Common Core. Mathematics Teacher Resource Book. Teacher Resource Book Table of Contents Pacing Guides Correlation Charts Sample Lessons

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1 204 Common Core Mathematics Teacher Resource Book 7 Teacher Resource Book Table of Contents Pacing Guides Correlation Charts Sample Lessons Sampler Includes For a complete Teacher Resource Book call

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3 Table of Contents Ready Common Core Program Overview Supporting the Implementation of the Common Core Answering the Demands of the Common Core with Ready The Standards for Mathematical Practice Depth of Knowledge Level Items in Ready Common Core Cognitive Rigor Matrix Using Ready Common Core Teaching with Ready Common Core Instruction Content Emphasis in the Common Core Standards Connecting with the Ready Teacher Toolbox Using i-ready Diagnostic with Ready Common Core Features of Ready Common Core Instruction Supporting Research Correlation Charts Common Core State Standards Coverage by Ready Instruction Interim Assessment Correlations Lesson Plans (with Answers) A6 A7 A8 A9 A0 A A2 A4 A6 A8 A20 A22 A8 A42 A46 CCSS Emphasis Unit : The Number System Lesson Understand Addition of Positive and Negative Integers M CCSS Focus - 7.NS.A.a, 7.NS.A.b Embedded SMPs Lesson 2 Understand Subtraction of Positive and Negative Integers M CCSS Focus - 7.NS.A.c Embedded SMPs - 4, 8 Lesson Add and Subtract Positive and Negative Integers 9 M CCSS Focus - 7.NS.A.d Embedded SMPs Lesson 4 Multiply and Divide Positive and Negative Integers 29 M CCSS Focus - 7.NS.A.2a, 7.NS.A.2b, 7.NS.A.2c Embedded SMPs - 2, 4, 7 Lesson 5 Terminating and Repeating Decimals 9 M CCSS Focus - 7.NS.A.2d Embedded SMPs -, 7 Lesson 6 Multiply and Divide Rational Numbers 49 M CCSS Focus - 7.NS.A.2a, 7.NS.A.2b, 7.NS.A.2c Embedded SMPs -, 2, 4, 7 M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards

4 Unit : The Number System (continued) CCSS Emphasis Lesson 7 Add and Subtract Rational Numbers 59 M CCSS Focus - 7.NS.A.a, 7.NS.A.b, 7.NS.A.c, 7.NS.A.d Embedded SMPs - 2, 4, 7 Lesson 8 Solve Problems with Rational Numbers 69 M CCSS Focus - 7.NS.A., 7.EE.B. Embedded SMPs - Unit Interim Assessment 79 Unit 2: Ratios and Proportional Relationships 82 Lesson 9 Ratios Involving Complex Fractions 84 M CCSS Focus - 7.RP.A. Embedded SMPs -, 6, 7 Lesson 0 Understand Proportional Relationships 94 M CCSS Focus - 7.RP.A.2a, 7.RP.A.2b Embedded SMPs -, 4 Lesson Equations for Proportional Relationships 02 M CCSS Focus - 7.RP.A.2c, 7.RP.A.2d Embedded SMPs -, 2, 4, 6, 8 Lesson 2 Problem Solving with Proportional Relationships 0 M CCSS Focus - 7.RP.A. Embedded SMPs - 4, 6 Lesson Proportional Relationships 20 M CCSS Focus - 7.RP.A. Embedded SMPs - 4, 6 Unit 2 Interim Assessment Unit : Expressions and Equations 4 Lesson 4 Equivalent Linear Expressions 7 M CCSS Focus - 7.EE.A. Embedded SMPs - 2, 6 8 Lesson 5 Writing Linear Expressions 47 M CCSS Focus - 7.EE.A.2 Embedded SMPs - 2, 4, 6 8 Lesson 6 Solve Problems with Equations 57 M CCSS Focus - 7.EE.B., 7.EE.B.4a Embedded SMPs - 7 Lesson 7 Solve Problems with Inequalities 67 M CCSS Focus - 7.EE.B., 7.EE.B.4b Embedded SMPs -, 2, 4, 6, 7 Unit Interim Assessment 77 Unit 4: Geometry 80 Lesson 8 Problem Solving with Angles 82 S/A CCSS Focus - 7.G.B.5 Embedded SMPs Lesson 9 Understand Conditions for Drawing Triangles 92 S/A CCSS Focus - 7.G.A.2 Embedded SMPs -, 2, 4 6 M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards

5 Unit 4: Geometry (continued) CCSS Emphasis Lesson 20 Area of Composed Figures 200 S/A CCSS Focus - 7.G.B.6 Embedded SMPs - 8 Lesson 2 Area and Circumference of a Circle 20 S/A CCSS Focus - 7.G.B.4 Embedded SMPs - 8 Lesson 22 Scale Drawings 222 S/A CCSS Focus - 7.G.A., 7.RP.A. Embedded SMPs - 8 Lesson 2 Volume of Solids 22 S/A CCSS Focus - 7.G.B.6 Embedded SMPs - 8 Lesson 24 Surface Area of Solids 242 S/A CCSS Focus - 7.G.B.6 Embedded SMPs - 8 Lesson 25 Understand Plane Sections of Prisms and Pyramids 252 S/A CCSS Focus - 7.G.A. Embedded SMPs - 2, 4, 5, 7 Unit 4 Interim Assessment 26 Unit 5: Statistics and Probability 264 Lesson 26 Understand Random Samples 267 S/A CCSS Focus - 7.SP.A. Embedded SMPs - 5 Lesson 27 Making Statistical Inferences 275 S/A CCSS Focus - 7.SP.A.2 Embedded SMPs -, 5 7 Lesson 28 Using Mean and Mean Absolute Deviation to Compare Data 285 S/A CCSS Focus - 7.SP.B. Embedded SMPs - 7 Lesson 29 Using Measures of Center and Variability to Compare Data 29 S/A CCSS Focus - 7.SP.B.4 Embedded SMPs - 7 Lesson 0 Understand Probability Concepts 0 S/A CCSS Focus - 7.SP.C.5 Embedded SMPs - 7 Lesson Experimental Probability 09 S/A CCSS Focus - 7.SP.C.6 Embedded SMPs - 5 Lesson 2 Probability Models 9 S/A CCSS Focus - 7.SP.C.7a, 7.SP.C.7b Embedded SMPs - 8 Lesson Probability of Compound Events S/A CCSS Focus - 7.SP.C.8a, 7.SP.C.8b, 7.SP.C.8c Embedded SMPs -, 2, 4, 5, 7, 8 Unit 5 Interim Assessment 4 M = Lessons that have a major emphasis in the Common Core Standards S/A = Lessons that have supporting/additional emphasis in the Common Core Standards

6 Answering the Demands of the Common Core with Ready THE DEMANDS OF THE COMMON CORE Focus: The Common Core Standards for Mathematics focus on fewer topics each year, allowing more time to truly learn a topic. Lessons need to go into more depth to help students to build better foundations and understanding. Coherent Connections (Building on Prior Knowledge): Instruction needs to provide logical ways for students to make connections between topics within a grade as well as across multiple grades. Instruction must build on prior knowledge and be organized to take advantage of the natural connections among standards within each cluster as well as connections across clusters or domains. This coherence is required for students to make sense of mathematics. Rigor and Higher-Order Thinking: To meet the Standards, equal attention must be given to conceptual understanding, procedural skill and fluency, and applications in each grade. Students need to use strategic thinking in order to answer questions of varying difficulty requiring different cognitive strategies and higher-order thinking skills. Conceptual Understanding: In the past, a major emphasis in mathematics was on procedural knowledge with less attention paid to understanding math concepts. The Common Core explicitly identifies standards that focus on conceptual understanding. Conceptual understanding allows students to see math as more than just a set of rules and isolated procedures and develop a deeper knowledge of mathematics. Mathematical Practices: The Standards for Mathematical Practice (SMP) must support content standards and be integrated into instruction. The content standards must be taught through intentional, appropriate use of the practice standards. Mathematical Reasoning: Mathematical reasoning must play a major role in student learning. Students must be able to analyze problems, determine effective strategies to use to solve them, and evaluate the reasonableness of their solutions. They must be able to explain their thinking, critique the reasoning of others, and generalize their results. HOW READY DELIVERS Ready lessons reflect the same focus as the Common Core standards. In fact, the majority of the lessons in each grade directly address the major focus of the year. Furthermore, each lesson was newly-written specifically to address the Common Core Standards. There is at least one lesson for each standard and only lessons that address the Common Core Standards are included. Ready units are organized by domains following the cluster headings of the Common Core. Each lesson starts by referencing prior knowledge and making connections to what students already know, particularly reinforcing algebraic thinking and problem-solving. These connections are highlighted for teachers in the Learning Progressions of the Teachers Resource Book so teachers can see at a glance how the lesson connects to previous and future learning. Ready lessons balance conceptual understanding, skill and procedural fluency, and applications. Students are asked higher-order thinking questions throughout the lessons. They are asked to understand, interpret, or explain concepts, applications, skills and strategies. Practice questions match the diversity and rigor of the Common Core standards. Ready includes conceptual understanding in every lesson through questions that ask students to explain models, strategies, and their mathematical thinking. In addition, a Focus on Math Concepts lesson is included for every Common Core standard that focuses on conceptual development those standards that begin with the word understand. The Standards for Mathematical Practice are fully integrated in an age-appropriate way throughout each lesson. The Teachers Resource Book includes SMP Tips that provide more in-depth information for select practice standards addressed in the lesson. See pages A9 and A27 for more details. Ready lessons build on problem-solving as a main component of instruction. Students work through a problem, discuss it, draw conclusions, make generalizations, and determine the reasonableness of their solutions. Guided Practice problems ask students to critique arguments presented by fictional characters and justify their own solutions. A8

7 The Standards for Mathematical Practice Mastery of the Standards for Mathematical Practice (SMP) is vital for educating students who can recognize and be proficient in the mathematics they will encounter in college and careers. As the chart below shows, the SMPs are built into the foundation of Ready Instruction.. Make sense of problems and persevere in solving them: Try more than one approach, think strategically, and succeed in solving problems that seem very difficult. Each Ready lesson leads students through new problems by using what they already know, demonstrates multiple approaches and access points, and gives encouraging tips and opportunities for cooperative dialogue. 2. Reason abstractly and quantitatively: Represent a word problem with an equation, or other symbols, solve the math, and then interpret the solution to answer the question posed. Ready lessons lead students to see mathematical relationships connecting equations, visual representations, and problem situations. Each lesson challenges students to analyze the connection between an abstract representation and pictorial or real-world situations.. Construct viable arguments and critique the reasoning of others: Discuss, communicate reasoning, create explanations, and critique the reasoning of others. In Ready, the teacher-led Mathematical Discourse feature guides students through collaborative reasoning and the exchange of ideas and mathematical arguments. Ready lessons also provide erroranalysis exercises that ask students to examine a fictional student s wrong answer, as well as multiple opportunities to explain and communicate reasoning. 4. Model with mathematics: Use math to solve actual problems. Students create a mathematical model using pictures, diagrams, tables, or equations to solve problems in each Ready lesson. In the Teacher Resource Book, the Real-World Connection feature adds another dimension to understanding application of a skill. 5. Use appropriate tools strategically: Make choices about which tools, if any, to use to solve a problem. Ready lessons model the use of a variety of tools, including diagrams, tables, or number lines; Guided Practice problems may be solved with a variety of strategies. 6. Attend to precision: Explain and argue, draw, label, and compute carefully and accurately. Ready lessons guide students to focus on precision in both procedures and communication, including special error-analysis tasks and group discussion questions that motivate students to employ precise, convincing arguments. 7. Look for and make use of structure: Build mathematical understanding by recognizing structures such as place value, decomposition of numbers, and the structure of fractions. Each Ready Focus on Math Concepts lesson builds understanding of new concepts by explicitly reviewing prior knowledge of mathematical structure. 8. Look for and express regularity in repeated reasoning: Recognize regularity in repeated reasoning and make generalizations or conjectures about other situations. Each Ready lesson leads students to focus attention on patterns that reflect regularity. Where appropriate, students draw a conclusion or make a generalization and explain their reasoning by referencing the observed pattern. A9

8 Depth of Knowledge Level Items in Ready Common Core The following table shows the Ready lessons and sections with higher-complexity items, as measured by Webb s Depth of Knowledge index. Depth of Knowledge Level Items in Ready Common Core Lesson Section Item Lesson Section Item Guided Practice 5 7 Common Core Practice 6 Guided Practice 6 Unit Interim Assessment PT Guided Practice 7 8 Guided Practice 8 Performance Task 8 8 Common Core Practice 4 2 Guided Practice 2 9 Guided Practice 2 Guided Practice 9 Guided Practice 2 2 Guided Practice 4 9 Performance Task 4 Guided Practice 9 20 Guided Practice 4 4 Guided Practice 2 20 Guided Practice 6 5 Guided Practice 8 2 Guided Practice 24 5 Common Core Practice 6 22 Guided Practice 6 6 Guided Practice 9 2 Guided Practice 8 7 Guided Practice Guided Practice 8 7 Common Core Practice 6 25 Guided Practice 2 8 Guided Practice 7 25 Guided Practice 8 Common Core Practice 6 25 Performance Task 4 Unit Interim Assessment PT Unit 4 Interim Assessment PT 9 Guided Practice Guided Practice 0 9 Common Core Practice 5 26 Guided Practice 0 Guided Practice 2 26 Guided Practice 2 0 Guided Practice 4 26 Performance Task 0 Performance Task 5 27 Guided Practice 5 Guided Practice 2 27 Common Core Practice 4 Common Core Practice 6 28 Guided Practice 0 2 Guided Practice Common Core Practice 4 Guided Practice 7 29 Guided Practice 9 Common Core Practice 6 29 Common Core Practice 4 Unit 2 Interim Assessment PT 0 Guided Practice 8 4 Guided Practice 8 0 Guided Practice 9 4 Common Core Practice 2 0 Performance Task 4 Common Core Practice 4 Guided Practice 6 4 Common Core Practice 5 Common Core Practice 4 5 Guided Practice 8 Common Core Practice 5 5 Common Core Practice 4 2 Guided Practice 24 5 Common Core Practice 5 2 Common Core Practice 6 Guided Practice 7 Guided Practice 2 6 Common Core Practice 4 Common Core Practice 5 6 Common Core Practice 5 Common Core Practice 6 7 Guided Practice 7 Unit 5 Interim Assessment PT A0

9 Cognitive Rigor Matrix The following table combines the hierarchies of learning from both Webb and Bloom. For each level of hierarchy, descriptions of student behaviors that would fulfill expectations at each of the four DOK levels are given. For example, when students compare solution methods, there isn t a lower-rigor (DOK or 2) way of truly assessing this skill. Depth of Thinking (Webb) Type of Thinking (Revised Bloom) DOK Level Recall & Reproduction DOK Level 2 Basic Skills & Concepts DOK Level Strategic Thinking & Reasoning DOK Level 4 Extended Thinking Remember Understand Apply Analyze Evaluate Create Recall conversations, terms, facts Evaluate an expression Locate points on a grid or number on number line Solve a one-step problem Represent math relationships in words, pictures, or symbols Follow simple procedures Calculate, measure, apply a rule (e.g.,rounding) Apply algorithm or formula Solve linear equations Make conversions Retrieve information from a table or graph to answer a question Identify a pattern/trend Brainstorm ideas, concepts, problems, or perspectives related to a topic or concept Specify, explain relationships Make basic inferences or logical predictions from data/observations Use models/diagrams to explain concepts Make and explain estimates Select a procedure and perform it Solve routine problem applying multiple concepts or decision points Retrieve information to solve a problem Translate between representations Categorize data, figures Organize, order data Select appropriate graph and organize and display data Interpret data from a simple graph Extend a pattern Generate conjectures or hypotheses based on observations or prior knowledge and experience Use concepts to solve non-routine problems Use supporting evidence to justify conjectures, generalize, or connect ideas Explain reasoning when more than one response is possible Explain phenomena in terms of concepts Design investigation for a specific purpose or research question Use reasoning, planning, and supporting evidence Translate between problem and symbolic notation when not a direct translation Compare information within or across data sets or texts Analyze and draw conclusions from data, citing evidence Generalize a pattern Interpret data from complex graph Cite evidence and develop a logical argument Compare/contrast solution methods Verify reasonableness Develop an alternative solution Synthesize information within one data set Relate mathematical concepts to other content areas, other domains Develop generalizations of the results obtained and the strategies used and apply them to new problem situations Initiate, design, and conduct a project that specifies a problem, identifies solution paths, solves the problem, and reports results Analyze multiple sources of evidence or data sets Apply understanding in a novel way, provide argument or justification for the new application Synthesize information across multiple sources or data sets Design a model to inform and solve a practical or abstract situation SBAC, 202; adapted from Hess et al., 2009 A

10 Using Ready Common Core Use Ready as Your Primary Instructional Program Because every Common Core Standard is addressed with clear, thoughtful instruction and practice, you can use Ready Common Core as your primary instructional program for a year-long mathematics course. The lesson sequence is based on the learning progressions of the Common Core to help students build upon earlier learning, develop conceptual understanding, use mathematical practices, and make connections among concepts. Instruct Teach one Ready Common Core Instruction lesson per week, using the Pacing Guides on pages A4 and A5 for planning. Use the web-based, electronic resources found in the Teacher Toolbox to review prerequisite skills and access on-level lessons as well as lessons from previous grades. See pages A8 and A9 for more information. Assess and Monitor Progress Assess student understanding using the Common Core Practice and Interim Assessments in Ready Common Core Instruction. See pages A29 and A46 for more information. Monitor progress using the benchmark tests in Ready Practice to assess cumulative understanding, identify student weaknesses for reteaching, and prepare for Common Core assessments. Differentiate Instruction Identify struggling students and differentiate instruction using the Assessment and Remediation pages at the end of each lesson in the Teacher Resource Book. See page A2 for a sample. Access activities and prerequisite lessons (including lessons from other grades) in the Teacher Toolbox to reteach and support students who are still struggling. See pages A8 and A9 for more details. Use Ready with the i-ready Diagnostic You can add the i-ready Diagnostic as part of your Ready solution. Administer the i-ready Diagnostic as a cross-grade-level assessment to pinpoint what students know and what they need to learn. Use the detailed individual and classroom diagnostic reports to address individual and classroom instructional needs using the lessons in Ready Common Core Instruction and the Teacher Toolbox. See pages A20 and A2 for more information. A2

11 Using Ready to Supplement Your Current Math Program If your instructional program was not written specifically to address the Common Core Standards, then your textbook likely does not include the concepts, skills, and strategies your students need to be successful. By supplementing with Ready Common Core Instruction, you ll be able to address these concerns: Filling gaps in mathematics content that has shifted from another grade Incorporating Common Core models and strategies into instruction Integrating the habits of mind that are in the Standards for Mathematical Practice Asking questions requiring students to engage in higher-level thinking, such as questions that ask students to explain effective strategies used to solve problems, critique the reasoning of others, and generalize their results Including lessons and questions that develop conceptual understanding Providing rigorous questions modeled on the latest Common Core assessment frameworks Step-by-Step Implementation Plan STEP IDENTIFY CONTENT NEEDS STEP 2 INTEGRATE READY STEP MEASURE STUDENT PROGRESS How do I know what to teach? Identify the Ready lessons you need to include in your instructional plan. First identify the Ready lessons that address standards that are a major emphasis in the Common Core. See page A6 or the Table of Contents to easily identify these Ready lessons. Next, identify the Common Core standards in the table on page A7 that are not addressed in your current math program. Identify the place in your scope and sequence to insert the Ready lessons. Focus on Math Concepts lessons should come before the lesson in your current book. How do I make time to teach the Ready lessons? Remove lessons or units from your current instructional plan that are no longer covered in the Common Core standards at that grade level. Replace lessons or units that do not teach topics using the models, strategies, and rigor of the Common Core with the appropriate Ready lessons. How can I address gaps in student knowledge? Use the Interim Assessments in Ready to make sure your students are successfully able to meet the rigorous demands of the Common Core. Use the benchmark tests in Ready Practice to identify student weaknesses and gaps in students knowledge. Use the Ready Teacher Toolbox to access activities, on-level lessons, and lessons from other grades to address gaps in students background and learning. See pages A8 and A9 for more on the Teacher Toolbox. A

12 Teaching with Ready Common Core Instruction Ready Instruction Year-Long Pacing Guide Week Ready Common Core Instruction Lesson Days Minutes/day Practice Test or i-ready Baseline Diagnostic 60 2 L: Understand Addition of Positive and Negative Integers L2: Understand Subtraction of Positive and Negative Integers L: Add and Subtract Positive and Negative Integers L4: Multiply and Divide Positive and Negative Integers L5: Terminating and Repeating Decimals L6: Multiply and Divide Rational Numbers L7: Add and Subtract Rational Numbers L8: Solve Problems with Rational Numbers Unit Interim Assessment L9: Ratios Involving Complex Fractions L0: Understand Proportional Relationships L: Equations for Proportional Relationships L2: Problem Solving with Proportional Relationships L: Proportional Relationships Unit 2 Interim Assessment L4: Equivalent Linear Expressions L5: Writing Linear Expressions L6: Solve Problems with Equations L7: Solve Problems with Inequalities Unit Interim Assessment Practice Test 2 or i-ready Interim Diagnostic L8: Problem Solving with Angles L9: Understand Conditions for Drawing Triangles L20: Area of Composed Figures L2: Area and Circumference of a Circle L22: Scale Drawings L2: Volume of Solids L24: Surface Area of Solids L25: Understand Plane Sections of Prisms and Pyramids Unit 4 Interim Assessment L26: Understand Random Samples L27: Making Statistical Inferences L28: Using Mean and Mean Absolute Deviation to Compare Data L29: Using Measures of Center and Variability to Compare Data L0: Understand Probability Concepts L: Experimental Probability L2: Probability Models L: Probability of Compound Events Unit 5 Interim Assessment Practice Test or i-ready Year-End Diagnostic 60 A4

13 Ready Instruction Weekly Pacing (One Lesson a Week) Use Ready Common Core Instruction as the foundation of a year-long mathematics program. The Year-Long Sample Week (below) shows a recommended schedule for teaching one lesson per week. Each day is divided into periods of direct instruction, independent work, and assessment. Use the Year-Long Pacing Guide on page A4 for a specific week-to-week schedule. Day Day 2 Day Day 4 Day 5 Introduction Modeled/Guided Instruction Modeled/Guided Instruction Guided Practice Common Core Practice Whole Class Introduction, including Vocabulary (0 minutes) Mathematical Discourse (0 min) Discuss graphic and verbal representations of a problem. Visual Support (5 minutes) Discuss graphic and verbal representations of a problem. Concept Extension (5 minutes) Discuss a sample problem. (0 minutes) Small Group/ Independent Hands-On Activity (where applicable) Work the math with a symbolic representation and practice with Try It problems. (20 minutes) Work the math with a symbolic representation and practice with Try It problems. (20 minutes) Work three problems independently, then Pair/Share. (20 minutes) Solve problems in test format or complete a Performance Task. (0 minutes) Assessment Discuss answer to the Reflect question. (5 minutes) Discuss solutions to the Try It problems. (0 minutes) Discuss solutions to the Try It problems. (0 minutes) Check solutions and facilitate Pair/ Share. (5 minutes) Review solutions and explanations. (5 minutes) Assessment and Remediation (time will vary) Ready Instruction Weekly Pacing (Two Lessons a Week) Target Ready Common Core Instruction lessons based on Ready Common Core Practice results to focus learning in a compressed time period. The chart below models teaching two lessons per week. The two lessons are identified as Lesson A and Lesson B in the chart below. In Class Lesson A Introduction (5 minutes) Modeled Instruction (0 minutes) Day Day 2 Day Day 4 Day 5 Lesson A Guided Instruction (5 minutes) Guided Practice (0 minutes) Lesson B Introduction (5 minutes) Modeled Instruction (0 minutes) Lesson B Guided Instruction (5 minutes) Guided Practice (0 minutes) Lesson A Review concepts and skills (20 minutes) Lesson B Review concepts and skills (20 minutes) Homework (optional) Lesson A Common Core Practice Lesson B Common Core Practice A5

14 Correlation Charts Common Core State Standards Coverage by Ready Instruction The table below correlates each Common Core State Standard to the Ready Common Core Instruction lesson(s) that offer(s) comprehensive instruction on that standard. Use this table to determine which lessons your students should complete based on their mastery of each standard. Common Core State Standards for Grade 7 Mathematics Standards Content Emphasis Ratios and Proportional Relationships Analyze proportional relationships and use them to solve real-world and mathematical problems. Ready Common Core Instruction Lesson(s) 7.RP.A. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 2 mile in each 4 hour, compute the unit rate as the complex fraction 2 miles per hour, equivalently 2 4 miles per hour. Major 9, 22 7.RP.A.2 Recognize and represent proportional relationships between quantities. Major 0, 7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.A.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t 5 pn. 7.RP.A.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (, r) where r is the unit rate. 7.RP.A. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. The Number System Apply and extend previous understandings of operations with fractions. 7.NS.A. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.A.a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 7.NS.A.b Understand p q as the number located a distance uqu from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. 7.NS.A.c Understand subtraction of rational numbers as adding the additive inverse, p 2 q 5 p (2q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Major 0 Major 0 Major Major Major 2, Major, 2,, 7 Major, 7 Major, 7 Major 2, 7 The Standards for Mathematical Practice are integrated throughout the instructional lessons. Common Core State Standards 200. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. A42

15 Common Core State Standards for Grade 7 Mathematics Standards The Number System (continued) Apply and extend previous understandings of operations with fractions. (continued) 7.NS.A.2 7.NS.A.d Apply properties ofoperations as strategies to add and subtract rational numbers. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (2)(2) 5 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then 2 p q 2 5 (2p) 5 p. Interpret quotients of q (2q) rational numbers by describing real-world contexts. 7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.A. Solve real-world and mathematical problems involving the four operations with rational numbers. Expressions and Equations Use properties of operations to generate equivalent expressions. 7.EE.A. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Content Emphasis Ready Common Core Instruction Lesson(s) Major, 7 Major 4, 5, 6 Major 4, 6 Major 4, 6 Major 4, 6 Major 5 Major 8 Major 4 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a 0.05a 5.05a means that increase by 5% is the same as multiply by.05. Major 5 Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.B. 7.EE.B.4 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 0% raise, she will make an additional of her salary an hour, or $2.50, for a 0 new salary of $ If you want to place a towel bar 9 inches long in the center of a door 4 that is 27 inches wide, you will need to place the bar about 9 inches from each edge; this 2 estimate can be used as a check on the exact computation. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 7.EE.B.4a Solve word problems leading to equations of the form px q 5 r and p(x q) 5 r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? 7.EE.B.4b Solve word problems leading to inequalities of the form px q. r or px q, r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $ per sale. This week you want your pay to be at least $00. Write an inequality for the number of sales you need to make, and describe the solutions. Major 8, 6, 7 Major 6, 7 Major 6 Major 7 A4

16 Geometry Common Core State Standards for Grade 7 Mathematics Standards Content Emphasis Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.A. 7.G.A.2 7.G.A. Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Supporting/ Additional Supporting/ Additional Supporting/ Additional Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve Supporting/ problems; give an informal derivation of the relationship between the circumference and area of a circle. Additional 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. 7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Statistics and Probability Use random sampling to draw inferences about a population. 7.SP.A. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. 7.SP.B. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 0 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. 7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. Investigate chance processes and develop, use, and evaluate probability models. 7.SP.C.5 Understand that the probability of a chance event is a number between 0 and that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 2 indicates an event that is neither unlikely nor likely, and a probability near indicates a likely event. 7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a or 6 would be rolled roughly 200 times, but probably not exactly 200 times. Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Ready Common Core Instruction Lesson(s) , 2, A44

17 Common Core State Standards for Grade 7 Mathematics Standards Statistics and Probability (continued) Investigate chance processes and develop, use, and evaluate probability models. (continued) 7.SP.C.7 7.SP.C.8 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. 7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. 7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 7.SP.C.8a 7.SP.C.8b 7.SP.C.8c Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the event. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Content Emphasis Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Supporting/ Additional Ready Common Core Instruction Lesson(s) A45

18 Interim Assessment Correlations The tables below show the depth-of-knowledge (DOK) level for the items in the Interim Assessments, as well as the standard(s) addressed, and the corresponding Ready Instruction lesson(s) being assessed by each item. Use this information to adjust lesson plans and focus remediation. Unit : The Number System Ready Common Core Interim Assessment Correlations Question DOK Standard(s) Ready Common Core Student Lesson(s) 2 7.NS.A.2a 4, NS.A.a, 7 7.NS.A.c 2, NS.A NS.A.d NS.A.a, 7.NS.A.b, 7.NS.A.c, 7.NS.A.d, 2,, NS.A.2d 5 PT 7.NS.A.b, 7.NS.A.c, 7.NS.A.2c, 7.NS.A., 2, 6, 8 Unit 2: Ratios and Proportional Relationships Question DOK Standard(s) Ready Common Core Student Lesson(s) 2 7.RP.A.2c 2 7.RP.A.2d 2 7.RP.A RP.A. 2, RP.A.2a, 7.RP.A.2b RP.A. 2, PT 7.RP.A. 2 Unit : Expressions and Equations Question DOK Standard(s) Ready Common Core Student Lesson(s) 2 7.EE.B. 6, EE.B.4b EE.B.4a EE.A EE.B. 6, EE.A. 4 PT 7.EE.B., 7.EE.B.4b 7 Depth of Knowledge levels:. The item requires superficial knowledge of the standard. 2. The item requires processing beyond recall and observation.. The item requires explanation, generalization, and connection to other ideas. A46

19 Unit 4: Geometry Ready Common Core Interim Assessment Correlations (continued) Question DOK Standard(s) Ready Common Core Student Lesson(s) 2 7.G.A G.A G.B G.B G.B.4, 7.G.B.6 20, G.B G.A.2 9 PT 7.G.B.6 20 Unit 5: Statistics and Probability Question DOK Standard(s) Ready Common Core Student Lesson(s) 2 7.SP.A SP.C.7b SP.C.8a, 7.SP.C.8b SP.B. 28 PT 7.SP.A., 7.SP.A.2 26, 27 A47

20 Develop Skills and Strategies Lesson 9 (Student Book pages 78 87) Ratios Involving Complex Fractions Lesson objectives Compute unit rates involving ratios with a fraction in the denominator. Compute unit rates involving ratios with a fraction in the numerator. Compute unit rates involving ratios with fractions in both the numerator and denominator. PReRequIsIte skills Compute unit rates involving ratios with whole numbers. Find equivalent fractions. Divide fractions. Write whole numbers as fractions. the LeaRnIng PRogRessIon Ratios (including rates, ratios, proportions, and percents) are commonplace in everyday life and critical for further study in math and science. In Grade 7, students extend the concepts of unit rate developed in Grade 6 to applications involving complex fractions. They transition from solving problems primarily with visual models to applying familiar algorithms. This lesson focuses on solving unit-rate problems that involve complex fractions. Students model real-world situations that involve ratios with fractions in the numerator and/or denominator. They learn to connect the process of simplifying complex fractions with the algorithm for the division of fractions. They learn how to interpret simplified ratios as unit rates to solve real-world problems. vocabulary unit rate: a rate in which the first quantity is compared to unit of the second quantity complex fraction: a fraction where either the numerator is a fraction, the denominator is a fraction, or both the numerator and the denominator are fractions Ready Lessons Teacher Toolbox Tools for Instruction Interactive Tutorials Prerequisite Skills Teacher-Toolbox.com 7.RP.A. CCss Focus 7.RP.A. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks mile in each 2 4 hour, compute the unit rate as the complex fraction 2 miles per 4 hour, equivalently 2 miles per hour. standards FoR MatheMatICaL PRaCtICe: SMP, 6, 7 (see page A9 for full text) 84 L9: Ratios Involving Complex Fractions

21 At A GlAnce Part : Introduction Students read a word problem and answer a series of questions designed to help them find a unit rate when one of the given quantities is a fraction. lesson 9 Part : Introduction Ratios Involving complex Fractions Lesson 9 Develop Skills and Strategies ccss 7.RP.A. SteP By SteP Tell students that this page models how to use a diagram to find a rate in miles per hour when the time is given as a number of minutes less than an hour. Have students read the problem at the top of the page. Work through Explore It as a class. Have students look at the diagram and explain how to figure out how many rectangles are needed to represent 5 minutes. Help students understand that in the diagram, 4 rectangles represent the ratio 4 miles : 5 minutes. Ask student pairs or groups to explain their answers for the remaining questions. SMP tip: Help students make sense of problems and persevere in solving them (SMP ) by asking them to explain what they are asked to find and to identify the needed information. Allow plenty of wait time. Visual Model Tell students that you will extend the diagram to show the number of miles per hour. Sketch the diagram on the board. Ask a volunteer to explain how many more rectangles you would need to draw to show 60 minutes instead of 45. [4] Add them to the diagram. Ask another volunteer to explain how to use the extended diagram to solve the problem. 78 In Grade 6, you learned about unit rates. take a look at this problem. Jana is training for a triathlon that includes a 2-mile bike ride. Today, she rode her bike 2 miles in 45 minutes. What is Jana s rate in miles per hour? explore It L9: Ratios Involving Complex Fractions 2 miles 45 minutes Use the math you already know to solve the problem. If Jana biked at a constant rate, how many miles did she bike in the first 5 minutes? 4 At the same rate, how many miles did she bike in the next 5 minutes? 4 At the same rate, how many miles did she bike in the last 5 minutes? 4 How many more minutes would Jana need to bike to total one hour? 5 At the same rate, how many miles would she bike in that amount of time? 4 Explain how you could find the number of miles Jana bikes in one hour. Possible answers: I could add to get 6; I could multiply 4 4 to get 6. Mathematical Discourse Why is it important that the first question says, If Jana biked at a constant rate? Listen for responses that indicate that a constant rate means the distance traveled is the same during each minute, so the problem can be solved with multiplication or division. The information is given in miles and minutes. Why might Jana want to know her rate in miles per hour instead of miles per minute? Listen for responses that note that she only rides a small part of a mile in one minute. L9: Ratios Involving Complex Fractions 85

22 At A GlAnce Part : Introduction Students revisit the problem on page 78 to learn how to model it using a ratio written as a complex fraction. Then students simplify the complex fraction by dividing. SteP By SteP Read Find Out More as a class. Review the meaning of unit rate. Have students look at the ratio 2. Ask, Why is it not a unit rate? [The number of hours must be one.] Have students describe how the ratio looks different from other fractions they have seen. Discuss the definition of a complex fraction. Ask students to give examples of complex fractions. Reinforce the idea that the fraction bar can mean division. Give other examples such as 5 and Work through the steps used to divide Have students assess the reasonableness of the answer. Note that hour is slightly more than 45 minutes and 6 miles is slightly more than 2 miles. ell Support Write the word per on the board. Next to it, write for each and in each. Give examples such as 5 crayons for each student means 5 crayons per student and driving 50 miles in each hour means 50 miles per hour. Give other examples and such as $.50 for each pound of peaches or cups of flour in each loaf of bread. Have students restate each using the word per. Then write unit rate on the board. Circle the word unit and write a above it. Say that in 50 miles per hour, the unit rate is 50 because it tells the number of miles in hour. The word per can mean in one or for one. Give more examples. Have students restate the ratio using the word per and then give the unit rate. 4 Part : Introduction Find Out More L9: Ratios Involving Complex Fractions Real-World connection Lesson 9 The number of miles Jana bikes in one hour is a unit rate. A unit rate compares two quantities where one of the quantities is. A unit rate tells you how many units of the first quantity correspond to one unit of the second quantity. The units in this problem are miles and hours. The problem tells us that Jana bikes 2 miles in 45 minutes. That s the same thing as 2 miles in hour. 4 number of miles number of hours The fraction is a complex fraction. A complex fraction is a fraction where either the 4 numerator is a fraction, the denominator is a fraction, or both the numerator and the denominator are fractions. You can simplify a complex fraction by dividing, just as you would do with whole numbers. The fraction bar represents division, so you can think of 6 miles as miles per hour. 2 hours 2 miles You can think about in the same way. 4 hour or 6 miles per hour The unit rate is 6. The number of miles Jana bikes is 6 times the number of hours. Reflect On another training ride, Jana bikes 5 miles in 50 minutes. Explain how you could find the number of miles she bikes in hour. Possible answers: Jana bikes miles every 0 minutes, so she would bike 5 8 miles in 60 minutes or hour; Write the ratio and then divide to get miles per hour. Encourage students to think of everyday situations in which measurements are given as fractions. Have volunteers share their ideas. Examples: Cooking 4 cup, 2 dozen 2 ; sewing yard, feet 2 ; traveling 2 2 blocks in 7 2 minutes 2 2 miles in 4 hour, lesson L9: Ratios Involving Complex Fractions