically, the data set will appear more normal. They will be given the definition of a normal distribution, and will be asked to analyze the percent of the data that falls within one, two, and three standard deviations. After this analysis they are asked to decide if the compiled data is more normal. Answers 1. Answers vary; should be 50 integers from 1 and 6. 99
Unit 1 data analysis Station Activities: Distributions and Estimating with Confidence 1a. Sample line plot: 1b. Answers vary based on simulated data. For sample data in 1a, data appears skewed to the right. 1c. Answers vary based on simulated data. For sample data in 1a: mean: 3.04; sample standard deviation: 1.666 2a. Distributions will vary by student. The range of all data will be the same, but line plots will differ. 2b. Means will likely be around 3, but student data will vary. Standard deviations will also likely be close to sample data in 1a. 3a. Answers will vary. Within one standard deviation of the sample data from 1a lies 28 of the data values, or 56% of the data. Within two standard deviations lies 44 of the data values, or 88% of the data, and within three standard deviations lies 100% of the data. There is not enough data, in this sample, within one standard deviation of the mean to be considered a normal distribution. 3b. The compiled data will be fairly evenly distributed among the 6 integers. Distribution will vary based on individual simulations. Below is a sample plot four group members. 100
Unit 1 data analysis Station Activities: Distributions and Estimating with Confidence Station 2 Students will work with simulated data. Students will describe the shape, mean, and standard deviation of the sample means. They will then determine if the sample means fulfill the requirements of a normal distribution. Answers 1a. 1b. The data set is too small to meaningfully analyze on a dot plot. The mean of the sample data from 1a is: 3.61, and the sample standard deviation is: 0.17. 2a. 2b. The shape of the sample means appears more normal. The mean is: 3.54, and the sample standard deviation is: 0.227. 2c. 21 of the 28 data points, or 75%, lie within one standard deviation. 26 of the 28 data points, or 93%, lie within two standard deviations, and all of the data, or 100%, lie within 3 standard deviations. This closely resembles the percentages of a normal distribution, and the data can be considered normally distributed. 101
Unit 1 data analysis Station Activities: Distributions and Estimating with Confidence Station 3 Students will analyze confidence intervals. As the sample size increases, and the confidence level remains constant, the margin of error will decrease. The confidence interval will also, respectively, decrease. Students can then compare the mean of a class s sample means which should closely resemble the mean of the population to the confidence interval for the mean of a group s sample means. They should see that the mean of the sample means falls within this interval, reinforcing that the confidence interval is a good indicator of the population mean. Answers 1a. Margin of error: 0.088 1b. Confidence interval: 3.45 < µ < 3.63 2a. Margin of error: 0.266 2a. Confidence interval: 3.34 < µ < 3.88 3a. The margin of error, with 95% confidence, of the population mean is much smaller when working with a sample of 200 data items versus 50. 3b. The mean of the sample means is: 3.54. The mean of the sample means is exactly in the center of the confidence interval, with 95% confidence, of our population mean. 3c. Based on the confidence interval and the mean of the sample means, the population mean of all die rolls of a six-sided die is approximately 3.5. Station 4 Students will examine the difference between an estimate of a population on different confidence intervals. As the confidence level decreases, the confidence interval decreases. Being less confident about the interval of a population parameter implies that fewer values are within that interval. Answers 1. Confidence interval: 27.01 < µ < 30.07 2. Yes. The estimate is within the range, so it would be acceptable for the car manufacturer to state the average mpg was 30. 3. Confidence interval, for 90% confidence: 27.59 < µ < 29.48. The stated mpg is not within this range, so it would not be a good estimate with 90% confidence. 4. For an identical data set, a lesser confidence level has a smaller confidence interval. This is because it is with less certainty that the mean is within a smaller range. A larger range of values allows for more error in the parameter, and allows us to be more certain. 102
Unit 1 data analysis Station Activities: Distributions and Estimating with Confidence Materials List/Set-Up Station 1 Station 2 Station 3 Station 4 student activity sheet; graphing calculators student activity sheet; graphing calculators student activity sheet; graphing calculators student activity sheet; graphing calculators 103
Unit 1 data analysis Station Activities: Distributions and Estimating with Confidence Discussion Guide To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions 1. How did the number of data points affect the distribution of the data? 2. Does the distribution of a collected set of data affect the distribution of the sample means? 3. How is the confidence interval of a population mean affected by an increase in sample size? 4. How does the confidence level affect the confidence interval? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in the same station group), and then discuss as a whole class. Suggested Appropriate Responses 1. The more data points, the more the data will represent data from an entire population. When trying to analyze sample means, it was difficult to discuss any distribution when only comparing 4 pieces of data. However, once all the class data was compiled, it was easy to determine if the data was normally distributed. 2. It shouldn t. The distribution of the data in Station 1 is fairly evenly distributed, but the sample means were normal. The data does not need to be normal for the distribution of the sample means to be normal. 3. The larger sample size greatly reduced the margin of error. The confidence interval was therefore much smaller, and narrowed down the possibilities for the population mean. 4. A higher confidence level means the interval is larger. Possible Misunderstandings/Mistakes Making general observations when only working with small numbers of data Omitting qualifying language, such as within my sample or in our group 104