Common Core Investigations Teacher’s Guide
Common Core InvestigationsTeacher’s GuideBoston, MassachusettsUpper Saddle River, New Jersey
Copyright by Pearson Education, Inc., publishing as Pearson Prentice Hall, Boston, Massachusetts 02116. All rightsreserved. Printed in the United States of America. This publication is protected by copyright, and permission should beobtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in anyform or by any means, electronic, mechanical, photocopying, recording, or likewise. The publisher hereby grants permissionto reproduce these pages, in part or in whole, for classroom use only, the number not to exceed the number of students ineach class. Notice of copyright must appear on all copies. For information regarding permission(s), write to: Rights andPermissions Department, One Lake Street, Upper Saddle River, New Jersey 07458.Common Core State Standards: Copyright 2010. National Governors Association Center for Best Practices and Council ofChief State School Officers. All rights reserved.Connected Mathematics is a trademark of Michigan State University.Pearson Prentice Hall is a registered trademark of Pearson Education, Inc.Pearson is a registered trademark of Pearson plc.Prentice Hall is a registered trademark of Pearson Education, Inc.ExamView is a registered trademark of eInstruction Corporation.TeacherExpress is a trademark of Pearson Education, Inc.13-digit ISBN 978-0-13-318383-210-digit ISBN 0-13-318383-11 2 3 4 5 6 7 8 9 10 V036 15 14 13 12 11
Common Core InvestigationsTeacher’s GuideGrade 7Common Core Investigations Content SupportCMP and the Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivCorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPacing Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiSupport for the Common Core InvestigationsCC Investigation 1: Graphing Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1At-A-Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Additional Practice, Skill Practice, and Check-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7CC Investigation 2: Equivalent Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11At-A-Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Additional Practice, Skill Practice, and Check-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15CC Investigation 3: Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19At-A-Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Additional Practice, Skill Practice, and Check-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23CC Investigation 4: Geometry Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27At-A-Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Additional Practice, Skill Practice, and Check-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33CC Investigation 5: Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37At-A-Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Additional Practice, Skill Practice, and Check-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Answers for Additional Practices, Skill Practices, and Check-Ups . . . . . 47Table of Contentsiii
Connected Mathematics (CMP) is a field-tested and research-validated program that focuses on afew big ideas at each grade level. Students explore these ideas in depth, thereby developing deepunderstanding of key ideas that they carry from one grade to the next.The sequencing of topicswithin a grade and from grade to grade, the result of lengthy field-testing and validation, helps toensure the development of students’ deep mathematical understanding and strong problem-solvingskills. By the end of grade 8, CMP students will have studied all of the content and skills in theCommon Core State Standards for Mathematics* (CCSSM) for middle grades (Grades 6-8). Thefocus on helping students develop deep mathematical understanding and strong problem solvingskills aligns well to the intent of the Common Core State Standards for Mathematics, whicharticulates 3 to 5 areas of emphasis at each grade level from Kindergarten through Grade 8.The sequence of content and skills in CMP varies in some instances from that in the CCSSM, so incollaboration with the CMP2 authors, Pearson has created a set of investigations for each grade levelto further support and fully develop students’ understanding of the content standards of theCCSSM.The authors are confident that the CMP2 curriculum supplemented with the additionalinvestigations at each grade level will address all of the content and skills of the CCSSM, but evenmore, will contribute significantly to advancing students’ mathematical proficiency as described inthe Standards for Mathematical Practices of the CCSSM.Through the in-depth exploration ofconcepts, students become confident in solving a variety of problems with flexibility, skill, andinsightfulness, and are able to communicate their reasoning and understanding in a variety of ways.In this supplement, you will find support for all of the Common Core (CC) Investigations. The At-A-Glance page includes Teaching Notes and answers to all problems and exercises for theCC Investigation. The Additional Practice and Skill Practice pages can be reproduced for your students. These offeropportunities for students to reinforce the core concepts of the CC Investigation. Use the Check-Up to assess your students’ understanding of the concepts presented in theinvestigation. The answers for all of the ancillary pages are found at the back of this book. The reduced student pages are provided for your convenience as you read through the teachingsupport and plan for implementing each investigation.In the Pacing Guide (pp. xii–xiii), we propose placement for teaching each CC Investigations.CC Investigations 1–3 build on the concepts of linear relationships, so they should be taught afterMoving Straight Ahead. CC Investigation 4 involves geometric concepts, so it should be used after theunit Filling and Wrapping. CC Investigation 5 on Sampling couples well with Data DistributionsInvestigation 1. Copyright 2010. National Governors Association Center for Best Practices and Council of ChiefState School Officers. All rights reserved.ivCommon Core Teacher’s Guide
The following alignment of the Common Core State Standards for Mathematics to Pearson’sConnected Mathematics 2 (CMP2) 2009 program includes the supplemental investigations thatcomplete the CMP2 program.COMMON CORE STATE STANDARDS GRADE 7CMP2 UNITSCONTENTRatios and Proportional RelationshipsAnalyze proportional relationships and use them to solve real-world and mathematical problems.7.RP.1Compute unit rates associated with ratios of fractions, including ratios oflengths, areas and other quantities measured in like or different units.CC InvestigationsCC Inv. 1: Graphing Proportions7.RP.2Recognize and represent proportional relationships between quantities.Comparing and ScalingInv. 1: Making ComparisonsInv. 2: Comparing Ratios,Percents, and FractionsInv. 3: Comparing and ScalingRatesInv. 4: Making Sense ofProportions7.RP.2.aDecide whether two quantities are in a proportional relationship, e.g., bytesting for equivalent ratios in a table or graphing on a coordinate planeand observing whether the graph is a straight line through the origin.Comparing and ScalingInv. 4: Making Sense ofProportionsCC InvestigationsCC Inv. 1: Graphing Proportions7.RP.2.bIdentify the constant of proportionality (unit rate) in tables, graphs,equations, diagrams, and verbal descriptions of proportional relationships.Comparing and ScalingInv. 3: Comparing and ScalingRatesInv. 4: Making Sense ofProportionsMoving Straight AheadInv. 1: Walking RatesInv. 2: Exploring LinearFunctions With Graphs andTablesInv. 3: Solving EquationsInv. 4: Exploring SlopeCommon Core State Standards Correlationsv
COMMON CORE STATE STANDARDS GRADE 77.RP.2.cRepresent proportional relationships by equations.7.RP.2.dExplain what a point (x, y) on the graph of a proportional relationshipmeans in terms of the situation, with special attention to the points (0, 0)and (1, r) where r is the unit rate.7.RP.3Use proportional relationships to solve multistep ratio and percentproblems. Examples: simple interest, tax, markups and markdowns,gratuities and commissions, fees, percent increase and decrease,percent error.viCommon Core Teacher’s GuideCMP2 UNITSCONTENTVariables and PatternsInv. 1: Variables, Tables, andCoordinate GraphsInv. 2: Analyzing Graphs andTablesInv. 3: Rules and EquationsInv. 4: ACE 13Comparing and ScalingInv. 4: Making Sense ofProportionsStretching andShrinkingInv. 4: Similarity and RatiosInv. 5: Using Similar Trianglesand RectanglesMoving Straight AheadInv. 1: Walking RatesInv. 2: Exploring LinearFunctions With Graphs andTablesInv. 3: Solving EquationsInv. 4: Exploring SlopeVariables and PatternsInv. 2: Analyzing Graphs andTablesComparing and ScalingInv. 3: ACE 9, 10Moving Straight AheadInv. 1: Walking RatesInv. 2: Exploring LinearFunctions With Graphs andTablesInv. 3: Solving EquationsInv. 4: Exploring SlopeCC InvestigationsCC Inv. 1: Graphing ProportionsVariables and PatternsInv. 4: ACE 12Comparing and ScalingInv. 1: Making ComparisonsInv. 2: Comparing Ratios,Percents, and FractionsInv. 3: Comparing and ScalingRatesInv. 4: Making Sense ofProportionsStretching andShrinkingInv. 4: Similarity and RatiosInv. 5: Using Similar Trianglesand Rectangles
COMMON CORE STATE STANDARDS GRADE 7CMP2 UNITSCONTENTThe Number SystemApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.7.NS.1Apply and extend previous understandings of addition and subtraction toadd and subtract rational numbers; represent addition and subtraction ona horizontal or vertical number line diagram.Accentuate theNegativeInv. 2: Adding and SubtractingIntegersInv. 4: Properties of Operations7.NS.1.aDescrie situations in which opposite quantities combine to make 0.Accentuate theNegativeInv. 2: Adding and SubtractingIntegers7.NS.1.bAccentuate theUnderstand p q as the number located a distance q from p, in the positive Negativeor negative direction depending on whether q is positive or negative. Showthat a number and its opposite have a sum of 0 (are additive inverses).Interpret sums of rational numbers by describing real-world contexts.Inv. 1: Extending the NumberSystemInv. 2: Adding and SubtractingIntegers7.NS.1.cUnderstand subtraction of rational numbers as adding the additiveinverse, p – q p (–q). Show that the distance between two rationalnumbers on the number line is the absolute value of their difference, andapply this principle in real-world contexts.Accentuate theNegativeInv. 2: Adding and SubtractingIntegers7.NS.1.dApply properties of operations as strategies to add and subtract rationalnumbers.Accentuate theNegativeInv. 2: Adding and SubtractingIntegersInv. 4: Properties of Operations7.NS.2Fluently divide multi-digit numbers using the standard algorithm.Accentuate theNegativeInv. 3: Multiplying and DividingIntegersInv. 4: Properties of Operations7.NS.2.aUnderstand that multiplication is extended from fractions to rationalnumbers by requiring that operations continue to satisfy the properties ofoperations, particularly the distributive property, leading to products suchas (–1)(–1) 1 and the rules for multiplying signed numbers. Interpretproducts of rational numbers by describing real-world contexts.Accentuate theNegativeInv. 3: Multiplying and DividingIntegersInv. 4: Properties of Operations7.NS.2.bUnderstand that integers can be divided, provided that the divisor is notzero, and every quotient of integers (with non-zero divisor) is a rationalnumber. If p and q are integers, then –(p/q) (–p)/q p/(–q). Interpretquotients of rational numbers by describing real-world contexts.Accentuate theNegativeInv. 3: Multiplying and DividingIntegers7.NS.2.cApply properties of operations as strategies to multiply and dividerational numbers.Accentuate theNegativeInv. 3: Multiplying and DividingIntegersInv. 4: Properties of Operations7.NS.2 .dConvert a rational number to a decimal using long division; know that thedecimal form of a rational number terminates in 0s or eventually repeatsComparing and ScalingInv. 3: Comparing and ScalingRatesAccentuate theNegativeInv. 3: Multiplying and DividingIntegersAccentuate theNegativeInv. 2: Adding and SubtractingIntegersInv. 3: Multiplying and DividingIntegersInv. 4: Properties of Operations7.NS.3Solve real-world and mathematical problems involving the fouroperations with rational numbers.NOTE: Computations with rational numbers extend the rules formanipulating fractions to complex fractions.Common Core State Standards Correlationsvii
COMMON CORE STATE STANDARDS GRADE 7CMP2 UNITSCONTENTExpressions and EquationsUse properties of operations to generate equivalent expressions.7.EE.1Apply properties of operations as strategies to add, subtract, factor, andexpand linear expressions with rational coefficients.Moving Straight AheadInv. 3: Solving EquationsInv. 4: Exploring SlopeCC InvestigationsCC Inv. 2: EquivalentExpressions7.EE.2Understand that rewriting an expression in different forms in a problemcontext can shed light on the problem and how the quantities in it arerelated.CC InvestigationsCC Inv. 2: EquivalentExpressions7.EE.3Solve multi-step real-life and mathematical problems posed with positiveand negative rational numbers in any form (whole numbers, fractions,and decimals), using tools strategically. Apply properties of operations tocalculate with numbers in any form; convert between forms asappropriate; and assess the reasonableness of answers using mentalcomputation and estimation strategies.Variables and PatternsInv. 2: Analyzing Graphs andTablesInv. 3: Rules and EquationsInv. 4: Calculator Tables andGraphsAccentuate theNegativeInv. 1: Extending the NumberSystemInv. 2: Adding and SubtractingIntegersInv. 3: Multiplying and DividingIntegersInv. 4: Properties of OperationsMoving Straight AheadInv. 1: Walking RatesInv. 2: Exploring LinearFunctions With Graphs andTablesInv. 3: Solving EquationsInv. 4: Exploring SlopeVariables and PatternsInv. 1: Variables, Tables, andCoordinate GraphsInv. 2: Analyzing Graphs andTablesInv. 3: Rules and EquationsMoving Straight AheadInv. 1: Walking RatesInv. 2: Exploring LinearFunctions With Graphs andTablesInv. 3: Solving EquationsInv. 4: Exploring SlopeVariables and PatternsInv. 1: Variables, Tables, andCoordinate GraphsInv. 2: Analyzing Graphs andTablesInv. 3: Rules and EquationsMoving Straight AheadInv. 1: Walking RatesInv. 2: Exploring LinearFunctions With Graphs andTablesInv. 3: Solving EquationsInv. 4: Exploring SlopeMoving Straight AheadInv. 2: ACE 44CC InvestigationsCC Inv. 3: Inequalities7.EE.4Use variables to represent quantities in a real-world or mathematicalproblem, and construct simple equations and inequalities to solveproblems by reasoning about the quantities.7.EE.4.aSolve word problems leading to equations of the form px q r andp(x q) r, where p, q, and r are specific rational numbers. Solveequations of these forms fluently. Compare an algebraic solution to anarithmetic solution, identifying the sequence of the operations used ineach approach.7.EE.4.bSolve word problems leading to inequalities of the form px q r orpx q r, where p, q, and r are specific rational numbers. Graph thesolution set of the inequality and interpret it in the context of theproblem.viiiCommon Core Teacher’s Guide
COMMON CORE STATE STANDARDS GRADE 7CMP2 UNITSCONTENTGeometryDraw, construct, and describe geometrical figures and describe the relationships between them.7.G.1Solve problems involving scale drawings of geometric figures, includingcomputing actual lengths and areas from a scale drawing andreproducing a scale drawing at a different scale.7.G.2Draw (freehand, with ruler and protractor, and with technology)geometric shapes with given conditions. Focus on constructing trianglesfrom three measures of angles or sides, noticing when the conditionsdetermine a unique triangle, more than one triangle, or no triangle.7.G.3Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms andright rectangular pyramids.Stretching andShrinkingInv. 1: Enlarging and ReducingShapesInv. 2: Similar FiguresInv. 3: Similar PolygonsInv. 4: Similarity and RatiosInv. 5: Using Similar Trianglesand RectanglesComparing and ScalingInv. 4: Making Sense ofProportionsFilling and WrappingInv. 1: Building BoxesInv. 2: Designing RectangularBoxesInv. 3: Prisms and CylindersInv. 4: Cones, Spheres, andPyramidsCC InvestigationsCC Inv. 4: Geometry TopicsCC InvestigationsCC Inv. 4: Geometry TopicsSolve real-life and mathematical problems involving angle measure, area, surface area, and volume.7.G.4Know the formulas for the area and circumference of a circle and usethem to solve problems; give an informal derivation of the relationshipbetween the circumference and area of a circle.CC InvestigationsCC Inv. 4: Geometry Topics7.G.5Use facts about supplementary, complementary, vertical, and adjacentangles in a multi-step problem to write and solve simple equations for anunknown angle in a figure.Stretching andShrinkingInv. 3: ACE 22-24CC InvestigationsCC Inv. 4: Geometry Topics7.G.6Solve real-world and mathematical problems involving area, volume andsurface area of two- and three-dimensional objects composed of triangles,quadrilaterals, polygons, cubes, and right prisms.Stretching andShrinkingInv. 2: Similar FiguresInv. 3: Similar PolygonsFilling and WrappingInv. 1: Building BoxesInv. 2: Designing RectangularBoxesInv. 3: Prisms and CylindersInv. 4: Cones, Spheres, andPyramidsInv. 5: Scaling BoxesCommon Core State Standards Correlationsix
COMMON CORE STATE STANDARDS GRADE 7CMP2 UNITSCONTENTStatistics and ProbabilityUse random sampling to draw inferences about a population.7.SP.1Understand that statistics can be used to gain information about apopulation by examining a sample of the population; generalizationsabout a population from a sample are valid only if the sample isrepresentative of that population. Understand that random samplingtends to produce representative samples and support valid inferences.CC InvestigationsCC Inv. 5: Variability7.SP.2Use data from a random sample to draw inferences about a populationwith an unknown characteristic of interest. Generate multiple samples (orsimulated samples) of the same size to gauge the variation in estimates orpredictions.CC InvestigationsCC Inv. 5: Variability7.SP.3Informally assess the degree of visual overlap of two numerical datadistributions with similar variabilities, measuring the difference betweenthe centers by expressing it as a multiple of a measure of variability.Data DistributionsInv. 2: Making Sense ofMeasures of CenterCC InvestigationsCC Inv. 5: Variability7.SP.4Use measures of center and measures of variability for numerical datafrom random samples to draw informal comparative inferences about twopopulations.Data DistributionsInv. 3: Comparing Distributions:Equal Numbers of DataValuesInv. 4: Comparing Distributions:Unequal Numbers of DataValuesDraw informal comparative inferences about two populations.Investigate chance processes and develop, use, and evaluate probability models.7.SP.5Understand that the probability of a chance event is a number between0 and 1 that expresses the likelihood of the event occurring. Largernumbers indicate greater likelihood. A probability near 0 indicates anunlikely event, a probability around 1/2 indicates an event that is neitherunlikely nor likely, and a probability near 1 indicates a likely event.What Do You Expect?Inv. 1: ACE 147.SP.6Approximate the probability of a chance event by collecting data on thechance process that produces it and observing its long-run relativefrequency, and predict the approximate relative frequency given theprobability.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial Outcomes7.SP.7Develop a probability model and use it to find probabilities of events.Compare probabilities from a model to observed frequencies; if theagreement is not good, explain possible sources of the discrepancy.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial Outcomes7.SP.7.aDevelop a uniform probability model by assigning equal probability to alloutcomes, and use the model to determine probabilities of events.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial Outcomes7.SP.7.bDevelop a probability model (which may not be uniform) by observingfrequencies in data generated from a chance process.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial OutcomesxCommon Core Teacher’s Guide
COMMON CORE STATE STANDARDS GRADE 7CMP2 UNITSCONTENT7.SP.8Find probabilities of compound events using organized lists, tables, treediagrams, and simulation.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial Outcomes7.SP.8.aUnderstand that, just as with simple events, the probability of acompound event is the fraction of outcomes in the sample space forwhich the compound event occurs.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial Outcomes7.SP.8.bRepresent sample spaces for compound events using methods such asorganized lists, tables and tree diagrams. For an event described ineveryday language (e.g., “rolling double sixes”), identify the outcomes inthe sample space which compose the event.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial Outcomes7.SP.8.cDesign and use a simulation to generate frequencies for compoundevents.What Do You Expect?Inv. 1: Evaluating Games ofChanceInv. 2: Analyzing SituationsUsing an Area ModelInv. 3: Expected ValueInv. 4: Binomial OutcomesCommon Core State Standards Correlationsxi
This Pacing Guide offers suggestions as you look to implement the Grade 7 Common Core StateStandards for Mathematics in the CMP2 classroom. The Chart shows placement recommendationsfor the Common Core Investigations provided in this supplement.Investigations labeled as Review (R) offer timely practice of concepts from earlier grades, helping toactivate students’ prior knowledge as they are introduced to new concepts that build on theseconcepts. Investigations labeled as Extending (*) offer students the opportunity to explore conceptsin greater depth or to extend their study of concepts.The suggested number of standard days for each unit is based on a 45-minute class period; a blockperiod is assumed to be 90 minutes of instructional time. Common Core students entering Grade 7were introduced to ratios and rates, expressions and equations, integers, and volumes and nets ofsolids in Grade 6. Because of these prior understandings, students will begin the units Variables andPatterns, Stretching and Shrinking, Comparing and Scaling, and Filling and Wrapping with toolsneeded to complete these units at an accelerated pace. This will leave time in the year to cover theCC Investigations needed for Grade 7.Variables and PatternsStandard 8 days Block 4 daysInv. 1 Variables, Tables, and Coordinate GraphsReviews 6.NS.6, 6.EE.9RInv. 2 Analyzing Graphs and TablesReviews 6.NS.6, 6.EE.9RInv. 3 Rules and Equations7.EE.4, 7.EE.4.a Inv. 4 Calculator Tables and GraphsExtends 7.EE.4, 7.EE.4.a*Stretching and ShrinkingStandard 22 days Block 11 daysInv. 1 Enlarging and Reducing Shapes7.G.1, 7.G.4 (ACE 10, 13) Inv. 2 Similar Figures7.G.1, 7.G.6 Inv. 3 Similar Polygons7.G.1, 7.G.5 (ACE 22–24), 7.G.6 Inv. 4 Similarity and Ratios7.RP.2.c, 7.RP.3, 7.G.1 Inv. 5 Using Similar Triangles and Rectangles7.RP.2.c, 7.RP.3, 7.G.1 Comparing and ScalingStandard 20 days Block 10 daysInv. 1 Making Comparisons7.RP.2, 7.RP.3 Inv. 2 Comparing Ratios, Percents, and Fractions7.RP.2, 7.RP.3 Inv. 3 Comparing and Scaling Rates7.RP.2, 7.RP.2.b, 7.RP.2.c, 7.RP.2.d (ACE 9, 10), 7.RP.3 Inv. 4 Making Sense of Proportions7.RP.2, 7.RP.2.a, 7.RP.2.b, 7.RP.3, 7.G.1 KEY Core ContentR Review* ExtendingxiiCommon Core Teacher’s Guide
Accentuate the NegativeStandard 18 days Block 9 daysInv. 1 Extending the Number System7.NS.1.b, 7.EE.3 Inv. 2 Adding and Subtracting Integers7.NS.1, 7.NS.1.a, 7.NS.1.b, 7.NS.1.c, 7.NS.1.d, 7.NS.3,7.EE.3 Inv. 3 Multiplying and Dividing Integers7.NS.2, 7.NS.2.a, 7.NS.2.b, 7.NS.2.c, 7.NS.2.d, 7.NS.3,7.EE.3 Inv. 4 Properties of Operations7.NS.1, 7.NS.1.d, 7.NS.2, 7.NS.2.c, 7.NS.3, 7.EE.3 Moving Straight AheadStandard 30 days Block 15 daysInv. 1 Walking Rates7.EE.3, 7.EE.4, 7.EE.4.a Inv. 2 Exploring Linear Functions With Graphs andTables7.EE.3, 7.EE.4, 7.EE.4.a, 7.EE.4.b (ACE 44) Inv. 3 Solving Equations7.EE.1, 7.EE.3, 7.EE.4, 7.EE.4.a Inv. 4 Exploring Slope7.EE.1, 7.EE.3, 7.EE.4, 7.EE.4.a CC Inv. 1 Graphing Proportional Relationships7.RP.1, 7.RP.2.a, 7.RP.2.d CC Inv. 2 Equivalent Expressions7.EE.1, 7.EE.2 CC Inv. 3 Inequalities7.NS.3, 7.NS.4.b Filling and WrappingStandard 24 days Block 12 daysInv. 1 Building Boxes7.G.6 Inv. 2 Designing Rectangular Boxes7.G.6 Inv. 3 Prisms and Cylinders7.G.4, 7.G.6 Inv. 4 Cones, Spheres, and Pyramids7.G.6 Inv. 5 Scaling Boxes7.G.6 CC Inv. 4 Geometry Topics7.G.2, 7.G.3, 7.G.4, 7.G.5 What Do You Expect?Standard 22 ½ days Block 11 daysInv. 1 Evaluating Games of Chance7.SP.5 (ACE 14), 7.SP.6, 7.SP.7. 7.a, b, 7.SP.8, 7.SP.8.a,7.SP.8.b, 7.SP.8.c Inv. 2 Analyzing Situations Using an Area Model7.SP.6, 7.SP.7, 7.SP.7.a, 7.SP.7.b, 7.SP.8, 7.SP.8.a,7.SP.8.b, 7.SP.8.c Inv. 3 Expected Value7.SP.6, 7.SP.7, 7.SP.7.a, 7.SP.7.b, 7.SP.8, 7.SP.8.a,7.SP.8.b, 7.SP.8.c Inv. 4 Binomial Outcomes7.SP.6, 7.SP.7, 7.SP.7.a, 7.SP.7.b, 7.SP.8, 7.SP.8.a,7.SP.8.b, 7.SP.8.c Data DistributionStandard 23 days Block 12 daysInv. 1 Making Sense of VariabilityReviews 6.SP.1, 6.SP.2, 6.SP.3RCC Inv. 5 Sampling and Variability7.SP.1, 7.SP.2, 7.SP.3 Inv. 2 Making Sense of Measures of Center7.SP.3, 7.SP.4 Inv. 3 Comparing Distributions: Equal Numbers ofData Values7.SP.4 Inv. 4 Comparing Distributions: Unequal Numbersof Data Values7.SP.4 Common Core State Standards Pacing Guidexiii
Content Standards 7.RP.1, 7.RP.2.a, 7.RP.2.dAt a GlanceCC Investigation 1: Graphing ProportionsPACING 2 daysMathematical GoalsDOMAIN: Ratios and Proportional Relationships Compute unit rates associated with ratios of fractions, including quantitiesmeasured in like or different units. Decide whether two quantities are in a proportional relationship. Explain what any point (x, y), including (0, 0) and (1, r) where r is aunit rate, on a graph of a proportional relationship means in terms ofthe situation.Teaching NotesIn this investigation, students will extend their understanding of ratios anddevelop understanding of proportionality to solve problems. In Problem 1.1,students will explore and graph y kx relationships. Discuss with them theimportance of the points (0, 0) and (1, r), where r represents the unit rate,on the graph. Students will convert rates to unit rates in Problem 1.2.Continue your discussions of y kx proportional relationships inProblem 1.3. Assist students as necessary in completing graphs of the data.Vocabulary rateunit rateproportionalrelationshipProblem 1.1Before students begin Problem 1.1, review with them equivalent fractions,1 23and how to simplify fractions. Ask: How are the fractions , , and2 4632related? How do you know? (They have equal values; andare461equivalent to .)2During Problem 1.1, ask: Which variable will you place on each axis of your graph? (Theindependent variable should go on the horizontal axis.) Does your choice of which variable to place on which axis affect thegraph? (The ra
COMMON CORE STATE STANDARDS GRADE 7 CMP2 UNITS CONTENT Ratios and Proportional Relationships Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
school economics course. Focus the investigations on intriguing economics questions to spark student and teacher interest. The Investigations Twelve investigations were created by teams of economics curriculum materials developers and high school economics teachers. The titles of each investigation identify its content area followed
Independent Personal Pronouns Personal Pronouns in Hebrew Person, Gender, Number Singular Person, Gender, Number Plural 3ms (he, it) א ִוה 3mp (they) Sֵה ,הַָּ֫ ֵה 3fs (she, it) א O ה 3fp (they) Uֵה , הַָּ֫ ֵה 2ms (you) הָּ תַא2mp (you all) Sֶּ תַא 2fs (you) ְ תַא 2fp (you
Six-Core Core i7 16-Core SPARC T3 10-Core Xeon Westmere-EX 8-Core POWER7 Quad-core z196 Quad-core Itanium Tukwila 8-Core Xeon Nehalem-EX Six-Core Xeon 7400 Dual-Core Itanium 2 Itanium 2 with 9MB cache POWER6 Cell Itanium 2 Barton curve shows
common core Performance common core Coach Performance Coach Performance Coach 8 English Language Arts Mathematics English Language Arts 8 common core Performance Coach 8 Student Edition www.triumphlearning.com Phone: (800) 338-6519 Fax: (866) 805-5723 E-mail: customerservice@triumphlearning.com ISBN-13: 978-1-62362-846-8 9 781623 628468 .
common core Performance common core Coach Performance Coach Performance Coach 5 Mathematics 5 common core Performance Coach 5 Student Edition www.triumphlearning.com
The Bottom Line New York has completed the fourth year of a 12-year Common Core phase-in. The Class of 2022: Had just completed kindergarten when the Common Core was adopted by the Board in summer 2010; Was enrolled in third grade when student progress on the Common Core was first measured in spring 2013; Will generally take the Common Core Algebra I Regents Exam in
2. Pittsburgh Modified Conners Teacher Rating Scale 3. Parent/Teacher DBD Rating Scale 4. Child Behavior Check List- Teacher Report Form 5. Narrative Description of Child -- Teacher 6. Academic and Behavioral Target Form 7. Classroom Management Techniques Generally, the teacher rating scales should be completed by the teacher who spends the .
Wendy Lloyd, Teacher Emily Johnson, Teacher Lora McFarland, Teacher Jenna Miller, Psychologist Anne Nelson, Teacher Lacie North, Teacher Tricia Pearson, SLP Sally Rogers, Teacher Kristen Sessions, Teacher Emily Shaw, SLP Bailee VanZeben, Teacher Kristen Walters, SLP L
