Common Core Math Units Grade 6Unit 1: Ratios I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 12This unit introduces students to the concept of a ratio and engages them in using ratio and rate reasoning to solve problems. It builds upon students' work in earlier grades with wholenumber multiplication and division. The multiplicative reasoning required for working with proportional relationships cuts across most Grade 6 mathematical content domains and istherefore a crucial starting point for the course.Common Core State Standards for 6.RP.1.1 Identify a ratio. G.RP.A.3a Students plotMathematical Content 6.RP.1.2 Write a ratio in different forms pairs of values (fraction, colon, word form). representing ratios inRatios and Proportional Relationships - 6.RP 6.RP.1.3 unit 5.A. Understand ratio concepts and use ratio 6.RP.1.4 Compare two quantities using a ratio.reasoning to solve problems. 6.RP.1.5 Explain a ratio by drawing a picture. The content standards for this Comprehend a given ratio by unit require that students make 1. Understand the concept of a ratio and use 6.RP.1.6 matching the ratio to a given sense of real-life and ratio language to describe a ratio illustration. mathematical problems (MP.1) by relationship between two quantities. For Use things/people in my classroom to modeling relationships with ratios example, \"The ratio of wings to beaks in the develop a ratio. (MP.4) using a variety bird house at the zoo was 2:1, because for of tools strategically (e.g. every 2 wings there was 1 beak.” \"For every 6.RP.3.1 Create a real-world ratio problem. equivalent ratios, tape diagrams, vote candidate A received, candidate C double number line diagrams, or received nearly three votes.” equations) (MP.5). 3. Use ratio and rate reasoning to solve real- 6.RP.3.2 Demonstrate a real-world ratio world and mathematical problems, e.g., by 6.RP.3.3 problem (not with a worksheet). reasoning about tables of equivalent ratios, Solve a real-world ratio problem. tape diagrams, double number line diagrams, or equations.a. Make tables of equivalent ratios relating 6.RP.3a.1 Identify equivalent ratios. quantities with whole- number measurements, find missing values in 6.RP.3a.2 Create a table by using equivalent the tables, and plot the pairs of values on 6.RP.3a.3 fractions. the coordinate plane. Use tables to 6.RP.3a.4 Find a missing value from a table of compare ratios. ratios. Solve proportions using equivalent ratios (ex. by using simple proportions ½ = c/4).Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 1

d. Use ratio reasoning to convert 6.RP.3d.1 Common Core Math Units measurement units; manipulate and 6.RP.3d.2 Grade 6 transform units appropriately when multiplying or dividing quantities. 6.RP.3d.3 Recall English and Metric units of 6.RP.3d.4 measurement.Common Core State Standards for 6.RP.3d.5Mathematical Practice Convert units using multiplication and division in multiple ways1. Make sense of problems and persevere in (proportions, multiplication, division,solving them. and moving decimals).4. Model with mathematics. Convert measurement units using ratio reasoning within customary 5. Use appropriate tools strategically. units. Convert measurement units using ratio reasoning within metric units. Convert measurement units using ratio reasoning between customary and metric units. 6.RP.3d.6 Use T-Charts, Base 10 blocks, and other manipulatives to convert units of measurement.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 2

Common Core Math Units Grade 6Unit 2: Rates, including percent I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 14This unit extends students' understanding of ratio reasoning as they investigate the idea that every ratio can be associated with unit rates.2 Students represent rate relationships withtape diagrams, tables, graphs, and equations. Students also are introduced to percent of a quantity as a rate per 100. Rate reasoning is central to the study of algebra and is a criticalfoundation for understanding slope, which is formalized in grades 7 and 8.Common Core State Standards for 6.NS.3.1 Fluently add multi-digit decimal 6.NS.B.3 Students build towardMathematical Content 6.NS.3.2 numbers. fluency throughout the grade: 6.NS.3.3 Fluently subtract multi-digit decimal therefore, this standardThe Number System - 6.NS 6.NS.3.4 numbers. is reinforced in future units ofB. Compute fluently with multi-digit numbers and 6.NS.3.5 Fluently multiply multi-digit decimal study, particularly units 3 and 5.find common factors and multiples. numbers. Teachers need to reinforce the Fluently divide multi-digit decimal use of the standard 3. Fluently add, subtract, multiply, and divide numbers. algorithm for each multi-digit decimals using the standard Recall estimation strategies for adding, operation throughout the algorithm for each operation. subtracting, multiplying and dividing year. decimals 6.NS.3.6 Recall estimation strategies for adding, As students work with unit rates subtracting, multiplying and dividing and interpret percent as a rate per 6.NS.8.1 decimals. 100, and as they analyze the 6.NS.8.2 6.NS.8.3 relationships among the values, 6.NS.8.4C. Apply and extend previous understandings of they look for and make use ofnumbers to the system of rational numbers. Graph points in all four quadrants of the structure (MP.7). As students 8. Solve real-world and mathematical problems by graphing points in all four coordinate plane. become more sophisticated in quadrants of the coordinate plane. Include use of coordinates and absolute value to Solve real-world problems by graphing their application of ratio reasoning, find distances between points with the they learn when it is best to solve same first coordinate or the same second points in all four quadrants of a plane problems with ratios, their coordinate. (maps, shapes, pictures). associated unit rates, or percents Solve mathematical problems by graphing (MP.S). Solving problems using points in all four quadrants of a plane ratio reasoning and rates calls for (maps, shapes, pictures) careful attention to the referents Calculate distances between two points for a given situation (MP.2). with the same first or second coordinates using absolute values, given only coordinates.Ratios and Proportional Relationships - 6.RPAdapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 3

A. Understand ratio concepts and use ratio 6.RP.2.1 Common Core Math Unitsreasoning to solve problems. 6.RP.2.2 Grade 6 2. Understand the concept of a unit rate alb 6.RP.2.3 Explain that a ratio is a fraction which associated with a ratio a:b with bO0, and 6.RP.2.4 is also a division problem. use rate language in the context of a ratio 6.RP.2.5 Write a ratio using letters a:b, a/b, a relationship. For example, \"This recipe has a to b – where b is not equal to 0 ratio of 3 cups of flour to 4 cups of sugar, so according to the definition of ratio. there is 3/4 cup of flour for each cup of Define unit rate. sugar.\" \"We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.\" 1 Explain in my own words what a unit NOTE: 1Expectations for unit rates in this grade rate is. Calculate unit rate when given are limited to non-complex fractions. certain quantities.3. Use ratio and rate reasoning to solve real- 6.RP.3.1 Create a real-world ratio problem. world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, 6.RP.3.2 Demonstrate a real-world ratio tape diagrams, double number line 6.RP.3.3 problem (not with a worksheet). diagrams, or equations. Solve a real-world ratio problem. b. Solve unit rate problems including 6.RP.3b.1 Solve for rate using real-world those involving unit pricing and 6.RP.3b.2 situations. constant speed. For example, if it took 7 6.RP.3b.3 Create a tape diagram using a unit hours to mow 4 lawns, then at that rate, rate. how many lawns could be mowed in 35 Interpret a rate from a tape diagram hours? At what rate were lawns being mowed? 6.RP.3b.4 Use a t-chart to compare 6.RP.3b.5 proportional quantities. Write a proportion correctly. 6.RP.3b.6 Solve an equation using proportions.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 4

c. Find a percent of a quantity as a rate per 100 6.RP.3c.1 Common Core Math Units(e.g., 30% of a quantity means 30l100 times the Grade 6quantity); solve problems involving finding thewhole, given a part and the percent. Explain that percent is out of 100.Common Core State Standards for 6.RP.3c.2 Explain how ratio and percent areMathematical Practice 6.RP.3c.3 related. Convert a rate to a percent.2. Reason abstractly and computationally. 6.RP.3c.4 Illustrate what I know about ratios &5. Use appropriate tools strategically. 6.RP.3c.5 fractions to define a percent using 6.RP.3c.6 Base 10 blocks.7. Look for and make use of structure. 6.RP.3c.7 Calculate percent given the rate and base (Rate of base = part) Design a model using Base 10 blocks showing how a percent is a ratio of 100. Solve problems involving finding the whole, given a part and the percent.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 5

Common Core Math Units Grade 6Unit 3: Multi-digit computation and finding I Can Statements Notes/Comments Unit Materials and Resourcescommon factors and multiplesSuggested number of days: 11Students build on their work in earlier grades in working with factors and multiples (4.0A.B.4, 5.NF.B.4b) as they formalize the concepts and uses of greatest common factor (includingthe distributive property) and least common multiple. Although students have learned all four operations with multi-digit whole numbers and decimals in earlier grades, in this unit theyformalize the related standard algorithms as they build proficiency and fluency.Common Core State Standards for 6.NS.2.1 Apply the rules of division to solve Students continue building fluencyMathematical Content 6.NS.2.2 multi-digit problems. with 6.NS.B.3.The Number System - 6.NS 6.NS.2.3 Explain my understanding of place As students compute fluentlyB. Compute fluently with multi-digit numbers and 6.NS.2.4 value when dividing a multi-digit with multi-digit numbers andfind common factors and multiples. number by another multi-digit number. find common factors and multiples, they need to look for 2. Fluently divide multi-digit numbers using the Use base 10 blocks to illustrate the and make use of structure (MP.7). standard algorithm. algorithm. They need to reason abstractly and computationally (MP.2) as they Use strategies such as T-chart, make sense of quantities and estimation, etc… to solve division relationships in problem situations. problems.3. Fluently add, subtract, multiply, and divide 6.NS.3.1 Fluently add multi-digit decimal multi-digit decimals using the standard 6.NS.3.2 numbers. algorithm for each operation. 6.NS.3.3 6.NS.3.4 Fluently subtract multi-digit decimal 6.NS.3.5 numbers. Fluently multiply multi-digit decimal numbers. Fluently divide multi-digit decimal numbers. Recall estimation strategies for adding, subtracting, multiplying and dividing decimals 6.NS.3.6 Recall estimation strategies for adding, subtracting, multiplying and dividing decimals.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 6

4. Find the greatest common factor of two 6.NS.4.1 Common Core Math Units whole numbers less than or equal to 100 Grade 6 and the least common multiple of two 6.NS.4.2 whole numbers less than or equal to 12. Use Determine the greatest common factor the distributive property to express a sum of 6.NS.4.3 (GCF) of two numbers less than or equal two whole numbers 1-100 with a common 6.NS.4.4 to 100. factor as a multiple of a sum of two whole 6.NS.4.5 numbers with no common factor. For Determine the least common multiple example, express 36 + 8 as 4 (9 + 2). 6.NS.4.6 (LCM) of two whole numbers less than or equal to 12.Common Core State Standards forMathematical Practice Explain the difference between LCM and GCF.2. Reason abstractly and quantitatively.7. Look for and make use of structure. Justify the LCM and GCF of two whole numbers using math. Prove that two whole numbers have a common factor based on the distributive property. Identify the distributive property. 6.NS.4.7 Use the distributive property to solve addition problems.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 7

Common Core Math Units Grade 6Unit 4: Dividing fractions I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 11This unit builds on students' Grade 5 work with dividing fractions by whole numbers and whole numbers by fractions as students investigate dividing fractions by fractions. This unit setsthe stage for students' future work in finding unit rates involving ratios of fractions (7.RP.A.1).Common Core State Standards for 6.NS.1.1 List the steps to determine the quotient Extending previous understandingsMathematical Content 6.NS.1.2 of fractions (1. Keep 2. Change 3. Flip). of multiplication and division to 6.NS.1.3 divide fractions by fractionsThe Number System - 6.NS Determine the quotient of fractions. requires that students look for andA. Apply and extend previous understandings of 6.NS.1.4 express regularity in repeatedmultiplication and division to divide fractions by 6.NS.1.5 Recognize a fraction as a division reasoning (MP.8). Dividingfractions. 6.NS.1.6 problem; naming the divisor, dividend, fractions typically has been a 6.NS.1.7 and quotient. challenging concept for students; 1. Interpret and compute quotients of Solve word problems involving division therefore, it is beneficial to require fractions, and solve word problems involving 6.NS.1.8 of fractions using different methods. that students construct and critique division of fractions by fractions, e.g., by arguments (MP.3) as they interpret using visual fraction models and equations to Compare quotients of fractions using and compute quotients of fractions represent the problem. For example, create a models. and that they persevere in solve story context for (2/3) ÷ (3/4) and use a visual word problems involving division of fraction model to show the quotient; use the Compute quotients of fractions using fractions (MP.1). This can support relationship between multiplication and models. their conceptual understanding division to explain that (2/3) ÷ (3/4) = 8/9 and development of procedural because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ Evaluate how to solve division problems fluency. (c/d) = ad/bc.) How much chocolate will each with fractions in real life and person get if 3 people share 1/2 lb of mathematical situations. chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How Find the reciprocal of a number. wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? 6.NS.1.9 Create a word problem involving division of fractions.Common Core State Standards forMathematical Practice 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 8. Look for and express regularity in repeatedreasoning.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 8

Common Core Math Units Grade 6Unit 5: Representing relationships I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 15The standards addressed in this unit represent major work of Grade 6 and is therefore placed in the first half in the school year. In this unit of study, students extend their understandingof ratio and rate reasoning to represent proportional relationships in multiple ways, including relationships among measurement units such as cups and gallons or meters andkilometers. Students also represent common non-proportional relationships explored in earlier grades, such as area of a square and volume of a cube. In investigating and representingrelationships among variables, students continue to build fluency with operations on multi-digit whole numbers, multi-digit decimals, and fractions.Common Core State Standards for 6.EE.6.1 Write an expression to represent a real- 6.RP.A.3.a Students apply theirMathematical Content world or mathematical situation. ratio reasoning from earlier unitsExpressions and Equations - 6.EE 6.EE.6.2 Represent a number with a variable. of study as they investigate andB. Reason about and solve one-variable equations 6.EE.6.3 represent proportionaland inequalities. Justify that a variable can represent an relationships. unknown number or any number in a 6. Use variables to represent numbers and specified set. 6.NS.B.3 In investigating and write expressions when solving a real-world representing relationships or mathematical problem; understand that a among variables, students variable can represent an unknown number, continue to build fluency with or, depending on the purpose at hand, any operations on number in a specified set. multi-digit whole numbers, multi- digit decimals, and fractions.C. Represent and analyze quantitativerelationships between dependent andindependent variables.9. Use variables to represent two quantities in 6.EE.9.1 Use variables to represent two quantities Students look for and express a real-world problem that change in 6.EE.9.2 regularity in repeated reasoning relationship to one another; write an in a real world situation that change in equation to express one quantity, thought of as the dependent variable, in terms of the relationship to one another. (MP.8) as they generate algebraic other quantity, thought of as the Distinguish between dependent and models (MP.4) to represent independent variables. relationships.independent variable. Analyze the 6.EE.9.3 Write an equation to express onerelationship between the dependent and 6.EE.9.4 quantity (dependent variable) in terms ofindependent variables using graphs and the other quantity (independenttables, and relate these to the equation. For variable).example, in a problem involving motion at Relate the data between the dependentconstant speed, list and graph ordered pairsof distances and times, and write the and independent variables using graphsequation d = 65t to represent the relationship and tables and relate these to thebetween distance and time. equation.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 9

Ratios and Proportional Relationships - 6.RP 6.RP.3.1 Common Core Math UnitsA. Understand ratio concepts and use ratio Grade 6reasoning to solve problems. Create a real-world ratio problem. 3. Use ratio and rate reasoning to solve real- world and mathematical problems, e.g., by 6.RP.3.2 Demonstrate a real-world ratio reasoning about tables of equivalent ratios, 6.RP.3.3 problem (not with a worksheet). tape diagrams, double number line diagrams, or equations. Solve a real-world ratio problem. a. Make tables of equivalent ratios relating quantities with whole- number 6.RP.3a.1 Identify equivalent ratios. measurements, find missing values in the tables, and plot the pairs of values on the 6.RP.3a.2 Create a table by using equivalent coordinate plane. Use tables to compare 6.RP.3a.3 fractions. ratios. 6.RP.3a.4 Find a missing value from a table of ratios. Solve proportions using equivalent ratios (ex. by using simple proportions ½ = c/4). The Number System - 6.NS 6.NS.3.1 Fluently add multi-digit decimal B. Compute fluently with multi-digit numbers and 6.NS.3.2 numbers. find common factors and multiples. 6.NS.3.3 6.NS.3.4 Fluently subtract multi-digit decimal 3. Fluently add, subtract, multiply, and divide 6.NS.3.5 numbers. multi-digit decimals using the standard algorithm for each operation. 6.NS.3.6 Fluently multiply multi-digit decimal numbers. Common Core State Standards for Mathematical Practice Fluently divide multi-digit decimal numbers. 4. Model with mathematics. 8. Look for and express regularity in repeated Recall estimation strategies for adding,reasoning. subtracting, multiplying and dividing decimals Recall estimation strategies for adding, subtracting, multiplying and dividing decimals.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 10

Common Core Math Units Grade 6Unit 6: Extending the number system I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 13This unit builds on students' earlier study of systems of numbers (natural numbers, whole numbers, and positive fractions and decimals) as students investigate values less than zero.They formalize an understanding of integers and their relationship to the set of rational numbers. Students develop an understanding of how integers are used in real-world contexts,including the meaning of absolute value. In Grade 7, students learn to operate with positive and negative rational numbers.Common Core State Standards for 6.NS.5.1 Understand the meaning of 0. The content standards for this unitMathematical Content 6.NS.5.2 of study require that students' 6.NS.5.3 Identify positive and negative exploration of rational numbersThe Number System - 6.NS 6.NS.5.4 numbers using a number line. include how they model real lifeC. Apply and extend previous understandings of 6.NS.5.5 situations (MP.4). Students reasonnumbers to the system of rational numbers. Label a number line with positive and abstractly and quantitatively 6.NS.6a.1 negative numbers. (MP.2) in precise ways when they 5. Understand that positive and negative 6.NS.6a.2 write, interpret, and explain (MP.3, numbers are used together to describe Interpret the use of negative and MP.6) statements of order in real- quantities having opposite directions or positive numbers to represent real world contexts. Number lines (both values (e.g., temperature above/below world situations. vertical and horizontal), the zero, elevation above/below sea level, Use positive and negative numbers to coordinate plane, and the content- credits/debits, positive/negative electric represent quantities in the real world specific notation within this unit charge); use positive and negative (ex. temperature above zero and are all tools with which students numbers to represent quantities in real- below zero) need explicit practice so that, over world contexts, explaining the meaning of time, they are able strategically 0 in each situation. Locate a number and its opposite select and use them to solve using a coordinate plane. problems with positive and 6. Understand a rational number as a negative rational numbers (MP.5). point on the number line. Extend Recognize that 0 is its own opposite. number line diagrams and coordinate axes familiar from previous grades to 6.NS.6a.3 Locate a rational number on a represent points on the line and in the 6.NS.6a.4 number line (ex. 5 and its opposite - plane with negative number 5). coordinates. Demonstrate that the opposite of the opposite of a number is the number a. Recognize opposite signs of numbers as itself (ie. –(-4)=4). indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 11

7. Understand ordering and absolute value of 6.NS.7a.1 Common Core Math Unitsrational numbers. 6.NS.7a.2 Grade 6 6.NS.7a.3 a. Interpret statements of inequality as 6.NS.7a.4 Compare negative and positive statements about the relative position of 6.NS.7a.5 numbers. two numbers on a number line diagram. 6.NS.7b.1 For example, interpret -3 > -7 as a Understand ordering of rational statement that -3 is located to the right 6.NS.7b.2 numbers. of -7 on a number line oriented from left 6.NS.7c.1 to right. Order rational numbers on a number line. b. Write, interpret, and explain statements of order for rational Understand absolute value of numbers in real-world contexts. For rational numbers. example, write -3 °C > -7 °C to express the fact that -3 °C is warmer than -7 Use a number line to find the °C. absolute value of any number. c. Understand the absolute value of a rational Write an inequality to show the number as its distance from 0 on the number relationship between rational line; interpret absolute value as magnitude numbers in real world situations. for a positive or negative quantity in a real- world situation. For example, for an account Explain in my own words how to balance of -30 dollars, write /-30/ = 30 to compare rational numbers (written describe the size of the debt in dollars. or spoken) d. Distinguish comparisons of absolute value Define and identify absolute value. from statements about order. For example, recognize that an account balance less than - 6.NS.7c.2 Use a number line to find the 30 dollars represents a debt greater than 30 6.NS.7c.3 absolute value of any number. dollars. 6.NS.7c.4 Understand absolute value as theCommon Core State Standards for 6.NS.7c.5 distance from zero on the numberMathematical Practice line. Apply absolute value to real world2. Reason abstractly and quantitatively. situations.3. Construct viable arguments and critique the Recognize the symbol (ll) asreasoning of others. representing absolute value.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 12

Common Core Math Units Grade 6Unit 7: Relationships in the coordinate plane I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 12Students expand their understanding of rational numbers as they extend the coordinate plane to include all four quadrants. Students apply their understanding of absolute value todetermine distances in the coordinate plane.The Number System - 6.NS 6.NS.6b.1 Choose the quadrant for any ordered As students explore how the signsC. Apply and extend previous understandings of 6.NS.6b.2 pair. of the coordinates indicatenumbers to the system of rational numbers. 6.NS.6b.3 Predict the quadrant for an ordered locations in the coordinate plane, 6.NS.6b.4 pair. they apply the practice of looking 6. Understand a rational number as a Use the signs of numbers (negative, for and making use of structure point on the number line. Extend 6.NS.6b.5 positive) to predict the quadrant of a (MP.7). number line diagrams and coordinate reflection. axes familiar from previous grades to represent points on the line and in the Justify that when two ordered pairs plane with negative number differ only by signs, the locations of coordinates. the points are related by reflections b. Understand signs of numbers in ordered across one or both axes. pairs as indicating locations in quadrants of the coordinate plane; recognize that Use reflections to correctly plot when two ordered pairs differ only by ordered pairs and their opposites. signs, the locations of the points are related by reflections across one or both 6.NS.6c.1 Order rational numbers on a number axes. 6.NS.6c.2 line horizontally or vertically. c. Find and position integers and 6.NS.6c.3 Plot all integers and other rational other rational numbers on a numbers on number lines (vertically horizontal or vertical number line and horizontally) and coordinate diagram; find and position pairs of planes. integers and other rational Find the position of integer pairs and numbers on a coordinate plane. other rational numbers on a coordinate plane.8. Solve real-world and mathematical 6.NS.8.1 Graph points in all four quadrants of problems by graphing points in all four the coordinate plane. quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with theAdapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 13

same first coordinate or the same second 6.NS.8.2 Common Core Math Units coordinate. 6.NS.8.3 Grade 6 6.NS.8.4Geometry - 6.G Solve real-world problems by graphingA. Solve real-world and mathematical problems points in all four quadrants of a planeinvolving area, surface area, and volume. (maps, shapes, pictures). 3. Draw polygons in the coordinate plane Solve mathematical problems by given coordinates for the vertices; use graphing points in all four quadrants of coordinates to find the length of a side a plane (maps, shapes, pictures) joining points with the same first coordinate or the same second Calculate distances between two coordinate. Apply these techniques in the points with the same first or second context of solving real-world and coordinates using absolute values, mathematical problems. given only coordinates.Common Core State Standards forMathematical Practice 7. Look for and make use of structure.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 14

Common Core Math Units Grade 6Unit 8: Algebraic expressions I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 12In earlier grades, students have worked extensively with numerical expressions and have been using letters to represent unknowns in problem solving situations since Grade 3(3.0A.D.8, 4.0A.A.3, S.MD.C.5b). Students also have experience in prior grades with investigating and applying properties of operations in numerical contexts, such as the associative,distributive, and commutative properties. In this unit, students build on their work from earlier grades by working systematically with algebraic expressions, thereby laying thegroundwork for more formal work in writing and solving equations in later grades.Common Core State Standards for 6.EE.1.1 Write a numerical expression involving Teachers should use cautionMathematical Content whole numbers using exponents. when introducing mnemonicExpressions and Equations - 6.EE 6.EE.1.2 Write and solve expressions involving devices such as PEMDASA. Apply and extend previous understandings of 6.EE.1.3 whole-number exponents. (Parentheses, Exponents,arithmetic to algebraic expressions. Evaluate a numerical expression Multiplication, Division, Addition, involving whole numbers using Subtraction)4 for remembering 1. Write and evaluate numerical expressions exponents. Order of Operations (6.EE.A.2c), involving whole-number exponents. because they have the potential to cause students to focus on the2. Write, read, and evaluate expressions in 6.EE.2a.1 Write numbers and variables to device, rather than on thewhich letters stand for numbers. represent desired operations. underlying mathematical meaning of an expression. This can lead toa. Write expressions that record 6.EE.2a.2 Rewrite a numerical expression using serious misconceptions in future operations with numbers and with variables. study. letters standing for numbers. Forexample, express the calculation \"Subtract y from 5\" as 5 - y. 6.EE.2b.1 Identify parts of an expression using As students apply properties ofb. Identify parts of an expression using 6.EE.2b.2 mathematical terms (sum, term, operations to generate product, factor, quotient, coefficient). equivalent expressions or mathematical terms (sum, term, product, identify factor, quotient, coefficient); view one or Identify one or more parts of an when two expressions are more parts of an expression as a single expression as a single entity. equivalent, they must reason entity. For example, describe the abstractly and computationally expression 2 (8 + 7) as a product of two 6.EE.2c.1 Write and evaluate algebraic (MP.2) while looking for and factors; view (8 + 7) as both a single 6.EE.2c.2 expressions from real world situations. making use of structure (MP.7). As entity and a sum of two terms. Apply order of operations to solve students construct and critiquec. Evaluate expressions at specific values of 6.EE.2c.3 expressions and equations. (no arguments (MP.3) related totheir variables. Include expressions that arise parenthesis) whether or not a given set offrom formulas used in real-world problems. expressions is equivalent, theyPerform arithmetic operations, including Substitute specific value for variables deepen their conceptualthose involving whole-number exponents, in in order to evaluate the expression. understanding and fluency withthe conventional order when there are no this content. Working withparentheses to specify a particular order(Order of Operations). For example, use theAdapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 15

Common Core Math Units Grade 6 formulas V = s3 and A = 6 s2 to find the volume 6.EE.3.1 Identify the properties of operations. exponents and the coefficients of and surface area of a cube with sides of length 6.EE.3.2 variable are both tools for s=M expressing regularity in repeated 6.EE.4.1 reasoning (MP.8).3. Apply the properties of operations to 6.EE.4.2 generate equivalent expressions. For 6.EE.4.3 Apply the properties of operations to example, apply the distributive property to the expression 3 (2 + x) to produce the generate equivalent expressions. equivalent expression 6 + 3x; apply the distributive property to the expression 24x + Identify when two expressions are l8y to produce the equivalent expression 6 equivalent. (4x + 3y); apply properties of operations to y Identify the properties of operations. + y + y to produce the equivalent expression Apply the properties of operations to 3y. generate equivalent expressions.4. Identify when two expressions are equivalent(i.e., when the two expressions name the samenumber regardless of which value is substitutedinto them). For example, the expressions y + y + yand 3y are equivalent because they name thesame number regardless of which number ystands for.Common Core State Standards forMathematical Practice2. Reason abstractly and quantitatively.3. Construct viable arguments and critique thereasoning of others.7. Look for and make use of structure.8. Look for and express regularity in repeatedreasoning.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 16

Common Core Math Units Grade 6Unit 9: Understanding, writing, and solving I Can Statements Notes/Comments Unit Materials and Resourcesequations and inequalitiesSuggested number of days: 16In earlier grades, students have written numerical expressions and equations. In this unit, students extend their learning to solving one-variable equations and inequalities. This unitfurther develops students' understanding of the meaning of variables, including determining whether a variable represents a specific number or a set of numbers.Common Core State Standards for 6.EE.5.1 Understand solving an equation or As with the previous unit, whenMathematical Content inequality as a process of answering a students reason about and solve 6.EE.5.2 question. one-variable equations andExpressions and Equations - 6.EE 6.EE.5.3 inequalities, they reason abstractlyB. Reason about and solve one-variable equations Recognize that the correct answer to an and computationally (MP.2) whileand inequalities. equation or inequality makes it true. looking for and making use of structure (MP.7)-for example, 5. Understand solving an equation or inequality Use substitution to check the answer or recognizing that solving the as a process of answering a question: which determine if a given number is a equation 5 + x = 10 values from a specified set, if any, make the solution is the same thing as x = 10 - 5. equation or inequality true? Use substitution Solving real- to determine whether a given number in a 6.EE.6.1 Write an expression to represent a real- world problems by writing and specified set makes an equation or inequality 6.EE.6.2 world or mathematical situation. solving equations is new at Grade 6 true. 6.EE.6.3 Represent a number with a variable. and requires students' Justify that a variable can represent an perseverance and careful analysis 6. Use variables to represent numbers and 6.EE.7.1 unknown number or any number in a of the given information (MP.1). write expressions when solving a real-world 6.EE.7.2 specified set. or mathematical problem; understand that a variable can represent an unknown number, 6.EE.7.3 Solve a simple equation and explain or, depending on the purpose at hand, any how I solved it. number in a specified set. Write an equation from real-world situations and then use inverse 7. Solve real-world and mathematical problems operations to solve the equation. by writing and solving equations of the form x + p = q and px = q for cases in which p, q Write and solve equations of the form and x are all nonnegative rational numbers. x+p and px=q (in which p,q, and x are non-negative rational numbers).Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 17

8. Write an inequality of the form x > c or x 6.EE.8.1 Common Core Math Units < c to represent a constraint or condition Grade 6 in a real-world or mathematical problem. 6.EE.8.2 Recognize that inequalities of the form x 6.EE.8.3 Write an inequality to represent a > c or x < c have infinitely many solutions; constraint or condition in a real-world represent solutions of such inequalities or mathematical situation. on number line diagrams. Recognize and justify that inequalitiesCommon Core State Standards for have infinitely many solutions.Mathematical Practice Represent solutions of inequalities on a1. Make sense of problems and persevere in number line (graph).solving them.2. Reason abstractly and computationally.7. Look for and make use of structure.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 18

Common Core Math Units Grade 6Unit 10: Problem solving with area in 2-D I Can Statements Notes/Comments Unit Materials and ResourcesshapesSuggested number of days: 11This unit builds upon students' learning in earlier grades with finding areas of rectangles (4.MD.A.3, S.NF.B.4b) as they investigate triangles, special quadrilaterals, and polygons. It alsoprovides students with opportunities to develop their understanding and use of equations in geometric contexts.Common Core State Standards for 6.G.1.1 List formulas for solving the area of The standards in this unit requireMathematical Content 6.G.1.2 various polygons. that students persevere in solving 6.G.1.3 Apply the formulas and calculate area problems (MP.1) and model real-Geometry - 6.G 6.G.1.4 of various polygons. world scenarios with mathematicalA. Solve real-world and mathematical problems 6.G.1.5 Determine the area of a triangle using models, including equationsinvolving area, surface area, and volume. a formula. (MP.4), with a degree of precision 6.G.1.6 Determine the area of a rectangle appropriate for the given situation 1. Find the area of right triangles, other using a formula (MP.6). triangles, special quadrilaterals, and 6.G.1.7 Recognize how to compose and polygons by composing into rectangles or 6.G.1.8 decompose polygons into triangles decomposing into triangles and other 6.G.1.9 and rectangles. shapes; apply these techniques in the context of solving real-world and 6.G.1.10 Find the area of certain polygons by mathematical problems. composing into rectangles and decomposing into triangles and other shapes. Determine the area of a triangle by constructing diagrams. Determine the area of real world shapes. Discuss the area of triangles and parallelograms in mathematical problems. Justify formulas of the area of a triangle.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 19

Expressions and Equations - 6.EE 6.EE.2c.1 Common Core Math UnitsA. Apply and extend previous understandings of 6.EE.2c.2 Grade 6arithmetic to algebraic expressions. 6.EE.2c.3 Write and evaluate algebraic 2. Write, read, and evaluate expressions in expressions from real world situations. which letters stand for numbers. Apply order of operations to solve c. Evaluate expressions at specific values of expressions and equations. (no their variables. Include expressions that parenthesis) arise from formulas used in real-world problems. Perform arithmetic operations, Substitute specific value for variables including those involving whole-number in order to evaluate the expression. exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = M.Common Core State Standards forMathematical Practice1. Make sense of problems and persevere insolving them.4. Model with mathematics.6. Attend to precision.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 20

Common Core Math Units Grade 6Unit 11: Problem solving with volume and I Can Statements Notes/Comments Unit Materials and Resourcessurface areaSuggested number of days: 11Building on their knowledge of volume from earlier grades (S.MD.C.S), students learn how to find volume of rectangular prisms with fractional side lengths. Students also model 3-Dfigures with nets and use the nets to find surface area. Students apply their ability to evaluate expressions as they calculate area, surface area, and volume using appropriate formulas.Common Core State Standards for 6.G.2.1 Recall the formula for finding the As students investigate volume ofMathematical Content 6.G.2.2 volume of a rectangular prism. surface area of 3-D figures, they 6.G.2.3 Calculate the volume of a right select and use appropriate tools-Geometry - 6.G 6.G.2.4 rectangle prism. such as formulas, nets, and unitA. Solve real-world and mathematical problems 6.G.2.5 Tell what volume is in relation to its cubes- strategically (MP.5).involving area, surface area, and volume. 6.G.2.6 formula. Understanding how net models of Construct unit cubes to find volume. 3-D figures relate to their surface 2. Find the volume of a right rectangular prism 6.G.2.7 Justify that the “modeled volume” is areas requires that students look with fractional edge lengths by packing it equal to the “formula volume”. for and make use of structure with unit cubes of the appropriate unit 6.G.4.1 Apply volume formulas for right (MP.7). Applying formulas to find fraction edge lengths, and show that the 6.G.4.2 rectangular prisms to solve real-world volumes with fractional edge volume is the same as would be found by 6.G.4.3 problems. lengths requires that students multiplying the edge lengths of the prism. 6.G.4.4 Model the volume of a right attend to the degree of precision Apply the formulas V = l w h and V = b h to rectangular prism with fractional edge appropriate for a given situation find volumes of right rectangular prisms with 6.G.4.5 lengths by packing it with unit cubes of (MP.6). fractional edge lengths in the context of the appropriate unit fraction edge solving real-world and mathematical lengths. problems. Give examples of 3 dimensional shapes. 4. Represent three-dimensional figures using Recognize that 3 dimensional shapes nets made up of rectangles and triangles, and can be represented using nets. use the nets Solve problems to determine the surface area of figures. to find the surface area of these figures. Construct three-dimensional figures Apply these techniques in the context of using nets made up of rectangles and solving real-world and mathematical triangles. problems. Use nets to calculate surface area of figures in a given situation.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 21

6.G.4.6 Common Core Math Units 6.G.4.7 Grade 6 Use area of a 2 dimensional shape to determine the surface area of 3 dimensional shapes. Explain the surface area of objects in the class room.Expressions and Equations - 6.EE 6.EE.2c.1 Write and evaluate algebraicA. Apply and extend previous understandings of 6.EE.2c.2 expressions from real world situations.arithmetic to algebraic expressions. Apply order of operations to solve 6.EE.2c.3 expressions and equations. (no 2. Write, read, and evaluate expressions in parenthesis) which letters stand for numbers. Substitute specific value for variables c. Evaluate expressions at specific values of in order to evaluate the expression. their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = M. Common Core State Standards for Mathematical Practice 5. Use appropriate tools strategically. 6. Attend to precision.7. Look for and make use of structure.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 22

Common Core Math Units Grade 6Unit 12: Understanding data distributions I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 9This unit is the first of two Grade 6 units addressing data and statistics. While this content could be organized coherently in different ways, this suggested sequence begins with this unitfocused on developing understanding of statistical variability. In this unit, students investigate and describe the variability of given data, including center, spread, and overall shape. Theyalso discuss and explore how measures of center can be used to summarize a set of data, using data sets and representations of data that are provided to them. This unit lays afoundation for the following unit in which students use their understanding to create the own representations for a given set of data, and analyze data, including determining measuresof center and variability.Common Core State Standards for 6.SP.1.1 Recognize variability. The development of students'Mathematical Content 6.SP.1.2 Ask statistical questions. understanding of statistical 6.SP.1.3 Recognize a statistical question as one variability requires that theyStatistics and Probability - 6.SP that anticipates variability. persevere in making sense of givenA. Develop understanding of statistical data sets (MP.1). They alsovariability. investigate how measures of center and variability model and 1. Recognize a statistical question as one that summarize data (MP.4). anticipates variability in the data related to the question and accounts for it in theanswers. For example, \"How old am 17\" isnot a statistical question, but \"How old arethe students in my school7\" is a statisticalquestion because one anticipates variabilityin students' ages.2. Understand that a set of data collected to 6.SP.2.1 Design a survey. answer a statistical question has a 6.SP.2.2 Collect data from a survey. distribution that can be described by its 6.SP.2.3 Interpret data gathered from a survey. center, spread, and overall shape. 6.SP.2.4 Construct a histogram. 6.SP.2.5 Use a histogram to answer a statistical question showing the distribution of its 6.SP.2.6 center, spread, and overall shape. Describe a set of data by its center, spread, and overall shape.3. Recognize that a measure of center for a 6.SP.3.1 Use mean to describe data. numerical data set summarizes all of its 6.SP.3.2 Understand that there is variation in values with a single number, while a data. measure of variation describes how its 6.SP.3.3 Distinguish between “measure of values vary with a single number. center” and “measure of variation”.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 23

Common Core Math Units Grade 6 Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them.4. Model with mathematics.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 24

Common Core Math Units Grade 6Unit 13: Analyzing data I Can Statements Notes/Comments Unit Materials and ResourcesSuggested number of days: 13This unit builds on the understanding developed in the previous unit as students represent and analyze data sets and determine the measures of variability and center.Common Core State Standards for 6.SP.4.1 Illustrate data using a variety of graph Summarizing and describingMathematical Content types. (number line, dot plot, distributions requires that histogram, & box plot) students use appropriateStatistics and Probability - 6.SP statistical measuresB. Summarize and describe distributions. 6.SP.5.1 Summarize data sets in relation to strategically (MP.5) and 6.SP.5a.1 their content. attend to precision in finding and 4. Display numerical data in plots on a number Report the number of observations. applying those measures (MP.6). line, including dot plots, histograms, and box Relating the choice of measures of plots. 6.SP.5b.1 Describe the data collected including center and variability to the shape 6.SP.5c.1 how it was measured and its unit of of the data distribution and the 5. Summarize numerical data sets in relation to 6.SP.5c.2 measurement. context in which the data were their context, such as by: 6.SP.5c.3 Explain data in terms of measures of gathered requires that students center and measures of variability. construct and critique arguments a. Reporting the number of observations. (MP.3). Students look for and apply Calculate quantitative measures of regularity in repeated reasoning b. Describing the nature of the attribute variance (ie. range, interquartile when they explore how adding or under investigation, including how it range, mean absolute deviation). removing data points from a data was measured and its units of set effects the measures of center measurement. Identify outliers in data. and variability (MP.8). c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State SchoolOfficers. All rights reserved. The Charles A. Dana Center at The University of Texas at Austin January 13, 2013 Page 25

# Common Core Math Units Grade 6

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**Description: ** Common Core Math Units Grade 6 Adapted from NGA Center CCSS are the sole owners and developers of the Common Core State Standards. © Copyright 2010 National ...

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