PROBABILITY & STATISTICS GRADE: 11 & 12 Prerequisite: Algebra 2 Credits: 2.5 ABSTRACT This half year, two and one half credit course is to allow students the opportunity to study higher level mathematics dealing with the concepts and methods of both descriptive and inferential statistics in addition to using probability as a tool for predicting the statistical characteristics of collected data. This one semester course is designed for students with a wide variety of future vocational and educational ambitions. It aims to give students an understanding of the most significant concepts of probability and statistics, and to enable students to be wiser users and more critical consumers of statistical data. Key topics include measures of central tendency, graphical and statistical displays and analysis, (i.e., frequency tables and scatter plots), random samplings, basic probability, normal distributions, confidence intervals and tests of significance. Due to the nature of the course, all problems allow students to apply statistical techniques to real world situations and career oriented applications. Extensive technology usage is recommended through the use of calculators and computers. A prerequisite for this course is the successful completion of Algebra I, Geometry and Algebra II. In preparation for the state assessments, students will be given formative and summative assessments throughout the course. Course content is aligned with the appropriate Common Core State Standards.
Unit 1: Data Collection Stage 1 Desired Results Established Goals 2 weeks September S-ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. N-VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. S-CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S-IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. S-IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Technology 8.1 21st Century Life and Careers 9.1.8.A.1-4 9.1.8.B.1-2,4,5 9.1.8.C.1-3 9.1.8.D.1-4 9.2 B2-3
9.2 C.1-2 9.2 D.1 MP1-8: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Enduring Understanding What is statistics? What does random mean? How can data be collected / measured? Essential Questions Can a sample misrepresent a population? How is statistics used in the real world? What is a population? What types of samples can be used to collect data? How can a random sample be obtained? How can errors be avoided or corrected? Stage 2 Assessment Evidence Performance tasks What is the best type of experiment design for a given study? Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others.
Other Evidence 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Pre-Assessment 9. Quizzes and tests(summative assessment) Checkpoint exercises Homework Oral questioning, closure, Projects Do-Nows/Exit tickets ( Formative assessment) Resources Benchmark Assessment Graphing Calculators Manipulatives(when applicable) Math Literature A wide variety of Online Resources Teacher Generated Materials Stage 3 Skills and Topics Define statistics Understand the process of statistics Distinguish between qualitative and quantitative variables Distinguish between discrete and continuous data Distinguish between an observational study and an experiment Obtain a simple random sample Understand sources of error in sampling Understand and apply the process of experiment design Applications of the various types of experiments Cross Curricular and Differentiation Individual Instruction Plans
Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities Integrate Technology: 1. Textbook online resources at my.hrw.com (assessments, learning tools) ipad (for above, and YouTube math videos, as appropriate) Special Education: Provide modifications and accommodations as listed in the student s IEP Position student near helping peer or have quick access to teacher Modify or reduce assignments/texts Reduce length of assignment for different mode of delivery Increase one-to-one time Utilize working contract between you and student at risk Prioritize tasks Provide manipulatives Use graphic organizers Use interactive math journals Use online resources for skill building Provide teacher notes Use collaborative grouping strategies such as small groups Use online resources ELL: Place student next to same-language speaker, if possible Provide text to speech for math problems Use of translation dictionary or software Implement strategy groups Confer frequently Provide graphic organizers
Modification plan Adapt a strategy-adjusting strategy for ELL: http://www.teachersfirst.com/content/esl/adaptstrat.cfm *ELL Students- Instruction will be based on language proficiency. At Risk: Tiered interventions following RtI framework Rtl Intervention Bank Use additional practice and textbook RTI resources Utilize online resources such as http://www.tenmarks.com or www.khanacademy.org Gifted: Process should be modified: higher-order-thinking skills, open-ended thinking, discovery Utilize project-based learning for greater depth of knowledge Utilize exploratory connections to higher grade concepts Contents should be modified: abstraction, complexity, variety, organization Products should be modified: real-world problems, audiences, deadlines, evaluation, transformations Learning environments should be modified: student-centered learning, independence, openness, complexity, groups varied Use of web based resources such as http://www.tenmarks.com www.khanacademy.org geogebra.org Unit 2: Organizing and Summarizing Data Stage 1 Desired Results Established Goals 8 weeks September -November S-ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S-CP.4. Construct and interpret two-way frequency tables of data when
two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S-ID.2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S-ID.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. S-ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots). CC F.TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems CC G.SRT.10 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Technology 8.1 21st Century Life and Careers 9.1.8.A.1-4 9.1.8.B.1-2,4,5 9.1.8.C.1-3 9.1.8.D.1-4 9.2 B2-3 9.2 C.1-2 9.2 D.1
MP1-8: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Enduring Understanding How can qualitative data be summarized? Why are graphs and plots useful to represent data? How should quantitative data be organized for understanding? What do measures of central tendency and dispersion help us understand? Essential Questions How can data points be compared within a set? What is qualitative data? What is quantitative data? How are frequency and relative frequency different? What is the difference between mean and median? Can one number greatly affect and entire set of data? Why are grouped data measures estimates? What do percentiles mean? (Think SAT) Stage 2 Assessment Evidence Performance tasks Why is the boxplot a great set summary tool? Mathematical Practices 1. Make sense of problems and persevere in solving them.
Other Evidence 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Pre-Assessment 9. Quizzes and tests(summative assessment) Checkpoint exercises Homework Oral questioning, closure, Projects Resources Do-Nows/Exit tickets ( Formative assessment) Graphing Calculators Manipulatives(when applicable) Math Literature A wide variety of Online Resources Teacher Generated Materials Stage 3 Skills and Topics Construct frequency and relative frequency distributions from qualitative data Construct bar graphs and pie charts Summarize discrete and continuous data in tables Construct histograms of discrete and continuous data Draw stem and leaf plots Identify the shape of a distribution Construct frequency polygons and ogives Create frequency and relative frequency tables Understand causes of misleading graphical representations of
Cross Curricular and Differentiation data Determine mean, median, and mode from raw data Use measures of central tendency to identify the shape of a distribution Compute the range, standard deviation, and variance of a data set Apply the Empirical Rule and Chebyshev s Theorem Compute measures of central tendency and dispersion for grouped data Apply z-scores, percentiles, and quartiles to data Check for outliers Visually represent data using boxplots Individual Instruction Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities Integrate Technology: 1. Textbook online resources at my.hrw.com (assessments, learning tools) ipad (for above, and YouTube math videos, as appropriate) Special Education: Provide modifications and accommodations as listed in the student s IEP Position student near helping peer or have quick access to teacher Modify or reduce assignments/texts Reduce length of assignment for different mode of delivery Increase one-to-one time Utilize working contract between you and student at risk Prioritize tasks Provide manipulatives Use graphic organizers
Use interactive math journals Use online resources for skill building Provide teacher notes Use collaborative grouping strategies such as small groups Use online resources ELL: Place student next to same-language speaker, if possible Provide text to speech for math problems Use of translation dictionary or software Implement strategy groups Confer frequently Provide graphic organizers Modification plan Adapt a strategy-adjusting strategy for ELL: http://www.teachersfirst.com/content/esl/adaptstrat.cfm *ELL Students- Instruction will be based on language proficiency. At Risk: Tiered interventions following RtI framework Rtl Intervention Bank Use additional practice and textbook RTI resources Utilize online resources such as http://www.tenmarks.com or www.khanacademy.org Gifted: Process should be modified: higher-order-thinking skills, open-ended thinking, discovery Utilize project-based learning for greater depth of knowledge Utilize exploratory connections to higher grade concepts Contents should be modified: abstraction, complexity, variety, organization Products should be modified: real-world problems, audiences, deadlines, evaluation, transformations Learning environments should be modified: student-centered learning, independence, openness, complexity, groups varied Use of web based resources such as http://www.tenmarks.com
www.khanacademy.org geogebra.org
Unit 3: Bivariate Data Stage 1 Desired Results Established Goals 2 Weeks November-December S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Informally assess the fit of a function by plotting and analyzing residuals. Fit a linear function for a scatter plot that suggests a linear association. S-ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit. S-ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Technology 8.1 21st Century Life and Careers 9.1.8.A.1-4 9.1.8.B.1-2,4,5 9.1.8.C.1-3 9.1.8.D.1-4 9.2 B2-3 9.2 C.1-2 9.2 D.1 MP1-8: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning. Enduring Understanding How can bivariate data be collected and organized? How can a regression line be used to gain more information about a data set? Essential Questions Why are predictions based on data necessary? How can bivariate data be graphed by hand, on the calculator, and using a computer or tablet? What does r tell us about the relation between variables? Do linear models work indefinitely? Stage 2 Assessment Evidence Performance tasks Other Evidence How can a regression model be used to make predictions? Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Pre-Assessment 9. Quizzes and tests(summative assessment) Checkpoint exercises Homework Oral questioning, closure, Projects Do-Nows/Exit tickets ( Formative assessment)
Resources Benchmark Assessment Graphing Calculators Manipulatives(when applicable) Math Literature A wide variety of Online Resources Teacher Generated Materials Stage 3 Skills and Topics Draw and interpret scatter diagrams Compute and interpret the linear correlation coefficient Find the least squares regression line Interpret the slope and y-intercept of the least squares regression line Interpolate and extrapolate data based on the least squares regression line Compute and interpret the coefficient of determination Perform residual analysis on a regression model Identify influential observations Cross Curricular and Differentiation Individual Instruction Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities Integrate Technology: 1. Textbook online resources at my.hrw.com (assessments, learning tools) ipad (for above, and YouTube math videos, as appropriate) Special Education: Provide modifications and accommodations as listed in the student s
IEP Position student near helping peer or have quick access to teacher Modify or reduce assignments/texts Reduce length of assignment for different mode of delivery Increase one-to-one time Utilize working contract between you and student at risk Prioritize tasks Provide manipulatives Use graphic organizers Use interactive math journals Use online resources for skill building Provide teacher notes Use collaborative grouping strategies such as small groups Use online resources ELL: Place student next to same-language speaker, if possible Provide text to speech for math problems Use of translation dictionary or software Implement strategy groups Confer frequently Provide graphic organizers Modification plan Adapt a strategy-adjusting strategy for ELL: http://www.teachersfirst.com/content/esl/adaptstrat.cfm *ELL Students- Instruction will be based on language proficiency. At Risk: Tiered interventions following RtI framework Rtl Intervention Bank Use additional practice and textbook RTI resources Utilize online resources such as http://www.tenmarks.com or www.khanacademy.org Gifted: Process should be modified: higher-order-thinking skills, open-ended thinking, discovery
Utilize project-based learning for greater depth of knowledge Utilize exploratory connections to higher grade concepts Contents should be modified: abstraction, complexity, variety, organization Products should be modified: real-world problems, audiences, deadlines, evaluation, transformations Learning environments should be modified: student-centered learning, independence, openness, complexity, groups varied Use of web based resources such as http://www.tenmarks.com www.khanacademy.org geogebra.org
Unit 4: Probability Stage 1 Desired Results Established Goals 2 Weeks December-January S-CP.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S-CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S-CP.4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S-CP.5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S-CP.6. Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model.
S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. S-CP.8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Technology 8.1 21st Century Life and Careers 9.1.8.A.1-4 9.1.8.B.1-2,4,5 9.1.8.C.1-3 9.1.8.D.1-4 9.2 B2-3 9.2 C.1-2 9.2 D.1 MP1-8: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Enduring Understanding How does probability influence our decisions? Why is it important to conduct experiments many times? How do related events affect probability? What is the relation between and & or and the multiplication and addition rules?
Essential Questions How can the number of possible outcomes be computed? How is simple probability computed? Does experimental data match computed probability? What does mutually exclusive mean? How are compound probabilities computed? Stage 2 Assessment Evidence Performance tasks Other Evidence How many outcomes are possible in the PowerBall Lottery? Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Pre-Assessment 9. Quizzes and tests(summative assessment) Checkpoint exercises Homework Oral questioning, closure, Projects Resources Do-Nows/Exit tickets ( Formative assessment) Graphing Calculators Manipulatives(when applicable) Math Literature
A wide variety of Online Resources Teacher Generated Materials Stage 3 Skills and Topics Understand the properties of probabilities Compute and interpret probabilities Obtain probabilities through experiments Understand subjective probabilities Use the addition rule Compute the probability of an event using complements Compute probabilities using the multiplication rule Compute conditional probabilities Use the multiplication rule to check for independence Solve counting problems using the multiplication principle, permutation, and combinations Cross Curricular and Differentiation Individual Instruction Plans Current-event problems Manipulatives Tiered Lessons Learning style adaptation R.A.F.T Project Based learning Compacting Multimedia presentations Open-ended responses Conclusions and analysis of exploratory activities Integrate Technology: 1. Textbook online resources at my.hrw.com (assessments, learning tools) ipad (for above, and YouTube math videos, as appropriate) Special Education: Provide modifications and accommodations as listed in the student s IEP Position student near helping peer or have quick access to teacher Modify or reduce assignments/texts Reduce length of assignment for different mode of delivery Increase one-to-one time Utilize working contract between you and student at risk
Prioritize tasks Provide manipulatives Use graphic organizers Use interactive math journals Use online resources for skill building Provide teacher notes Use collaborative grouping strategies such as small groups Use online resources ELL: Place student next to same-language speaker, if possible Provide text to speech for math problems Use of translation dictionary or software Implement strategy groups Confer frequently Provide graphic organizers Modification plan Adapt a strategy-adjusting strategy for ELL: http://www.teachersfirst.com/content/esl/adaptstrat.cfm *ELL Students- Instruction will be based on language proficiency. At Risk: Tiered interventions following RtI framework Rtl Intervention Bank Use additional practice and textbook RTI resources Utilize online resources such as http://www.tenmarks.com or www.khanacademy.org Gifted: Process should be modified: higher-order-thinking skills, open-ended thinking, discovery Utilize project-based learning for greater depth of knowledge Utilize exploratory connections to higher grade concepts Contents should be modified: abstraction, complexity, variety, organization Products should be modified: real-world problems, audiences, deadlines, evaluation, transformations
Learning environments should be modified: student-centered learning, independence, openness, complexity, groups varied Use of web based resources such as http://www.tenmarks.com www.khanacademy.org geogebra.org