Grade 6 2017 for Montgomery County, Maryland C2.0 Mathematics 6
envisionmath2.0 Grade 6, 2017 for Montgomery County C2.0 Mathematics 6 Introduction envisionmath2.0 Grades 6-8 provides a rigorous scope and sequence that addresses the latest information on the market, aligning to the next generation assessments and major content emphases requirements. This document provides a concise alignment of envisionmath2.0 Grade 6 to the concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines with the fundamental lesson(s) for each concept denoted in bold text. We also provide a correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics. While maintaining the research-based instructional model of the envisionmath2.0 elementary series, the Grades 6-8 program drives the effective use of new technology to personalize learning and improve outcomes. Dynamic, interactive content objects offer robust support of the instructional model as an authentic learning experience. This allows for the curriculum to be taught in a variety of classroom models. Problem-Based Learning, key to learners conceptual development, is the first step of every lesson in the student print component and as a digital experience at every grade. Visual learning, supported with engaging interactivities through Visual Learning Plus animations, formalizes mathematical learning. Point-of-use supports in MathXL for School online homework provide on-demand support and tutorials, enhancing independent practice. envisionmath2.0 offers a rich library of differentiation resources for every lesson that include STEM projects, enrichment activities, and a robust mathematical literacy thread, to name just a few. Combined with a great variety of system- and teacher-driven engaging experiences for all levels of learners through digital tools, games, and interactive workspaces, the program truly meets the needs of all students. Print components include a write-in Student s Edition, an Additional Practice Workbook, a 2-volume Teacher s Edition, and a variety of supporting components. envisionmath2.0 Grades 6-8 is powered by Pearson s Realize learning management system, providing teachers with the ability to customize content, auto-assign differentiation, and use assessment data quickly and easily. In addition, online assessments are in the format of the new high-stakes assessments. Copyright 2017 Pearson Education, Inc. or its affiliate(s). All rights reserved 2
envisionmath2.0 Grade 6, 2017 for Montgomery County C2.0 Mathematics 6 Table of Contents Montgomery County C2.0 Mathematics 6... 1 Common Core State Standards for Mathematics... 16 Copyright 2017 Pearson Education, Inc. or its affiliate(s). All rights reserved 3
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 1 Course Outline Ratios, Fractions, and Decimals envisionmath2.0 Topic 1: Understanding Ratios and Rates Students investigate the concepts of ratio and rate. They use multiple forms of ratio language and ratio notation and make connections between ratios and fractions. Students relate their understanding of ratios to explain and classify different types of ratios; including rate and unit rate. They use reasoning about multiplication and division to solve ratio and rate problems about quantities. Describe and represent a ratio relationship between two quantities. 5-1 Understand Ratios, 5-4 Represent and Graph Ratios Recognize ratios in a variety of contexts and translate ratios into words, and words into ratios. 5-1 Understand Ratios, 5-4 Represent and Graph Ratios Identify the similarities and differences between ratios and fractions. 5-1 Understand Ratios, 5-2 Generate Equivalent Ratios, 5-3 Compare Ratios Understand that a ratio is a multiplicative comparison of two quantities. 5-1 Understand Ratios Explain and classify different types of ratios, including rate and unit rate. 5-4 Represent and Graph Ratios, 5-5 Understand Rates and Unit Rates Interpret problems in order to describe the relationship between quantities and identify unit rate. 5-6 Compare Unit Rates, 5-7 Solve Unit Rate Problems Topic 2: Applications of Ratios and Rates Students apply reasoning when solving collections of ratio problems in real world contexts using various tools. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in a table, students connect their understanding of multiplication and division with ratios and rates. Thus, students expand the scope of problems for which they can use multiplication and division to solve problems. Students solve a wide variety of problems involving ratios and rates, discovering that a percent of a quantity is a rate per 100. Students express a fraction as a percent and find a percent of a quantity in real world concepts, supporting their reasoning with familiar representations they used previously. Although it is traditional to move students quickly to solving proportions by cross multiplying, it is imperative for students to reason multiplicatively with ratios to aid their understanding of future mathematics. Make and interpret tables of equivalent ratios. 5-2 Generate Equivalent Ratios, 5-4 Represent and Graph Ratios 1 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 1 Course Outline Ratios, Fractions, and Decimals Use multiplicative relationships to find missing values in tables of equivalent ratios. envisionmath2.0 5-2 Generate Equivalent Ratios, 5-4 Represent and Graph Ratios Display a series of equivalent ratios on a coordinate grid. 5-4 Represent and Graph Ratios Solve rate or ratio problems using multiple representations such as tape diagrams, double number lines, or equations. 5-4 Represent and Graph Ratios, 5-7 Solve Unit Rate Problems Solve unit rate problems including unit pricing and constant speed. 5-6 Compare Unit Rates, 5-7 Solve Unit Rate Problems Convert customary and metric units using ratio reasoning. 5-8 Ratio Reasoning: Convert Customary Units, 5-9 Ratio Reasoning: Convert Metric Units Describe a ratio as a fraction, decimal, and percent. 5-1 Understand Ratios; 6-2 Relate Fractions, Decimals, and Percents Compare fractions, decimals, and percents. 5-3 Compare Ratios; 6-2 Relate Fractions, Decimals, and Percents Solve problems by finding a percent of a quantity. 6-5 Find the Percent of a Number, Topic 6 3-Act Mathematical Modeling: Ace the Test Solve percent problems involving finding the part or the whole. 6-6 Find the Whole Given a Part and the Percent, Topic 6 3-Act Mathematical Modeling: Ace the Test 2 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 1 Course Outline Ratios, Fractions, and Decimals envisionmath2.0 Topic 3: Division of Fractions In this topic, students conclude the work with fraction operations from previous grade levels by computing quotients of fractions. Students develop a general understanding of fraction division. Students investigate two interpretations of division: measurement and sharing. They can use story contexts and visual models to develop an understanding of computing quotients of fractions, but also begin to move towards using the relationship between division and multiplication to determine the standard algorithm of dividing fractions. Divide fractions by whole numbers using a 1-4 Understand Division with Fractions model. Use models to divide whole numbers by fractions. 1-4 Understand Division with Fractions Use models to divide fractions by unit fractions. 1-4 Understand Division with Fractions, 1-5 Divide Fractions by Fractions Use models to divide fractions. 1-4 Understand Division with Fractions, 1-5 Divide Fractions by Fractions Interpret division of a fraction by fraction. 1-4 Understand Division with Fractions, 1-5 Divide Fractions by Fractions Reason about the inverse relationship between division and multiplication of fractions. 1-4 Understand Division with Fractions, 1-5 Divide Fractions by Fractions Apply the inverse relationship between division and multiplication of fractions. 1-4 Understand Division with Fractions, 1-5 Divide Fractions by Fractions Divide fractions by using the standard algorithm. 1-5 Divide Fractions by Fractions, 1-6 Divide Mixed Numbers Create and solve word problems involving division of a fraction by a fraction. 1-4 Understand Division with Fractions, 1-5 Divide Fractions by Fractions 3 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 1 Course Outline Ratios, Fractions, and Decimals envisionmath2.0 Topic 4: Multi-Digit Computation In Grade 6 students consolidate the work of earlier grades on operations with whole numbers and decimals by becoming fluent in the four operations on these numbers. Much of the foundation for this fluency has been laid in earlier grades. Students are expected to demonstrate fluency in two computation standards by the end of Grade 6. During the first marking period, students divide multidigit numbers using the standard algorithm. They also add and subtract multi-digit decimals using the standard algorithm. Students use estimation and place-value understandings to explain computation with the standard algorithms. The standard algorithms become parts of a repertoire of strategies used throughout the year. In subsequent marking periods, problems include the four operations. A spectrum of multi-step word problems of varying complexity levels are purposefully incorporated throughout the year as new learning is developed and applied. Solve division word problems using properties of multiplication and division. 1-2 Fluently Divide Whole Numbers and Decimals Apply place value to develop understanding of the standard algorithm for division. 1-2 Fluently Divide Whole Numbers and Decimals Solve word problems using the standard algorithm for division. 1-2 Fluently Divide Whole Numbers and Decimals Add or subtract decimals using models and relate strategies to the standard algorithm. 1-1 Fluently Add, Subtract, and Multiply Decimals Add and subtract decimals to solve word problems. 1-1 Fluently Add, Subtract, and Multiply Decimals, Topic 1 3-Act Mathematical Modeling: Stocking Up 4 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 2 Course Outline Number Relationships envisionmath2.0 Topic 1: Multi-Digit Computation (multiplying and dividing decimals) Students are expected to demonstrate fluency in two standards by the end of Grade 6. During the second marking period, students use their knowledge of equivalent fractions to fluently multiply and divide multi-digit decimals using the standard algorithm. A spectrum of multi-step word problems of varying complexity levels are purposefully incorporated throughout the year as new learning is developed and applied. Multiply decimals using fractions and the standard algorithm. 1-1 Fluently Add, Subtract, and Multiply Decimals; 1-3 Multiply Fractions Multiply decimals using the standard algorithm. 1-1 Fluently Add, Subtract, and Multiply Decimals Divide decimals by reasoning about place value and fractions. 1-2 Fluently Divide Whole Numbers and Decimals, 1-4 Understand Division with Fractions Use the standard algorithm to divide decimals. 1-2 Fluently Divide Whole Numbers and Decimals Fluently multiply and divide multi-digit decimals. 1-1 Fluently Add, Subtract, and Multiply Decimals; 1-2 Fluently Divide Whole Numbers and Decimals Topic 2: Extending the Number Line In Grade 6 the number line is extended to include negative numbers. In this topic, students are focused on developing an understanding of the meaning of negative numbers in real-world contexts. Students extend their previous understandings of number and ordering of numbers to the rational number system, which includes negative rational numbers. Understand that positive and negative integers 2-1 Understand Integers are used together to describe quantities having opposite directions or values. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line Understand that positive and negative rational numbers are used to describe quantities having opposite directions or values. 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 5 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 2 Course Outline Number Relationships Analyze and create real-world scenarios based on relative position on a number line. Use positive and negative rational numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. Recognize that the opposite of the opposite of the number is the number itself. Position a number, its opposite, and the opposite of the opposite of the number on a horizontal or vertical number line diagram. Find and position rational numbers on a horizontal or vertical number line diagram. envisionmath2.0 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 2-1 Understand Integers 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line Topic 3: Order and Absolute Value The number line serves as a model for students to relate integers and other rational numbers to statements of order in real-world contexts. Students describe the relationship between rational numbers in real-world scenarios with respect to numbers positions on the number line. Students use the concept of absolute value and its notation to show a number s distance from zero on the number line and recognize that opposite numbers have the same absolute value. Using real-world scenarios, students interpret absolute value. They apply their understanding to distinguish comparisons of absolute value from statements about order. Order rational numbers and interpret comparisons about the relative position of numbers on a number line. Interpret and explain statements of order for rational numbers. Write and explain statements of order for rational numbers in real world contexts. Describe absolute value of a rational number as the distance from 0 on a number line. 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 2-1 Understand Integers, 2-2 Represent Rational Numbers on the Number Line 2-3 Absolute Values of Rational Numbers 6 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 2 Course Outline Number Relationships Interpret absolute value as the magnitude of the number from 0 in a real-world situation. envisionmath2.0 2-3 Absolute Values of Rational Numbers Distinguish comparisons of absolute value from statements about order. 2-2 Represent Rational Numbers on the Number Line, 2-3 Absolute Values of Rational Numbers Use comparisons of absolute value and statements about order to solve real-world problems. 2-2 Represent Rational Numbers on the Number Line, 2-3 Absolute Values of Rational Numbers Topic 4: Coordinate Relationships Students will apply and extend previous understandings of numbers to the system of rational numbers. Students extend their understanding of the ordering of rational numbers on a number line to the coordinate plane. They construct the vertical and horizontal axes of a plane and determine the relationship between the four quadrants. Students also determine the relationship between the signs of numbers in an ordered pair in relation to the location of the point. They build upon their foundational understanding of plotting points in the first quadrant and transition to locating points in all four quadrants. Students apply the concept of absolute value to find the distance between points located on vertical or horizontal lines and solve real-world problems related to distance, segments, and shapes. Determine how a coordinate plane is designed using two number lines and identify parts of the coordinate plane. 2-4 Represent Rational Numbers on the Coordinate Plane Use ordered pairs to name points in a grid and locate points on a coordinate plane. 2-4 Represent Rational Numbers on the Coordinate Plane Determine the relationship between coordinates reflected over the x and y axes. 2-4 Represent Rational Numbers on the Coordinate Plane Create a coordinate plane using an appropriate scale. 2-4 Represent Rational Numbers on the Coordinate Plane Find horizontal and vertical distances on the coordinate plane using absolute value. 2-5 Find Distances on the Coordinate Plane Use the coordinate plane to graph geometric shapes and reason about absolute value to find the related distances. 2-6 Represent Polygons on the Coordinate Plane 7 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 2 Course Outline Number Relationships Solve real-world and mathematical problems using the coordinate plane. envisionmath2.0 2-4 Represent Rational Numbers on the Coordinate Plane, 2-5 Find Distances on the Coordinate Plane Topic 5: Multi-Digit Computation (divide whole numbers and decimal operations) Students are expected to demonstrate fluency in two standards by the end of Grade 6. During the second marking period, students use their knowledge of equivalent fractions to fluently multiply and divide multi-digit decimals using the standard algorithm. A spectrum of multi-step word problems of varying complexity levels are purposefully incorporated throughout the year as new learning is developed and applied. Fluently add, subtract, multiply, divide multi-digit decimals. 1-1 Fluently Add, Subtract, and Multiply Decimals; 1-2 Fluently Divide Whole Numbers and Decimals Topic 6: Exponents Students apply prior knowledge of place value and powers of ten and extend this knowledge to any positive rational number using repeated multiplication. Students develop the conceptual understanding that a whole number exponent is equivalent to repeated multiplication. In order to develop this conceptual understanding, students write and evaluate numerical expressions that are limited to whole number exponents. This understanding will be extended when students begin to solve algebraic expressions in marking period 3. Identify and model the relationship between 3-1 Understand and Represent Exponents exponents and repeated multiplication. Find the value of an expression using whole number exponents. 3-1 Understand and Represent Exponents, 3-3 Write and Evaluate Numerical Expressions Write and evaluate numerical expressions involving whole-number exponents. 3-1 Understand and Represent Exponents, 3-3 Write and Evaluate Numerical Expressions 8 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 3 Course Outline Expressions and Equations envisionmath2.0 Topic 1: Introducing Algebraic Expressions In this topic, students begin working systematically with algebraic expressions and use mathematical terms to clarify parts of an expression. They write, evaluate, and interpret expressions that correspond to given situations. Students use variables or letters to stand for unknowns to write expressions. When given the value of the variable, students use conventions about the order in which arithmetic operations are performed in the absence of parentheses (Order of Operations) to evaluate expressions. Students write expressions by abstracting patterns thus summarizing a calculation that can be carried out repeatedly with different numbers. They recognize how one part of the expression can impact the other parts and they view those parts as a single entity. Create, interpret and evaluate numerical expressions involving exponents using order of operations. Identify parts of numerical and algebraic expressions to interpret the meaning of the expression. Reason about mathematical terminology to create and interpret algebraic expressions. 3-3 Write and Evaluate Numerical Expressions 3-3 Write and Evaluate Numerical Expressions, 3-4 Write Algebraic Expressions 3-4 Write Algebraic Expressions Reason about situations to interpret, create, and evaluate algebraic expressions. Create and evaluate an expression for an unknown based on a context. 3-4 Write Algebraic Expressions, 3-5 Evaluate Algebraic Expressions 3-4 Write Algebraic Expressions, Topic 3 3-Act Mathematical Modeling: The Field Trip 9 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 3 Course Outline Expressions and Equations envisionmath2.0 Topic 2: Properties and Equivalency In this topic, students now consider common factors and multiples and find greatest common factors and least common multiples. Students have previously developed proficiency with multiplication and division and have been using properties of operations to write expressions in different ways. Students now apply their knowledge of common factors and multiples by using the distributive property. Students apply their previous understandings of equivalent numerical expressions to algebraic expressions. A firm grasp on variables as numbers helps students extend their work with properties of operations from arithmetic to algebra. Identify common factors and common multiples of a given pair of numbers. 3-2 Find Greatest Common Factor and Least Common Multiple Determine the greatest common factor and least common multiple of two whole numbers. 3-2 Find Greatest Common Factor and Least Common Multiple Identify the common factors from two whole numbers and rewrite the sum using the distributive property. 3-2 Find Greatest Common Factor and Least Common Multiple, 3-6 Generate Equivalent Expressions Determine the greatest common factor from two whole numbers and rewrite the sum using the distributive property. 3-2 Find Greatest Common Factor and Least Common Multiple, 3-6 Generate Equivalent Expressions Create multiple representations of equivalent algebraic expressions. 3-6 Generate Equivalent Expressions, 3-7 Simplify Algebraic Expressions Generate and identify equivalent numerical and algebraic expressions. 3-6 Generate Equivalent Expressions, 3-7 Simplify Algebraic Expressions 10 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 3 Course Outline Expressions and Equations envisionmath2.0 Topic 3: Equations and Inequalities In this topic, students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations and inequalities. Students understand the solution of an equation is the value of the variable that makes the equation true. It is important for students to state precisely the meaning of variables they use when setting up equations. Students solve real-world problems and represent inequalities on number line diagrams. Reason about the role of the variable by examining the structure of two given equivalent expressions. 3-6 Generate Equivalent Expressions, 3-7 Simplify Algebraic Expressions Reason about the solution by examining the structure of a given equation. 4-1 Understand Equations and Solutions Solve equations, precisely state the meaning of the variable, and represent the solution. 4-3 Write and Solve Addition and Subtraction Equations, 4-4 Write and Solve Multiplication and Division Equations Represent real-world problems by writing and solving equations. 4-4 Write and Solve Multiplication and Division Equations, 4-5 Write and Solve Equations with Rational Numbers Compare and interpret the solution(s) of equations and inequalities created from a realworld problem. 4-5 Write and Solve Equations with Rational Numbers, 4-6 Understand and Write Inequalities Model, graph, and interpret real world inequalities. 4-6 Understand and Write Inequalities, 4-7 Solve Inequalities 11 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 3 Course Outline Expressions and Equations envisionmath2.0 Topic 4: Relationships Between Variables In addition to constructing and solving equations in one variable, students will use equations in two variables to express relationships between two quantities that vary together. This prepares students for work with functions in later grades. As they work with two-variable equations, students begin to develop a dynamic understanding of variables and an appreciation that they can stand for any number from some domain. In this topic, students apply their previous knowledge to solve equations in realworld, contextual problems. Students will use prior knowledge from earlier marking period topics to construct tables of independent and dependent values in order to analyze equations with two variables from real-life contexts. They will represent equations by plotting values from the tables on a coordinate grid. Analyze the relationship between two variables to choose an independent variable and dependent variable. 4-8 Understand Dependent and Independent Variables, 4-9 Use Patterns to Write and Solve Equations Reason about equations in two variables and the solutions. 4-9 Use Patterns to Write and Solve Equations; 4-10 Relate Tables, Graphs, and Equations Given an equation, represent the relationship between the independent and dependent variables on a table and a graph. 4-10 Relate Tables, Graphs, and Equations Write an equation in two variables given a realworld context. 4-9 Use Patterns to Write and Solve Equations; 4-10 Relate Tables, Graphs, and Equations 12 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 4 Course Outline Geometric and Statistical Relationships envisionmath2.0 Topic 1: Relating Area and Volume Students develop the formula for the area of triangles. They apply their knowledge of the area of triangular and/or rectangular regions to determine the area of quadrilaterals, polygons, and composite figures in real-life contexts. Students build on their understanding of how to find the volume of a right rectangular prism to include those with fractional edge lengths. They explore how the volume of a shape can be found by composing or decomposing shapes including when the edge lengths are fractional. Students apply formulas and use their previous experience with solving equations to find the values of missing volumes or missing dimensions. Explore rectangles and the attributes of other twodimensional figures. 2-6 Represent Polygons on the Coordinate Plane, 7-4 Find Areas of Polygons Develop the area formulas for right triangles and parallelograms. 7-1 Find Areas of Parallelograms and Rhombuses, 7-2 Solve Triangle Area Problems Determine the area of any triangle. 7-2 Solve Triangle Area Problems Determine the area of special quadrilaterals. 7-1 Find Areas of Parallelograms and Rhombuses, 7-3 Find Areas of Trapezoids and Kites Determine the area of polygons by decomposing the region into triangles and rectangles. 7-4 Find Areas of Polygons Determine the area of composite figures using decomposition and composition of polygons. 7-4 Find Areas of Polygons Model volume. 7-8 Find Volumes of Prisms with Fractional Edge Lengths Examine the relationship between the area of the base and the height of a prism to derive a formula for volume. Apply the volume formula. 7-8 Find Volumes of Prisms with Fractional Edge Lengths 7-8 Find Volumes of Prisms with Fractional Edge Lengths 13 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 4 Course Outline Geometric and Statistical Relationships envisionmath2.0 Topic 2: Surface Area Students deconstruct solid figures into nets to identify the measurement of the solids face edges and area. They use nets to develop, then apply the formula for surface area of prisms. Construct three-dimensional figures using nets. 7-5 Represent Solid Figures Using Nets Construct nets of three-dimensional objects using measurements of a solid s edges. 7-5 Represent Solid Figures Using Nets Use nets to determine the surface area of threedimensional figures. Determine the surface area of rectangular and triangular prisms using nets. Determine the surface area of three-dimensional figures. Topic 7 3-Act Mathematical Modeling: That's a Wrap, 7-6 Find Surface Areas of Prisms, Find Surface Areas of Pyramids Topic 7 3-Act Mathematical Modeling: That's a Wrap, 7-6 Find Surface Areas of Prisms Topic 7 3-Act Mathematical Modeling: That's a Wrap, 7-6 Find Surface Areas of Prisms, Find Surface Areas of Pyramids Topic 3: Multi-Digit Computation Students are expected to demonstrate fluency with multi-digit numbers and decimals by the end of Grade 6. During the fourth marking period, students use the standard algorithm to fluently operate with multi-digit numbers and decimals. A spectrum of multi-step word problems of varying complexity levels are purposefully incorporated in this topic as new learning is developed and applied. Use the standard algorithm to fluently add, subtract, multiply, divide multi-digit decimals and divide whole numbers. 1-1 Fluently Add, Subtract, and Multiply Decimals; 1-2 Fluently Divide Whole Numbers and Decimals Topic 4: Statistical Questions In Grade 6, students progress from representing data to interpreting data. Students begin by differentiating between statistical and non-statistical types of questions. Statistical questions anticipate variability in the question which needs to be accounted for in the answer. Variability is explored in terms of the context, the statistical question being asked, measures of center and spread, and the display of the data. Explore data displays. 8-3 Display Data in Box Plots, 8-4 Display Data in Frequency Tables and Histograms Distinguish between statistical and non-statistical questions. 8-1 Recognize Statistical Questions 14 Bold Text = Fundamental Lesson
A Concise Alignment of envisionmath2.0 Grade 6 to the Concepts of the Montgomery County C2.0 Mathematics 6 Marking Period Course Outlines Montgomery County C2.0 Mathematics 6 Marking Period 4 Course Outline Geometric and Statistical Relationships Write examples of statistical questions. envisionmath2.0 8-1 Recognize Statistical Questions Topic 5: Analyzing Data Distributions Students recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Students display numerical data in plots on a number line, including dot plots, histograms, and box plots. They learn to describe and summarize numerical data sets by identifying clusters, peaks, gaps, symmetry, skew, and outliers considering the context in which the data were collected. Exploring variation with the mean absolute deviation (MAD) sets the stage for introducing the standard deviation in later grades. Reason about measures of center and spread. Determine and analyze measures of center and spread. 8-2 Summarize Data Using Mean, Median, and Mode; 8-5 Summarize Data Using Measures of Variability Investigate possible data sets when provided with measures of center and spread. 8-2 Summarize Data Using Mean, Median, and Mode; 8-5 Summarize Data Using Measures of Variability Explore the mean absolute deviation. 8-5 Summarize Data Using Measures of Variability, 8-7 Summarize and Compare Data Distributions Construct and interpret histograms to analyze a data set. 8-4 Display Data in Frequency Tables and Histograms Analyze variability by exploring box plots. 8-3 Display Data in Box Plots, 8-5 Summarize Data Using Measures of Variability Analyze box plots. 8-3 Display Data in Box Plots, 8-7 Summarize and Compare Data Distributions Identify the data display that best describes measures of center, spread, and the overall shape of the distribution. 8-3 Display Data in Box Plots, 8-4 Display Data in Frequency Tables and Histograms 15 Bold Text = Fundamental Lesson
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Mathematical Practices Make sense of problems and persevere in solving them. envisionmath2.0, 2017 Lesson 1-4: Understand Division with Fractions Lesson 1-5: Divide Fractions by Fractions Lesson 1-6: Divide Mixed Numbers Lesson 1-7: Solve Problems with Rational Numbers Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 3-4: Write Algebraic Expressions Lesson 3-5: Evaluate Algebraic Expressions Lesson 4-3: Write and Solve Addition and Subtraction Equations Lesson 4-4: Write and Solve Multiplication and Division Equations Lesson 4-5: Write and Solve Equations with Rational Numbers Lesson 5-7: Solve Unit Rate Problems Lesson 5-8: Ratio Reasoning: Convert Customary Units Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 6-5: Find the Percent of a Number Lesson 6-6: Find the Whole Given a Part and the Percent Lesson 7-1: Find Areas of Parallelograms and Rhombuses Lesson 7-2: Solve Triangle Area Problems Lesson 8-2: Summarize Data Using Mean, Median, Mode, and Range Lesson 8-5: Summarize Data Using Measures of Variability 16
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Reason abstractly and quantitatively. envisionmath2.0, 2017 Lesson 1-2: Fluently Divide Whole Numbers and Decimals Lesson 1-4: Understand Division with Fractions Lesson 1-5: Divide Fractions by Fractions Lesson 1-7: Solve Problems with Rational Numbers Lesson 2-1: Understand Integers Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 2-6: Represent Polygons on the Coordinate Plane Lesson 3-5: Evaluate Algebraic Expressions Lesson 4-1: Understand Equations and Solutions Lesson 4-2: Apply Properties of Equality Lesson 4-6: Understand and Write Inequalities Lesson 5-3: Compare Ratios Lesson 5-5: Understand Rates and Unit Rates Lesson 5-6: Compare Unit Rates Lesson 5-8: Ratio Reasoning: Convert Customary Units Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 5-10: Relate Customary and Metric Units Lesson 6-3: Represent Percents Greater Than 100 or Less Than 1 Lesson 6-4: Estimate to Find Percent Lesson 7-2: Solve Triangle Area Problems Lesson 7-3: Find Areas of Trapezoids and Kites Lesson 8-5: Summarize Data Using Measures of Variability 17
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Construct viable arguments and critique the reasoning of others. envisionmath2.0, 2017 Lesson 1-7: Solve Problems with Rational Numbers Lesson 2-1: Understand Integers Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 2-3: Absolute Values of Rational Numbers Lesson 2-6: Represent Polygons on the Coordinate Plane Lesson 3-1: Understand and Represent Exponents Lesson 3-4: Write Algebraic Expressions Lesson 3-6: Generate Equivalent Expressions Lesson 4-1: Understand Equations and Solutions Lesson 4-8: Understand Independent and Dependent Variables Lesson 5-6: Compare Unit Rates Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 5-10: Relate Customary and Metric Units Lesson 7-6: Find Surface Areas of Prisms Lesson 7-7: Find Surface Areas of Pyramids Lesson 8-5: Summarize Data Using Measures of Variability 18
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Model with mathematics. envisionmath2.0, 2017 Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 2-6: Represent Polygons on the Coordinate Plane Lesson 3-3: Write and Evaluate Numerical Expressions Lesson 3-4: Write Algebraic Expressions Lesson 3-6: Generate Equivalent Expressions Lesson 4-3: Write and Solve Addition and Subtraction Equations Lesson 4-4: Write and Solve Multiplication and Division Equations Lesson 4-5: Write and Solve Equations with Rational Numbers Lesson 5-8: Ratio Reasoning: Convert Customary Units Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 6-6: Find the Whole Given a Part and the Percent Lesson 7-1: Find Areas of Parallelograms and Rhombuses Lesson 8-3: Display Data in Box Plots Lesson 8-4: Display Data in Frequency Tables and Histograms 19
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Use appropriate tools strategically. envisionmath2.0, 2017 Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 2-4: Represent Rational Numbers on the Coordinate Plane Lesson 2-5: Find Distances on the Coordinate Plane Lesson 2-6: Represent Polygons on the Coordinate Plane Lesson 4-10: Relate Tables, Graphs, and Equations Lesson 5-2: Generate Equivalent Ratios Lesson 5-4: Represent and Graph Ratios Lesson 6-5: Find the Percent of a Number Lesson 7-1: Find Areas of Parallelograms and Rhombuses Lesson 7-3: Find Areas of Trapezoids and Kites Lesson 7-4: Find Area of Polygons Lesson 7-6: Find Surface Areas of Prisms Lesson 8-3: Display Data in Box Plots Lesson 8-4: Display Data in Frequency Tables and Histograms Lesson 8-7: Summarize Data Distributions 20
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Attend to precision. envisionmath2.0, 2017 Lesson 1-4: Understand Division with Fractions Lesson 1-5: Divide Fractions by Fractions Lesson 1-6: Divide Mixed Numbers Lesson 1-7: Solve Problems with Rational Numbers Lesson 3-5: Evaluate Algebraic Expressions Lesson 4-3: Write and Solve Addition and Subtraction Equations Lesson 4-4: Write and Solve Multiplication and Division Equations Lesson 4-5: Write and Solve Equations with Rational Numbers Lesson 5-7: Solve Unit Rate Problems Lesson 5-8: Ratio Reasoning: Convert Customary Units Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 6-5: Find the Percent of a Number Lesson 6-6: Find the Whole Given a Part and the Percent Lesson 7-1: Find Areas of Parallelograms and Rhombuses Lesson 7-2: Solve Triangle Area Problems Lesson 8-2: Summarize Data Using Mean, Median, Mode, and Range 21
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 Look for and make use of structure. Look for and express regularity in repeated reasoning. envisionmath2.0, 2017 Lesson 1-5: Divide Fractions by Fractions Lesson 1-6: Divide Mixed Numbers Lesson 2-3: Absolute Values of Rational Numbers Lesson 2-5: Find Distances on the Coordinate Plane Lesson 3-1: Understand and Represent Exponents Lesson 3-4: Write Algebraic Expressions Lesson 3-5: Evaluate Algebraic Expressions Lesson 3-6: Generate Equivalent Expressions Lesson 3-7: Simplify Algebraic Expressions Lesson 4-10: Relate Tables, Graphs, and Equations Lesson 5-2: Generate Equivalent Ratios Lesson 5-6: Compare Unit Rates Lesson 6-2: Relate Fractions, Decimals, and Percents Lesson 6-3: Represent Percents Greater Than 100 or Less Than Lesson 7-8: Find Volume with Fractional Edge Lengths Lesson 8-2: Summarize Data Using Mean, Median, Mode, and Range Lesson 1-2: Fluently Divide Whole Numbers and Decimals Lesson 1-6: Divide Mixed Numbers Lesson 3-1: Understand and Represent Exponents Lesson 3-2: Find Greatest Common Factor and Least Common Multiple Lesson 3-6: Generate Equivalent Expressions Lesson 5-1: Understand Ratios Lesson 5-2: Generate Equivalent Ratios Lesson 5-3: Compare Ratios Lesson 5-7: Solve Unit Rate Problems Lesson 7-6: Find Surface Areas of Prisms Lesson 8-2: Summarize Data Using Mean, Median, Mode, and Range Lesson 8-5: Summarize Data Using Measures of Variability 22
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 envisionmath2.0, 2017 Ratios and Proportional Relationships 6.RP Understand ratio concepts and use ratio reasoning to solve problems. 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Lesson 5-1: Understand Ratios Topic 5 Review: Understand and Use Ratio and Rate 2. Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. Lesson 5-5: Understand Rates and Unit Rates Topic 5 Review: Understand and Use Ratio and Rate 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Lesson 5-2: Generate Equivalent Ratios Lesson 5-3: Compare Ratios Lesson 5-4: Represent and Graph Ratios Lesson 5-5: Understand Rates and Unit Rates Lesson 5-6: Compare Unit Rates Lesson 5-7: Solve Unit Rate Problems Lesson 5-8: Ratio Reasoning: Convert Customary Units Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 5-10: Relate Customary and Metric Units Topic 5 Review: Understand and Use Ratio and Rate a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Lesson 5-2: Generate Equivalent Ratios Lesson 5-3: Compare Ratios Lesson 5-4: Represent and Graph Ratios Lesson 5-5: Understand Rates and Unit Rates Lesson 5-6: Compare Unit Rates Topic 5 Review: Understand and Use Ratio and Rate b. Solve unit rate problems including those involving unit pricing and constant speed. Lesson 5-5: Understand Rates and Unit Rates Lesson 5-6: Compare Unit Rates Lesson 5-7: Solve Unit Rate Problems Topic 5 Review: Understand and Use Ratio and Rate 23
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. envisionmath2.0, 2017 Lesson 6-1: Understand Percent Lesson 6-2: Relate Fractions, Decimals, and Percents Lesson 6-3: Represent Percents Greater Than 100 or Less Than 1 Lesson 6-4: Estimate to Find Percent Lesson 6-5: Find the Percent of a Number Lesson 6-6: Find the Whole Given a Part and the Percent Topic 6 Review: Understand and Use Percent d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Lesson 5-8: Ratio Reasoning: Convert Customary Units Lesson 5-9: Ratio Reasoning: Convert Metric Units Lesson 5-10: Relate Customary and Metric Units Topic 5 Review: Understand and Use Ratio and Rate The Number System 6.NS Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Lesson 1-4: Understand Division with Fractions Lesson 1-5: Divide Fractions by Fractions Lesson 1-6: Divide Mixed Numbers Lesson 1-7: Solve Problems with Rational Numbers Topic 1 Review: Use Positive Rational Numbers Compute fluently with multi-digit numbers and find common factors and multiples. 2. Fluently divide multi-digit numbers using the standard algorithm. Lesson 1-2: Fluently Divide Whole Numbers and Decimals Topic 1 Review: Use Positive Rational Numbers 3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Lesson 1-1: Fluently Add, Subtract, and Multiply Decimals Lesson 1-2: Fluently Divide Whole Numbers and Decimals Topic 1 Review: Use Positive Rational Numbers 24
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. envisionmath2.0, 2017 Lesson 3-2: Find Greatest Common Factor and Least Common Multiple Topic 3 Review: Numeric and Algebraic Expressions Apply and extend previous understandings of numbers to the system of rational numbers. 5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. Lesson 2-1: Understand Integers Topic 2 Review: Integers and Rational Numbers 6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Lesson 2-1: Understand Integers Lesson 2-4: Represent Rational Numbers on the Coordinate Plane Topic 2 Review: Integers and Rational Numbers Lesson 7-4: Find Area of Polygons Topic 7 Review: Solve Area, Surface Area and Volume Problems a. Recognize opposite signs of numbers as 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. Lesson 2-1: Understand Integers Topic 2 Review: Integers and Rational Numbers b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. Lesson 2-4: Represent Rational Numbers on the Coordinate Plane Topic 2 Review: Integers and Rational Numbers 25
A Correlation of envisionmath2.0 Grade 6 to the Common Core State Standards for Mathematics Grade 6 Common Core State Standards for Mathematics Grade 6 c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 7. Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. envisionmath2.0, 2017 Lesson 2-1: Understand Integers Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 2-4: Represent Rational Numbers on the Coordinate Plane Topic 2 Review: Integers and Rational Numbers Lesson 7-4: Find Area of Polygons Topic 7 Review: Solve Area, Surface Area and Volume Problems Lesson 2-2: Represent Rational Numbers on the Number Line Lesson 2-3: Absolute Values of Rational Numbers Topic 2 Review: Integers and Rational Numbers Lesson 2-2: Represent Rational Numbers on the Number Line Topic 2 Review: Integers and Rational Numbers b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. d. Distinguish comparisons of absolute value from statements about order. 8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Lesson 2-2: Represent Rational Numbers on the Number Line Topic 2 Review: Integers and Rational Numbers Lesson 2-3: Absolute Values of Rational Numbers Topic 2 Review: Integers and Rational Numbers Lesson 2-3: Absolute Values of Rational Numbers Topic 2 Review: Integers and Rational Numbers Lesson 2-5: Find Distances on the Coordinate Plane Lesson 2-6: Represent Polygons on the Coordinate Plane Topic 2 Review: Integers and Rational Numbers Lesson 7-4: Find Area of Polygons Topic 7 Review: Solve Area, Surface Area and Volume Problems 26