Curriculum and Instruction – Mathematics Quarter 2 Finite Math

Introduction In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025,  80% of our students will graduate from high school college or career ready  90% of students will graduate on time  100% of our students who graduate college or career ready will enroll in a post-secondary opportunity

In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, College and Career Ready standards- aligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and Career Ready Standards are rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor. Focus Coherence Rigor

• The TN Standards call for a greater focus in Thinking across grades: Conceptual understanding: mathematics. Rather than racing to cover topics • The TN Standards are designed • The TN Standards call for in a mile-wide, inch-deep curriculum, the around coherent progressions from grade to conceptual understanding of key concepts, such Standards require us to significantly narrow and grade. Learning is carefully connected across as place value and ratios. Students must be able deepen the way time and energy is spent in the grades so that students can build new to access concepts from a number of math classroom. We focus deeply on the major understanding onto foundations built in perspectives so that they are able to see math as concepts of each subject so that students can previous years. Each standard is not a new more than a set of mnemonics or discrete gain strong foundations: solid conceptual event, but an extension of previous learning. procedures. understanding, a high degree of procedural skill Procedural skill and fluency: and fluency, and the ability to apply the math Linking to major topics: • The Standards call for speed and they know to solve problems inside and outside • Instead of allowing additional or accuracy in calculation. While the high school the math classroom. supporting topics to detract from course, these standards for math do not list high school concepts serve the course focus. For example, fluencies, there are suggested fluency standards instead of data displays as an end in themselves, for algebra 1, geometry and algebra 2. they are an opportunity to do grade-level word problems. Application: • The Standards call for students to use math flexibly for applications in problem- solving contexts. In content areas outside of Shelby County Schools 2016/2017 Revised 8/31/16 1 of 12 math, particularly science, students are given the opportunity to use math to make meaning of and access content.

The Standards for Mathematical Practice describe varieties of expertise, habits of minds and productive dispositions that mathematics educators at all levels should seek to develop in their students. These practices rest on important National Council of Teachers of Mathematics (NCTM) “processes and proficiencies” with longstanding importance in mathematics education. Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice.

This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that, ultimately our students, can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their practice so that it is in alignment with the three mathematics instructional shifts.

Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around the content standards and mathematical practice standards that teachers should consistently access:

The TN Mathematics Standards The Tennessee Mathematics Standards: Teachers can access the Tennessee State standards, which https://www.tn.gov/education/article/mathematics- are featured throughout this curriculum map and represent standards college and career ready learning at reach respective grade level. Standards for Mathematical Practice Mathematical Practice Standards Teachers can access the Mathematical Practice Standards, https://drive.google.com/file/d/0B926oAMrdzI4RUpMd1 which are featured throughout this curriculum map. This link pGdEJTYkE/view contains more a more detailed explanation of each practice along with implications for instructions.

2 Curriculum and Instruction – Mathematics Quarter 2 Finite Math

Purpose of the Mathematics Curriculum Maps

This curriculum framework or map is meant to help teachers and their support providers (e.g., coaches, leaders) on their path to effective, college and career ready (CCR) aligned instruction and our pursuit of Destination 2025. It is a resource for organizing instruction around the TN State Standards, which define what to teach and what students need to learn at each grade level. The framework is designed to reinforce the grade/course-specific standards and content—the major work of the grade (scope)—and provides a suggested sequencing and pacing and time frames, aligned resources—including sample questions, tasks and other planning tools. Our hope is that by curating and organizing a variety of standards-aligned resources, teachers will be able to spend less time wondering what to teach and searching for quality materials (though they may both select from and/or supplement those included here) and have more time to plan, teach, assess, and reflect with colleagues to continuously improve practice and best meet the needs of their students.

The map is meant to support effective planning and instruction to rigorous standards; it is not meant to replace teacher planning or prescribe pacing or instructional practice. In fact, our goal is not to merely “cover the curriculum,” but rather to “uncover” it by developing students’ deep understanding of the content and mastery of the standards. Teachers who are knowledgeable about and intentionally align the learning target (standards and objectives), topic, task, and needs (and assessment) of the learners are best- positioned to make decisions about how to support student learning toward such mastery. Teachers are therefore expected--with the support of their colleagues, coaches, leaders, and other support providers--to exercise their professional judgement aligned to our shared vision of effective instruction, the Teacher Effectiveness Measure (TEM) and related best practices. However, while the framework allows for flexibility and encourages each teacher/teacher team to make it their own, our expectations for student learning are non-negotiable. We must ensure all of our children have access to rigor—high-quality teaching and learning to grade- level specific standards, including purposeful support of literacy and language learning across the content areas.

Additional Instructional Support Shelby County Schools adopted our current math textbooks for grades 6-8 in 2010-2011. The textbook adoption process at that time followed the requirements set forth by the Tennessee Department of Education and took into consideration all texts approved by the TDOE as appropriate. We now have new standards; therefore, the textbook(s) have been vetted using the Instructional Materials Evaluation Tool (IMET). This tool was developed in partnership with Achieve, the Council of Chief State Officers (CCSSO) Shelby County Schools 2016/2017 Revised 8/31/16 3 of 12 and the Council of Great City Schools. The review revealed some gaps in the content, scope, sequencing, and rigor (including the balance of conceptual knowledge development and application of these concepts), of our current materials.

The additional materials purposefully address the identified gaps in alignment to meet the expectations of the CCR standards and related instructional shifts while still incorporating the current materials to which schools have access. Materials selected for inclusion in the Curriculum Maps, both those from the textbooks and external/supplemental resources (e.g., EngageNY), have been evaluated by district staff to ensure that they meet the IMET criteria.

How to Use the Mathematics Curriculum Maps

Overview An overview is provided for each quarter. The information given is intended to aid teachers, coaches and administrators develop an understanding of the content the students will learn in the quarter, how the content addresses prior knowledge and future learning, and may provide some non-summative assessment items.

Tennessee State Standards The TN State Standards are located in the left column. Each content standard is identified as the following: Major Work, Supporting Content or Additional Content.; a key can be found at the bottom of the map. The major work of the grade should comprise 65-85% of your instructional time. Supporting Content are standards that supports student’s learning of the major work. Therefore, you will see supporting and additional standards taught in conjunction with major work. It is the teacher’s responsibility to examine the standards and skills needed in order to ensure student mastery of the indicated standard.

Content Teachers are expected to carefully craft weekly and daily learning objectives/ based on their knowledge of TEM Teach 1. In addition, teachers should include related best practices based upon the TN State Standards, related shifts, and knowledge of students from a variety of sources (e.g., student work samples, MAP, etc.). Support for the development of these lesson objectives can be found under the column titled ‘Content’. The enduring understandings will help clarify the “big picture” of the standard. The essential questions break that picture down into smaller questions and the objectives provide specific outcomes for that standard(s). Best practices tell us that clearly communicating and making objectives measureable leads to greater student mastery.

Instructional Support and Resources District and web-based resources have been provided in the Instructional Resources column. Throughout the map you will find

4 Curriculum and Instruction – Mathematics Quarter 2 Finite Math instructional/performance tasks, i-Ready lessons and additional resources that align with the standards in that module. The additional resources provided are supplementary and should be used as needed for content support and differentiation.

Topics Addressed in Quarter  Random Variables, Averages, and Statistics  Logic for Finite Math  Systems of Linear Equations

Overview In quarter two students begin to bridge the study of probability and the application of probability theory to the field of statistics. Students solve problems and use the concepts of random variable, mean, and standard deviation. The normal random variable and the binomial random variable is also studied. Student study a few concepts involving logic including truth tables, statements and deduction. Students conclude the quarter by reviewing systems of linear equations and applying these concepts to analyze and solve real-world problems involving linear systems.

Fluency

The high school standards do not set explicit expectations for fluency, but fluency is important in high school mathematics. Fluency in algebra can help students get past the need to manage computational and algebraic manipulation details so that they can observe structure and patterns in problems. Such fluency can also allow for smooth progress toward readiness for further study/careers in science, technology, engineering, and mathematics (STEM) fields. These fluencies are highlighted to stress the need to provide sufficient supports and opportunities for practice to help students gain fluency. Fluency is not meant to come at the expense of conceptual understanding. Rather, it should be an outcome resulting from a progression of learning and thoughtful practice. It is important to provide the conceptual building blocks that develop understanding along with skill toward developing fluency.

The fluency recommendations for Algebra I listed below should be incorporated throughout your instruction over the course of the school year. Shelby County Schools 2016/2017 Revised 8/31/16 5 of 12 . A/G Solving characteristic problems involving the analytic geometry of lines . A-APR.A.1 Fluency in adding, subtracting, and multiplying polynomials . A-SSE.A.1b Fluency in transforming expressions and seeing parts of an expression as a single object

References:  http://www.corestandards.org/  http://www.nctm.org/  http://achievethecore.org/

TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Chapter 4 Random Variables, Averages, and Statistics (Allow approximately 3 weeks for instruction, review, and assessment) Conceptual Category: Data Enduring Understanding(s): Tennessee Finite Math Textbook Important Terms & Concepts Analysis, Statistics, and Probability (Chapter 4)  Interpretation of 4.1 Random Variables and Domain: Organize and Interpret data is dependent upon the Probability Density Functions Approximation method, binomial data graphical displays and random variable, density function, expected value, expected value of a D-ID.1 Organize data for problem numerical summaries. Additional Resources binomial random variable, mean, solving.  The question to be Khan Academy Video – Probability normal approximation to a binomial D-ID .2 Use a variety of counting answered determines the Density Functions random variable, normal random methods to organize information, data to be collected and Khan Academy Video – Random variable, random variable, standard determine probabilities, and solve how best to collect it. Variables deviation, standard normal curve, problems.  The normal distribution is a FiniteHelp Practice Problems standard normal random variable, fundamental component of variance, variance of a binomial FiniteHelp Notation Guide statistical inference. random variable  Density curves are used to Finite Help mimic probability. Finite Math Student Resources Writing in Math  The normal distribution is used Explain the difference between to model the spread of data. positively skewed, negatively  Probability models are useful skewed, and normally distributed tools for making decisions and sets of data and describe an example predictions. of each.  Probability is the basis of statistical inference. Find a real-world data set that appears to represent a normal 6 Curriculum and Instruction – Mathematics Quarter 2 Finite Math

TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES  The notion and behavior of a distribution and one that does not. random variable is Describe the characteristics of each foundational to understanding distribution. Create a visual probability distributions. representation of each set of data.  Probability models are useful tools for making decisions and Glencoe Reading & Writing in the predictions. Mathematics Classroom Graphic Organizers (9-12) Essential Question(s): Graphic Organizers (dgelman)  What is a density function?  How do density functions relate Literacy Skills and Strategies for to probability? Content Area Teachers  How are measures of central tendency relevant to density functions?  How can density functions be used to express relative standing?  What is a normal distribution?  How does one assess normality?  What does a normal distribution imply about the spread of data? Objectives: Students will:  Construct a probability distribution for random variables.

Shelby County Schools 2016/2017 Revised 8/31/16 7 of 12 TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Conceptual Category: Data Objectives: Tennessee Finite Math Textbook Analysis, Statistics, and Probability Students will: 4.2 Expected Values and Domain: Organize and Interpret  Determine the expected Standard Deviations of data value of a random variable. Random Variables D-ID .6 Calculate expected value, e.g.,  Determine the standard to determine the fair price of an deviation of a random Additional Resources investment. variable. Finite Math Student Resources  Use the expected value to FiniteHelp Practice Problems determine the average payoff TI Activity: It’s To Be Expected or loss in a game of chance.

Conceptual Category: Data Objectives: Tennessee Finite Math Textbook Analysis, Statistics, and Probability Students will: 4.3 Normal Random Variables and Domain: Organize and Interpret the Normal Approximation to the  Determine the probabilities for a data Binomial normally distributed variable D-ID.4 Calculate and interpret using a z-score. statistical problem using measures of Additional Resources central tendency and graphs.  Determine the binominal probabilities for a normally Wolfram: Normal Approximation to D-ID .6 Calculate expected value, e.g., distributed variable. a Binomial Random Variable to determine the fair price of an investment.

Chapter 11 Logic for Finite Math (Allow approximately 3 weeks for instruction, review, and assessment) Conceptual Category: Geometry Essential Question(s): Tennessee Finite Math Textbook Important Terms & Concepts and Measurement (Chapter 11)  Can you describe the method 11.1 Statements, Connectives, and Domain: Investigate logic used in mathematics to assign a Negation Biconditional, Conditional, G-L.1 Define the order of operations truth value to a compound Conjunction, Counterexample, Deduction, Disjunction, Equivalence, for the logical operators. statement? Additional Resources Implication, Logical connective G-L.2 Define conjunction, disjunction,  What is the importance of the Wolfram: Logic “and”, Logical connective “not”, negation, conditional, and concepts implication and proof? Logical connective ”or”, Negative, biconditional. Truth table, Valid argument

8 Curriculum and Instruction – Mathematics Quarter 2 Finite Math

TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES Objectives: Students will: Writing in Math  Define the order of operations for the logical operators. Glencoe Reading & Writing in the Mathematics Classroom  Define conjunction, disjunction, negation, conditional, and Graphic Organizers (9-12) biconditional. Graphic Organizers (dgelman)

Literacy Skills and Strategies for Content Area Teachers

Conceptual Category: Geometry Objectives: Tennessee Finite Math Textbook and Measurement Students will: 11.2 Truth Tables Domain: Investigate logic  Construct and use a truth table Additional Resources G-L.4 Construct and use a truth table to draw conclusions about a Truth Tables Video to draw conclusions about a statement. statement.  Analyze arguments with G-L. 7 Analyze arguments with quantifiers through the use of quantifiers through the use of Venn Venn diagrams diagrams. Conceptual Category: Geometry Objectives: Tennessee Finite Math Textbook Writing in Math and Measurement Students will: 11.3 Equivalence, Implication, and Explain the difference between Domain: Investigate logic  Construct and use a truth table Deduction inductive and deductive reasoning G-L.2 Construct and use a truth table to draw conclusions about a Additional Resources and give an example of each. to draw conclusions about a statement. Logic Puzzles Logic is used in electrical statement. engineering in designing circuits.  Solve a variety of Find another example of the use of G-L. 3 Solve a variety of logic puzzles. logic puzzles and apply the logic in the real world and write a Shelby County Schools 2016/2017 Revised 8/31/16 9 of 12 TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES G-L. 5 Apply the laws of logic laws of logic to judge the brief report about your findings. to judge the validity of validity of arguments. arguments.  Give counterexamples to G-L. 6 Give counterexamples to disprove statements. disprove statements. Chapter 5 Systems of Linear Equations (Allow approximately 3 weeks for instruction, review, and assessment) HSF.IF.C.7.A Enduring Understanding(s): Tennessee Finite Math Textbook Important Terms & Concepts (Chapter 5) Graph linear functions .  A mathematical model consists 5.1 Review of Equations and Graphs HSF.LE.A.1.B of both symbolic notations and of Lines Algorithm for solving a system of linear equations, augmented matrix, Recognize situations in which one relations among symbols. Additional Resources coefficient matrix, consistent system, quantity changes at a constant FiniteHelp Video- 5.1  Many situations in which coordinates in the plane and in the rate per unit interval relative to mathematical concepts and Finite Math Student Resources three-dimensional space, Echelon another. methods are applied in real world method, equation of a plane, HSF.LE.A.2 situations involve quantities function, function notation, Gaussian Construct linear functions given a related to each other through elimination, Gauss-Jordan graph, a description of a one or more equations. elimination, general equation of a relationship, or two input- output Essential Question(s): line, inconsistent system, intercept, pairs (include reading these from a line, linear extrapolation, linear  How can you model a simulation table). interpolation, linear model, reduced to represent a real life situation? row-echelon form, reduction method, row-echelon form, slope, slope- Objectives: intercept equation of a line, solution Students will learn: of a system of equations, Theorem on the solution of a system of linear  Create and graph linear functions in equations real-world problems.  Recognize situations in which one Writing in Math quantity changes at a constant rate Explain how the rate of change and per unit interval relative to another. slope are related and how to find the slope of a line.

HAS.REI.C.6 Objectives: Tennessee Finite Math Textbook Writing in Math Solve systems of linear Students will: 5.2 Formulation and Solution of Explain why you would solve an equations exactly and Systems of Linear Equations in Two equation like r = ax + b by solving 10 Curriculum and Instruction – Mathematics Quarter 2 Finite Math

TN STATE STANDARDS CONTENT INSTRUCTIONAL SUPPORT & RESOURCES approximately (e.g., with  Solve systems of linear equations Variables the system of equations y = r and y graphs), focusing on pairs of exactly and approximately (e.g., Additional Resources = ax + b. linear equations in two with graphs), focusing on pairs of FiniteHelp Video- 5.2 variables. linear equations in two variables.

Conceptual Category: Algebra  Identify and write the general Domain: Linear systems, matrices, solution to a system of linear and their applications equations; in the case of an infinite A-LM.5 Identify and write the general solution set, select various solution to a system of linear particular solutions given specific equations; in the case of an infinite properties. solution set, select various particular solutions given specific properties. Conceptual Category: Algebra Objectives: Tennessee Finite Math Textbook Writing in Math Domain: Linear systems, matrices, Students will: 5.3 Formulation and Solution of Describe the advantages and and their applications Systems of Linear Equations in disadvantages of using an  Identify and write the general Three or More Variables augmented matrix to solve a system 2. A-LM.5 Identify and write the solution to a system of linear of equations. general solution to a system of equations; in the case of an infinite Additional Resources linear equations; in the case of an solution set, select various FiniteHelp Video- 5.3 infinite solution set, select particular solutions given specific various particular solutions given properties. Finite Math Student Resources specific properties.

RESOURCE TOOLBOX Textbook Resources CCSS/PARCC Videos Tennessee Finite Math Common Core Standards - Mathematics Khan Academy by Dan Maki and Maynard Thompson Common Core Standards - Mathematics Illuminations (NCTM) Published by McGraw Hill 2011 Appendix A Discovery Education The Mathematics Common Core Toolbox The Futures Channel

Shelby County Schools 2016/2017 Revised 8/31/16 11 of 12 Finite Help State Academic Standards (Finite Math) The Teaching Channel TN Department of Education Math Standards Teachertube.com Edutoolbox (formerly TNCore) FiniteHelp Lecture Videos

Calculator Interactive Manipulatives Additional Sites Texas Instruments Education Rossmanchance.com NCTM Math Illuminations TI-Nspired Core Math Tools http://www.atomiclearning.com/ti_84 Math is Fun TICommonCore.com Wolfram Math World http://www.casioeducation.com/educators Nrich ST atistics E ducation W eb Online Algebra and Trigonometry Tutorial

Literacy ACT Tasks/Lessons Glencoe Reading & Writing in the Mathematics TN ACT Information & Resources UT Dana Center Classroom ACT College & Career Readiness Mathematics Mars Tasks Standards Inside Math Tasks Graphic Organizers (9-12) Math Vision Project Tasks Graphic Organizers (dgelman) Better Lesson Edutoolbox (formerly TNCore) Literacy Skills and Strategies for Content Area Teachers

12