The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that they assess grade-level content.

Each Topic contains diagnostic, formative, and summative assessments. Summative assessments include: Topic Assessments Forms A and B, Topic Performance Tasks Forms A and B, and Cumulative/Benchmark Assessments. Assessments can be administered online or printed for paper/pencil format. No above grade-level assessment questions are present. Examples of grade-level assessment aligned to standards include: 

• Topic 1, Assessment Form A, Question 6, “Raven is making pillows. Each pillow requires $$\frac{3}{5}$$ yard of fabric. Raven has $$6\frac{2}{3}$$ yards of fabric. Find the number of pillows Raven can make. A. 11 pillows, B. 10 pillows, C. 5 pillows, D. 4 pillows.” (6.NS.1)
• Topic 3, Performance Task Form B, Question 3, Part A, “Mr. Jones is going to build a garden in back of the restaurant to have fresh produce available. The garden will be rectangular, with a length of 2x + 3 feet and a width of x feet. Part A: Fencing material costs $3 per foot including delivery. Write an expression to show the amount of fencing that Mr. Jones will need and then write an expression for the cost of the fence.” (6.EE.2 and 6.EE.6)
• Topic 6, Assessment Form B, Question 5, “A survey found that 78% of high school freshmen have Internet access at home. Of the 754 freshmen at one high school, about how many would be expected to have Internet access at home? Explain.” (6.RP.3)
• Topics 1-6, Cumulative /Benchmark Assessment, Question 5, “Last month, Tara worked 16.5 hours the first week, 19 hours the second week, 23 hours the third week, and 15.75 hours the fourth week. She plans to work more hours this month than last month. Write an inequality to represent the number of hours, h, Tara plans to work this month.” (6.EE.8 and 6.NS.3)
• Topic 7, Assessment Form A, Question 1, “Curtis is making a triangular frame with a base of 12 feet. The perpendicular distance from the base of the frame to its vertex is 6 feet. What is the area of the frame? A. 144 ft$$^2$$, B. 72 ft$$^2$$, C. 36 ft$$^2$$, D. 18 ft$$^2$$.” (6.G.1 and 6.EE.2)
• Topic 8, Performance Task Form A, Question 3, “The Red Team decides to practice for the next competition. Their goal is to get their mean time to 80 but also keep the variability low. Assess whether you think the goal is reasonable, or whether it should be modified. If it should be modified, offer your own goal. Justify your answer.” (6.SP.3 and 6.SP.5)

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations for spending a majority of instructional time on major work of the grade. For example:

• The approximate number of Topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6.5 out of 8, which is approximately 81%.
• The number of lessons (content-focused lessons, 3-Act Mathematical Modeling tasks, projects, Topic Reviews, and assessments) devoted to major work of the grade (including supporting work connected to the major work) is 75 out of 93, which is approximately 81%.
• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 161 out of 194, which is approximately 83%. 

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each Topic. As a result, approximately 81% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. Examples from the Teacher Resource include:

• Lesson 2-6, Represent Polygons on the Coordinate Plane, Visual Learning, Example 2, students graph polygons on a four quadrant coordinate plane and find distances using absolute value, “A rancher maps the coordinates for a holding pen for his cows. How much fencing does the rancher need to enclose the cows’ holding pen?” This example connects the supporting work of 6.G.3, draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate to the major work of 6.NS.7, understand ordering and absolute value of rational numbers and 6.NS.8, solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.
• Lesson 3-2, Find Greatest Common Factor and Least Common Multiple, Visual Learning, Examples 2, 3 and 4, students find common factors and multiples of two whole numbers while using the properties of operations, such as the Distributive Property, to generate equivalent algebraic expressions. Example 3, “Use the GCF and the Distributive Property to find the sum of 18 and 24. Can you use the Distributive Property to rewrite the sum of any two numbers? Explain. Why do you want to identify the common multiples?” These examples connect the supporting work of 6.NS.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 to the major work of 6.EE.3, apply the properties of operations to generate equivalent expressions.
• Lesson 7.5, Represent Solid Figures Using Nets, Visual Learning, Example 3, students use understanding of the coordinate plane to draw nets of three-dimensional figures, “How can you draw a net of a rectangular prism that has a height of 2 units and bases that are 4 units long and 2 units wide?” This example connects the supporting work of 6.G.4, represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures to the major work 6.NS.8, solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.
• Lesson 7.8, Find Volume with Fractional Edge Lengths, Visual Learning, Example 2, students write algebraic expressions to find volume, “Sean bought the fish tank shown. What is the volume of Sean’s fish tank?” This example connects the supporting work of 6.G.2, find the volume of a right rectangular prism with fractional edge lengths to the major work of 6.EE.2, write, read, and evaluate expressions in which letters stand for numbers.
• Lesson 8-3, Display Data in Box Plots, Visual Learning, Do You Know How?, Item 7, students use understanding of number lines to represent data using box plots, “Sarah’s scores on tests were 79, 75, 82, 90, 73, 82, 78, 85, and 78. Draw a box plot that shows the distribution of Sarah’s test scores.” This question connects the supporting work of 6.SP.4, display numerical data in plots on a number line, including dot plots, histograms, and box plots to the major work of 6.NS.6, understand a rational number as a point on the number line.

Note: In the Cross-Cluster Connection in the Teacher Resource a connection between the supporting work of 6.SP.A and major work of 6.EE.C is identified. This connection is not supported by student work. The two instances where this occurs include:

• Lesson 8-1, Lesson Overview, Coherence, Cross-Cluster Connection, “Prior work with recognizing and analyzing quantitative relationships between dependent and independent variables in Lesson 4-8 (6.EE.C) connects to students recognizing a statistical question and understanding statistical variability (6.SP.A). Student work with independent and dependent variables is not present.
• Lesson 8-2, Lesson Overview, Coherence, Cross-Cluster Connection, “Prior work with recognizing and analyzing quantitative relationships between dependent and independent variables in lesson 4-8 (6.EE.C) connects to students recognizing how to summarize a data set by its measure of center of variability (6.SP.A). Student work with independent and dependent variables is not present.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the amount of content designated for one grade-level is viable for one year. As designed, the instructional materials can be completed in 170-194 days.

According to the Pacing Guide in the Teacher Resource, Program Overview, “enVision Mathematics 6-8 was designed to provide students rich opportunities to build understanding of important new mathematical concepts, develop fluency with key skills necessary for success in algebra, and to gain proficiency with the habits of mind and thinking dispositions of proficient mathematical students. To achieve these goals, the program includes content-focused lessons, 3-Act Mathematical Modeling lessons, STEM projects, and Pick a Project. All of these instructional activities are integral to helping students achieve success, and the pacing of the program reflects this. Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 day for the enVisions STEM project and/or Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessment.” For example:

• There are 8 Topics with 61 content-focused lessons for a total of 122 instructional days.
• Each of the 8 Topics contains a 3-Act Mathematical Modeling Lesson for a total of 8-16 instructional days.
• Each of the 8 Topics contains a STEM Project/Pick a Project for a total of 8 instruction days.
• Each of the 8 Topics contains a Topic Review and Topic Assessment for a total of 16 instructional days. 
• Materials allow 16-32 additional instructional days for remediation, fluency practice, differentiation, and other assessments.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations for the materials being consistent with the progressions in the Standards.

The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. According to the Teacher Resource, Program Overview, “Connections to content in previous grades and in future grades are highlighted in the Coherence page of the Topic Overview in the Teacher’s Edition.” These sections are labeled Look Back and Look Ahead. Examples from the Teacher Resource include:

• Topic 1 Overview, Use Positive Rational Numbers, Math Background, Coherence, “In Grade 5, students added, subtracted, and multiplied decimals through hundredths. They used place-value strategies to divide decimals by whole numbers and decimals. They estimated quotients to place the first digit in the quotient and to determine whether their calculations were reasonable. The work students do in this Topic connects directly to Topic 5: Understand and Use Ratio and Rate as students solve problems involving rates, Topic 6: Understand and Use Percent as students relate decimals to percents, Topic 7: Solve Area, Surface Area, and Volume Problems as students compute areas and volumes of figures and summarizes in Topic 8: Display, Describe, and summarize data when students summarize data. Beyond sixth grade, students extend their understanding of ratios and rates to investigate proportional relationships in seventh grade. In Grade 7, students will apply their understanding of fraction computation to proportional relationships and percents.” 
• Topic 2 Overview, Integers and Rational Numbers, Math Background, Coherence, “In Grade 5, students extended value to the thousandths place. They graphed decimals on a number line to help them compare and round decimals. They also extended their ability to do computations with rational numbers to include adding, subtracting, multiplying, and dividing decimals and fractions. Students learned about the coordinate plane and graphed points in the first quadrant to solve real-world and mathematical problems. In Grade 7, students will add, subtract, multiply, and divide both positive and negative rational numbers. They will solve multistep problems involving operations of rational numbers, renaming numbers as appropriate.”
• Topic 5 Overview, Understand and Use Ratio and Rate, Math Background, Coherence, “In Grade 5, students used equivalent fractions to add and subtract fractions and mixed numbers with unlike denominators, and to multiply with fractions. They also learned to divide two whole numbers and get a quotient expressed as either a fraction or mixed number. Students learned to convert measurements within a given measurement system by using multiplication and division. Students learned to graph points in the first quadrant of the coordinate plane. In Topic 6, students will understand percent as a rate in which the comparison is to 100. They will use this understanding to relate fractions, decimals, and percents to solve problems. In Topic 8, students will use percents as they summarize data distributions. In Grade 7, students will compute unit rates associated with ratios of fractions. Students will apply their understanding of the ratio between the circumference and diameter of a circle when they solve problems involving area and circumference of a circle. Students will recognize and represent proportional relationships between quantities. They will also use proportional relationships to solve multistep ratio and percent problems.” 

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The Solve & Discuss It! section presents students with high-interest problems that embed new math ideas, connect prior knowledge to new learning and provide multiple entry points. Example problems are highly visual, provide guided instruction and formalize the mathematics of the lesson. Try It! provides problems that can be used as formative assessment following Example problems and Convince Me! provides problems that connect back to the Essential Understanding of the lesson. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept and skill application. Examples from the Teacher Resource include:

• Lesson 1-5 Divide Fractions By Fractions, Solve & Discuss It!, students use a model and an algorithm to divide fractions by fractions, “A granola bar was cut into 6 equal pieces. Someone ate part of the granola bar so that $$\frac{2}{3}$$ of the original bar remains. How many $$\frac{1}{6}$$ parts are left? Use the picture to draw a model to represent and find $$\frac{2}{3}$$ ÷ $$\frac{1}{6}$$.” (6.NS.1)
• Lesson 3-6, Generate Equivalent Expressions, Visual Learning, Example 1, students use properties of operations to generate equivalent algebraic expressions, “Which of the expressions below are equivalent? 8x - 4, 4x, and 4(2x - 1).” (6.EE.3, and 6.EE.4)
• Lesson 4-5, Write and Solve Equations with Rational Numbers, Do You Know How?, Items 6-8, students write and solve equations that contain fractions, mixed numbers, and decimals using inverse relationships and properties of equality, “t - $$\frac{2}{3}$$ = 25$$\frac{3}{4}$$; $$\frac{f}{2}$$=$$\frac{5}{8}$$; 13.27 = t - 24.45.” (6.EE.7) 
• Lesson 5-3, Visual Learning, Example 2, Try It, students use ratio tables to compare ratios and solve ratio problems, “Tank 3 has a ratio of 3 guppies for every 4 angelfish. Complete the ratio table to find the number of angelfish in Tank 3 with 12 guppies. Using the information in Example 2 and the table at the right, which tank with guppies has more fish?” (6.RP.3)
• Lesson 7-8, Find Volume with Fractional Edge Lengths, Visual Learning, Example 1, Convince Me!, students find the volume of a rectangular prism built with appropriate unit fraction edge length cubes, “Suppose that the length of the rectangular prism in the Try It! were 3 $$\frac{1}{2}$$ inches instead of 2 $$\frac{1}{2}$$ inches. How many cubes would there be in the prism? What would be the volume of the prism?” (6.G.2)
• Lesson 8-5, Summarize Data Using Measures of Variability, Visual Learning, Example 3, Try It!, students summarize numerical data in relation to a given context, “Jonah’s team scored 36, 37, 38, 38, 41, 46, 47, 47, and 48 points in the last nine games. Find the IQR and range of the points Jonah’s team scored in its last nine games. Are these good measures for describing the points scored?” (6.SP.5)

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Each Lesson Overview contains a Coherence section that connects learning to prior grades. Examples include:

• Lesson 2-4, Represent Rational Numbers on the Coordinate Plane, Lesson Overview, Coherence, “Students will be able to identify and graph points with rational coordinates on the coordinate plane and reflect points with rational coordinates across both axes.” (6.NS.6b, 6.NS.6c) “In Grade 5, students represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane.”
• Lesson 5-1, Understand Ratios, Lesson Overview, Coherence, “Students will be able to use ratios to describe the relationship between two quantities and use bar diagrams and double number line diagrams to model ratio relationships.” (6.RP.1, 6.RP.3). “In Grade 5, students analyzed patterns and relationships and used models to represent fractional relationships.”
• Lesson 7-1 Find Areas of Parallelograms and Rhombuses, Lesson Overview, Coherence, “Students will be able to use a formula to find the areas of parallelograms and rhombuses and find the base or height of a parallelogram or rhombus when the area and the height or base are known.” (6.G.1, 6.EE.2) “In Grades 4 and 5, students used the formula for the area of a rectangle to solve problems.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Topics are divided into Lessons focused on domains. Grade 6 standards are clearly identified in each Topic Planner found in the Topic Overview. Additionally, each lesson identifies the Content Standards in the Mathematics Overview. Examples from the Teacher Resource include:

• Lesson 1-3, Multiply Fractions, Lesson Overview, Mathematics Objective, “Use models to multiply fractions. Multiply the numerators and then the denominators to find the product of two fractions. Multiply mixed numbers.” (6.NS.1)
• Lesson 3-5, Evaluate Algebraic Expressions, Lesson Overview, Mathematics Objective, “Evaluate algebraic expressions, including those with whole numbers, decimals, and fractions.” (6.EE.2c, 6.EE.6)
• Lesson 5-5, Understand Rates and Unit Rates, Lesson Overview, Mathematics Objective, “Use rates to describe ratios in which the terms have different units. Use rates and unit rates to solve problems.” (6.RP.2, 6.RP.3a, 6.RP.3b)
• Lesson 7-3, Find Areas of Trapezoids and Kites, Lesson Overview, Mathematics Objective, “Find the areas of trapezoids. Find the areas of kites.” (6.G.1, 6.EE.2c) 
• Lesson 8-5, Summarize Data Using Measures of Variability, Lesson Overview, Mathematics Objective, “Display numerical data in plots on a number line, including dot plots, histograms, and box plots.” (6.SP.4)

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples from the Teacher Resource include:

• Lesson 2-6, Represent Polygons on the Coordinate Plane, Solve and Discuss It!, students graph points on a coordinate plane and use the side lengths to find the perimeter of polygons, “Draw a polygon with vertices at A(-1, 6), B(-7, 6), C(-7, -3), and D(-1, -3). Then find the perimeter of the polygon.” This question connects the work of 6.NS.C, solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane to 6.G.A, draw polygons in the coordinate plane given coordinates for the vertices.
• Lesson 3-4, Write Algebraic Expressions, Practice and Problems Solving, Item 22, students write an algebraic expression and identify the parts in order to represent a real world problem with a variable to represent an unknown number, “Last month, a truck driver made 5 round-trips to Los Angeles and some round-trips to San Diego. Write an expression that shows how many miles he drove in all. Identify and describe the part of the expression that shows how many miles he drove and trips he made to San Diego.” This question connects the work of 6.EE.A, apply and extend previous understandings of arithmetic to algebraic expressions to 6.EE.B, reason about and solve one-variable equations and inequalities.     
• Lesson 6-5, Find the Percent of a Number, Do You Know How?, Item 11, students find the percent from real-world problems that involve one variable, “The original price of a computer game is $45. The price is marked down by $18. What percent of the original price is the markdown?” This question connects the work of 6.RP to the work of 6.EE. 
• Lesson 7-4, Find Areas of Polygons, Visual Learning, Example 3, students use their understanding of integers to represent and find the area of polygons on the coordinate plane, “The floor plan for a new stage at a school is sketched on a coordinate plane. A flooring expert recommends bamboo flooring for the stage floor. How much bamboo flooring, in square meters does the school need?” This example connects the work of 6.G to the work of 6.NS.
• Lesson 8-7, Summarize Data Distributions, Visual Learning, Example 2, students begin to understand spread and variability of data sets, “The fat content, in grams, was measured for one slice ($$\frac{1}{8}$$ pizza) of 24 different 12-inch pizzas. The data are displayed in the dot plot. How can the data be used to describe the fat content of a slice of pizza?” This example connects the work of 6.SP.A, develop understanding of statistical variability to the work 6.SP.B, summarize and describe distributions.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Visual Learning, Example 3, students develop conceptual understanding when using integers to represent real-world quantities and explain the meaning of 0 in each context, “Integers describe many real-world situations including altitude, elevation, depth, temperature, and electrical charges. Zero represents a specific value in each situation. Which integer represents sea level, The airplane? The whale?” (6.NS.5 and 6.NS.6)
• Lesson 4-6, Understand and Write Inequalities, Solve & Discuss It!, students develop conceptual understanding of inequality symbols and use them to write inequalities to describe mathematical or real-world situations, “The record time for the girls’ 50-meter freestyle swimming competition is 24.49 seconds. Camilla has been training and wants to break the record. What are some possible times Camilla would have to swim to break the current record?” (6.EE.5)
• Lesson 5-1, Understand Ratios, Visual Learning, Example 2, students develop conceptual understanding when introduced to ratios through the use of a bar diagram, “The ratio of footballs to soccer balls at a sporting goods store is 5 to 3. If the store has 100 footballs in stock, how many soccer balls does it have? Use a bar diagram to show the ratio 5:3. Use the same diagram to represent 100 footballs.” (6.RP.1)
• Lesson 7-1, Find Area of Parallelograms and Rhombuses, Solve & Discuss It!, students develop conceptual understanding of how to find the area of a parallelogram by decomposing the parallelogram and then composing shapes into a rectangle, “Sofia drew the grid below and plotted the points A, B, C, and D. Connect point A to B, B to C, C to D, and D to A. Then find the area of the shape and explain how you found it. Using the same grid, move points B and C four units to the right. Connect the points to make a new parallelogram ABCD. What is the area of this shape?” (6.G.1 and 6.EE.2)
• Lesson 8-7, Summarize Data Distributions, Visual Learning, Example 1, students develop conceptual understanding of describing the center, spread, and overall shape of a data set, “A science class is testing how different types of fertilizer affect the growth of plants. The dot plot shows the heights of the plants being grown in the science lab. How can you describe the data?” (6.SP.2)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice and Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept. Examples from the Teacher Resource include:

• Lesson 2-2, Represent Rational Numbers on the Number Line, Practice & Problem Solving, Items 24, students independently demonstrate conceptual understanding of rational numbers on a number line, “What is the least number of points you must plot to have examples of all four sets of numbers, including at least one positive integer and one negative integer? Explain.” A rational number diagram is shown. (6.NS.7a)
• Lesson 4-5, Do You Understand?, Item 1, students independently demonstrate conceptual understanding of solving equations involving multiplication or division, “How can you write and solve a multiplication and division equation.” (6.EE.7)
• Lesson 5-5, Understand Rates and Unit Rates, Do You Understand?, Item 3, students independently demonstrate conceptual understanding of problems involving rate and unit rate, “A bathroom shower streams 5 gallons of water in 2 minutes. a. Find the unit rate for gallons per minute and describe it in words. b. Find the unit rate for minutes per gallon and describe it in words.” (6.RP.2 and 6.RP.3)
• Lesson 7-2, Solve Triangle Problems, Practice & Problem Solving, Item 15, students independently demonstrate conceptual understanding of how to find the area of a triangle by using dimensions given, “The dimensions of the sail for Erica’s sailboat are shown. Find the area of the sail.” (6.G.1 and 6.EE.2)
• Lesson 8-6, Choose Appropriate Statistical Measures, Visual Learning, Example 1, Try It!, students independently demonstrate conceptual understanding of choosing the best measure of center to describe the data, “If Gary scored a 70 on his next weekly quiz, how would that affect his mean score? Gary says that he usually scores 98 on his weekly quiz. What measures of center did Gary use? Explain.” (6.SP.5)

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples from the Teacher Resource Include:

• Lesson 1-2, Fluently Divide Whole Numbers and Decimals, Visual Learning, Example 3, Try It!, students use the division algorithm to develop and maintain fluency in dividing whole numbers and decimals, “Divide. a. 658; b. 14.48; c. 128.81.4.” (6.NS.2 and 6.NS.3)
• Lesson 3-3, Write and Evaluate Numerical Expressions, Do You Know How?, Item 6, students use order of operations to evaluate numerical expressions, “Evaluate. (8.2 + 5.3) ÷ 5.” (6.EE.1)
• Lesson 6-1, Find Percents, Do You Know How?, Item 11, students use ratio and rate reasoning for percents, “Find the percent of the line segment that point D represents in Example 2.” (6.RP.3)
• Lesson 7-4, Find Areas of Polygons, Visual Learning, Example 3, Try It, students find areas of polygons, “Find the area of the shaded region in square units.” (6.G.3)
• Lesson 8-4, Display Data in Frequency Tables and Histograms, Do You Know How?, Item 5, students organize data with equal intervals into frequency tables and histograms. “A data set contains ages ranging from 6 to 27. (6, 11, 9, 13, 18, 15, 21, 15, 17, 24, 24, 12) Complete the frequency table and histogram.” (6.SP.4) 

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Fluency Practice page which engages students in fluency activities. Examples include:

• Topic 1 Review, Fluency Practice, students fluently multiply and divide decimals, “Pathfinder: Shade a path from START to FINISH. Follow the solutions in which the digit in the hundredths place is greater than the digit in the tenths place. You can only move up, down, right, or left.” The first row of problems are, “22.04 x 9, 42.12 ÷ 7.2, 53.08 x 2.4, 0.18 x 1.5, and 0.28 ÷ 7.” (6.NS.3)
• Lesson 2-2, Fluently Divide Whole Numbers and Decimals, Practice & Problem Solving, Item 15, students plot rational numbers on a number line, “In 15-20, write the number positioned at each point. Point A.” (6.NS.6 and 6.NS.7)
• Lesson 3-6, Generate Equivalent Expressions, Practice & Problem Solving, Item 18, students generate equivalent expressions, “Write equivalent expressions. 2x + 4y.” (6.EE.4)
• Lesson 5-2, Generate Equivalent Ratios, Practice & Problem Solving, Item 14, students write equivalent ratios, “Write three ratios that are equivalent to the given ratio: 8:14.” (6.RP.3)
• Lesson 7-1, Find Areas of Parallelograms and Rhombuses, Practice & Problem Solving, Item 12, students use the formula A = bh to find the missing measurement in various parallelograms, “The area of a parallelogram is 132 in$$^2$$. What is the height of the parallelogram?” A visual is provided with a base of 11 in. (6.G.1)
• Lesson 8-5, Summarize Data Using Measures of Variability, Practice & Problem Solving, Item 11, students find measures of variability of a given data set, “Use the data shown in the data plot. What are the mean and the MAD?” (6.SP.5)

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied. 

The instructional materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade-level. According to the Teacher Resource Program Overview, “In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues. For example:

• Topic 2, 3-Act Mathematical Modeling: The Ultimate Throw, Question 14, students determine how far each person threw a disc and who threw the disc farther, "Suppose each person walks to the other person's disc. They throw each other’s discs toward the starting point. Where do you think each disc will land?" (6.NS.5 and 6.NS.7)
• Topic 3, STEM Project, Design a Bridge, students use equations and inequalities to represent measurements for building a bridge, "Now that you have defined the problem, identified the criteria and constraints and performed some data collection, it is time to focus on the solution. You and your classmates will continue to be engineers as you brainstorm solutions and develop prototypes for your bridge.” (6.EE.5 and 6.EE.9)
• Topic 5, STEM Project, Get into Gear, students examine sample pairings of gears and write gear ratios, "Cyclists strive to achieve efficiency during continuous riding. But, which pairing of gears is the best or most efficient? And does the answer change depending on the terrain? You and your classmates will explore gear ratios and how they can affect pedaling and riding speeds.” (6.RP.1, 6.RP.3)
• Topic 7, 3-Act Mathematical Modeling: That's a Wrap, Question 15, students determine how many stickers are needed to cover the surface area of a box, "A classmate says that if all dimensions of the gift were doubled, you would need twice as many squares. Do you agree? Justify his reasoning or explain his error." (6.G.4 and 6.EE.2)
• Topic 7, STEM Project, Pack It, students determine how the volume of packing food items relates to the volume of the food items being packed, "Food packaging engineers consider many elements related to both form and function when designing packaging. How do engineers make decisions about package designs as they consider constraints, such as limited dimensions or materials? You and your classmates will use the engineering design process to explore and propose food packaging that satisfies certain criteria.” (6.G.2 and 6.G.4)
• Topic 8, 3-Act Mathematical Modeling: Vocal Range, Question 12, students use informal arguments and statistical reasoning to decide who should win a singing competition, "Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?" (6.SP.2, 6.SP.3, and 6.SP.5)

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Lesson 1-5, Divide Fractions by Fractions, Practice & Problem Solving, Item 26, students use models to divide fractions by fractions, “A large bag contains $$\frac{12}{15}$$ pound of granola. How many $$\frac{1}{3}$$ pound bags can be filled with this amount of granola? How much granola is left over?” (6.NS.1)
• Lesson 4-3, Write and Solve Addition and Subtraction Equations, Practice & Problem Solving, Item 17, students write and solve addition and subtraction equations, “You have some baseball cards. You give 21 baseball cards to a friend and have 9 left for yourself. How many baseball cards were in your original deck? Write and solve an equation to find t, the number of baseball cards in your original deck.” (6.EE.7)
• Topic 4, Pick a Project 4C, students make a model of a staircase using tables and equations, “Think about what you need in order to make a model of a staircase. Design a staircase following these rules: The staircase must follow a linear pattern, Use identical blocks to model the staircase, Keep track of the number of blocks you need for each step, Make a table of data for the number of blocks used for any number of steps, Write an equation to represent your staircase pattern.” (6.EE.9)
• Topic 7, Pick a Project 7C, students calculate surface area, “Suppose you ordered four gifts online. These items would be delivered to you in boxes that needed to be wrapped. Find the amount of wrapping paper you would need to wrap all four gifts. Use four different sized boxes to represent the four gifts. Calculate the surface area of each box. Determine the amount of wrapping paper, in square units, that you will need.” (6.G.4)
• Lesson 8-4, Display Data in Frequency Tables and Histograms, Practice & Problem Solving, Item 15, students apply their understanding of frequency tables and histograms to solve problems in real-world contexts, “Todd wants to know how many people took 20 seconds or more to stop a bike safely. Would a frequency table or a histogram be the better way to show this? Explain.” (6.SP.4, and 6.SP.5)

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

All three aspects of rigor are present independently throughout the program materials. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, and application include:

• Unit 2 Review, Fluency Practice, students add and subtract multi-digit decimals, “Hidden Clue: For each ordered pair, simplify the two coordinates. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help answer the riddle below.” Ordered pair B states, “(9.65 + 0.4, 16.058 - 12).” (6.NS.3)
• Lesson 5-5, Understand Rates and Unit Rates, Practice & Problem Solving, Item 22, students use rates and unit rates to solve application problems, “An elephant charges an object that is 0.35 kiliometer away. How long will it take the elephant to reach the object?” A picture of an elephant with the caption, “Elephants can charge at speeds of 0.7 km per minute” is shown. (6.RP.2 and 6.RP.3)
• Lesson 8-6, Choose Appropriate Statistical Measures, Visual Learning, Example 1, students develop conceptual understanding of the characteristics to consider when choosing measures to describe a data set, “Gary reviews the scores on his weekly quizzes. What measure should Gary use to get the best sense of how well he is doing on his weekly quizzes?” The teacher asks, “How does a dot plot make analyzing the spread and clustering of data in a set easier? Why is 65 considered an outlier? The mean and median are close in value. How else could you argue for the median as the best measure of center?” (6.SP.5)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• Lesson 1-7, Solve Problems with Rational Numbers, Practice & Problem Solving, Item 15, students apply their understanding of decimal and fraction operations to solve multistep problems, “Kelly buys three containers of potato salad at the deli. She brings $$\frac{4}{5}$$ of the potato salad to a picnic. How many pounds of potato salad does Kelly bring to the picnic? Describe two different ways to solve the problem.” This question develops conceptual understanding and application of 6.NS.1, interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.
• Lesson 2-3, Absolute Values of Rational Numbers, Do You Know How?, Item 15, students find and interpret absolute values, “In 15-17, use the absolute value of each account balance to determine which account has the greater overdrawn amount. Account A: -$5.42; Account B: -$35.76.” This question develops conceptual understanding and procedural skill of 6.NS.7, understand ordering and absolute value of rational numbers.
• Lesson 4-1, Understand Equations and Solutions, Practice & Problem Solving, Item 21, students solve problems in real world contexts, “Gerard spent $5.12 for a drink and a sandwich. His drink cost $1.30. Did he have a ham sandwich for $3.54, a tuna sandwich for $3.82, or a turkey sandwich for $3.92? Use the equation s + 1.30 = 5.12 to justify your answer.” This question develops application and conceptual understanding of 6.EE.5, understand solving an equation or inequality as a process of answering a question.
• Lesson 5-3, Compare Ratios, Do You Know How?, Item 4, students compare ratios using a common term, “To make plaster, Kevin mixes 3 cups of water with 4 pounds of plaster powder. Complete the ratio table. How much water will Kevin mix with 20 pounds of powder?” This question develops application and fluency of 6.RP.3, use ratio and rate reasoning to solve real-world and mathematical problems.
• Lesson 7-1, Find Areas of Parallelograms and Rhombuses, Visual Learning, Example 3, students find missing dimensions of a parallelogram when given the area. “A. The area of the parallelogram is 72 m$$^2$$. What is the height of the parallelogram? B. The area of the parallelogram is 135 in$$^2$$. What is the base of the parallelogram?” This question develops conceptual understanding and procedural skill of 6.G.1, find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. 

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All 8 MPs are clearly identified throughout the materials, with few or no exceptions. Math Practices identification in this program according to the Teacher Resource Program Overview include:

• Materials provide a Math Practices and Problem Solving Handbook for students, “A great resource to help students build on and enhance their mathematical thinking and habits of mind.” This handbook explains math practices in student-friendly language and digital animation videos for each math practice are also available.
• Opportunities to apply math practices are found in the Explore It, Explain It, and Solve & Discuss It portions of the lesson. “The Solve & Discuss It calls on students to draw on nearly all of the math practices, but especially sense-making and solution formulation as well as abstract and quantitative reasoning. The Explore It focuses students on mathematical modeling, generalizations, and structure of mathematical models. The Explain It emphasizes mathematical reasoning and argumentation. Students construct arguments to defend a claim or critique an argument defending a claim.”
• The Math Practices and Problem Solving Handbook Teacher Pages, “provide overviews of the math practices, offer instructional strategies to help students refine and enhance their thinking habits, and include student behaviors to listen and look for for each standard.”
• Each Topic Overview contains Math Practices Teacher Pages which include, “Two highlighted math practices with student behaviors to look for, and questions to help students become more proficient with these thinking habits.” For example, in Topic 3, Model with mathematics suggested question state, “How can you change the value of an expression by adding grouping symbols? How can expressions be rewritten without changing their values?”
• Math Practices boxes found in the student text provide, “Reminders to be thinking about the application of the math practices as they solve problems.”
• Math Practices Run-in Heads such as “Construct Arguments and Reasoning” are found in the Practice & Problem Solving questions, “Remind students to apply the math practices as they solve problems.”

The majority of the time the MPs are used to enrich the mathematical content and are not treated separately. Examples include:

• MP1: Make sense of problems and persevere in solving them. Lesson 3-7, Simplify Algebraic Expressions, Teacher's Edition, Solve & Discuss It!, students make sense of problems by using prior knowledge and applying properties of operations to rewrite algebraic expressions through simplifying, “Write an expression equivalent to x + 5 + 2x + 2. Students might use the Commutative Property to reorder the terms, rewrite 2x using repeated addition, and then rewrite the sum using multiplication to give the simplified version of the expression. They also might rewrite one or more terms using repeated addition to give a non-simplified equivalent expression. If needed, ask: How could you rewrite 2x using a different operation?” 
• MP2: Reason abstractly and quantitatively. Lesson 8-3, Display Data in Box Plots, Practice & Problem Solving, Item 17, students use reasoning as they determine the importance of ordering when they find the median, “The price per share of Electric Company’s stock during 9 days, rounded to the nearest dollar, was as follows: $16, $17, $16, $16, $18, $18, $21, $22, $19. Use a box plot to determine how much greater the third quartile’s price per share was than the first quartile’s price per share.” 
• MP4: Model with mathematics. Lesson 5-6, Compare Unit Rates, Practice & Problem Solving, Item 19, students use unit rates to model the relationships between quantities presented in real-world problems, as well as identifying important quantities and using them to complete a double number line diagram to model ratio relationships, “Katrina and Becca exchanged 270 text messages in 45 minutes. An equal number of texts was sent each minute. The girls can send 90 more text messages before they are charged additional fees. Complete the double number line diagram. At this rate, for how many more minutes can the girls exchange texts before they are charged extra?” Students have opportunities to take different approaches, organize and explain their strategies so that others, who may have taken a different approach, can follow their line of thinking.
• MP5: Use appropriate tools strategically. Lesson 4-7, Solve Inequalities, Practice and Problem Solving, Item 29, students use a number line to write and represent solutions of inequalities, “Graph the inequalities x > 2 and x < 2 on the same number line. What value, if any, is not a solution of either inequality? Explain.”
• MP6: Attend to precision. Lesson 1-1, Fluently Add, Subtract, and Multiply Decimals, Practice & Problem Solving, Item 36, students use algorithms to add, subtract, and multiply decimals efficiently, accurately, and fluently, “The fastest speed a table tennis ball has been hit is about 13.07 times as fast as the speed for the fastest swimming. What is the speed for the table tennis ball?”
• MP7: Look for and make use of structure. Lesson 8-2, Summarize Data Using Mean, Median, Mode, and Range, Practice & Problem Solving, Item 14, students use structure to find and analyze statistical measures, “Does increasing the 3 to 6 change the mode? If so, how?” Students are provided a data set of states lived in and visited. Students use the structure of a data set to analyze statistical measures. 
• MP8: Look for and express regularity in repeated reasoning. Lesson 6-2, Relate Fractions, Decimals, and Percents, Practice & Problem Solving, Item 24, students generalize the existence of equivalence in multiple forms of a number, “What are the attributes of fractions that are equivalent to 100%?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

The materials do not attend to the full meaning of MP5: Use appropriate tools strategically. Examples include: 

• Lesson 4-3: Write and Solve Addition and Subtraction Equations, Solve & Discuss It!, “A group of students were on a school bus. How many students were on the bus before the last stop?” Use Appropriate Tools box states, “You can use a pan balance to help solve for the unknown.” 
• Lesson 4-7, Solve Inequalities, Solve & Discuss It!, “Henry is thinking of a number that is less than 17. What number could he be thinking of?” Teacher directions state, “Provide blank number lines, as needed.” Students are provided number lines to show inequality solutions.
• Lesson 6-5, Find the Percent of a Number, Do You Understand?, “How can you use a calculator to find the percent of 180 is 108?” Students are told to use a calculator to find the percent of a number. 
• Lesson 8-2, Draw Geometric Figures, Visual Learning, Example 1, students draw a quadrilateral with given conditions, “The school’s landscaping club is designing a 4-sided patio and garden. The patio has 2 perpendicular sides that each measure 4 yards, and a third side that is perpendicular to one of the sides but twice as long. One angle of the patio measures 135. Make a scale drawing of the patio using a scale of 1 cm = 1 yd.” Students do not choose an appropriate tool as an image of a protractor, ruler, pencil, and straight edge is given to the students. Also, a side box, Use Appropriate Tools, states, “You can use rulers and protractors to construct precise drawings. Step 1: Use a ruler to draw three sides that meet the given conditions, 8cm, 4cm. Step 2: Use a protractor to draw a 135 angle that connects and completes the shape.” 

The materials do attend to the full meaning of the following MPs. For example:

• MP1: Make sense of problems and persevere in solving them. Lesson 7-6, Find Surface Areas of Prisms, Solve & Discuss It!, “Suppose Marianne has only one large sheet of green paper that is 15 inches by 30 inches. Is the area of this sheet of paper great enough to cover all the faces of one box? Explain.” Students analyze a multistep problem involving surface area of prisms and consider different ways to find solutions.
• MP2: Reason abstractly and quantitatively. Lesson 5-3, Compare Ratios, Teacher Resource, Solve & Discuss It!, “Scott is making a snack mix using almonds and raisins. For every 2 cups of almonds in the snack mix, there are 3 cups of raisins. Ariel is making a snack mix that has 3 cups of almonds for every 5 cups of sunflower seeds. If Scott and Ariel each use 6 cups of almonds to make a bag of snack mix, who will make a larger batch?” Students use quantitative reasoning when they attend to the meaning of the quantities in problems and determine what needs to be done to find a solution.
• MP4: Model with mathematics. Lesson 4-3, Absolute Values of Rational Numbers, Practice & Problem Solving, Item 40, “Find the distance from Alberto’s horseshoe to Rebecca’s horseshoe. Explain.” Students write and solve equations to represent real-life situations.
• MP6: Attend to precision. Lesson 2-4, Represent Rational Numbers on the Coordinate Plane, Practice & Problem Solving, Item 33, “Write the ordered pair to locate the end of the hiking trail in two different ways.” Students use precision as they locate and identify points on a coordinate plane.
• MP7: Look for and make use of structure. Lesson 3-6, Generate Equivalent Expressions, Practice & Problem Solving, Item 28, “Write an algebraic expression to represent the area of the rectangular rug. Then use properties of operations to write an equivalent expression.” Dimensions provided in illustration are l = 2(x - 1) and w = 5. Students use structure and properties of operations to determine the equivalence of expressions.
• MP8: Look for and express regularity in repeated reasoning. Lesson 6-2, Relate Fractions Decimals and Percents, Practice & Problem Solving, Item 17, “How could you write $$\frac{4}{8}$$ as a percent without dividing?” Students write fractions as decimals and percentages.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Lesson 1-3, Multiply Fractions, Do You Understand?, Item 3, students use their understanding of fraction multiplication to construct arguments to support their response, “Why is adding $$\frac{3}{9}$$ and $$\frac{6}{9}$$ different from multiplying the two fractions?”
• Lesson 3-2, Find the Greatest Common Factor and Least Common Multiple, Do You Understand?, Item 4, students use their understanding of least common multiple to construct an argument to support their response, “In Example 4, Grant finds applesauce that comes in packages of 8, but now he finds the juice bottles in only packages of 3. Will the LCM change? Explain.” 
• Lesson 4-2, Apply Properties of Equality, Practice & Problem Solving, Item 20, students use their understanding of properties of equality to write equivalent equations to construct arguments as they form the basis for the procedure used to solve algebraic equations, “John wrote that 5 + 5 = 10. Then he wrote that 5 + 5 + n = 10 + n. Are the equations John wrote equivalent? Explain.”
• Lesson 5-3, Compare Ratios, Visual Learning, Example 1, Try It! students use their understanding of ratios to construct arguments, “Dustin had 3 hits for every 8 at bats. Adrian had 4 hits for every 10 at bats. Marlon had 6 hits in 15 at bats. Based on their hits to at bats ratios, who would you expect to have more hits in a game, Marlon or Dustin? Explain.”
• Lesson 8-7, Summarize Data Distributions, Explain It!, students use their understanding of data distributions to construct arguments, “George tosses two six-sided number cubes 20 times. A. Describe the shape of the data distribution. B. George says that he expects to roll a sum of 11 on his next roll. Do you agree? Justify your reasoning. Construct Arguments: Suppose George tossed the number cubes 20 more times and added the data to his dot plot. Would you expect the shape of the distribution to be different? Construct an argument that supports your reasoning.”

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Lesson 1-2, Fluently Divide Whole Numbers and Decimals, Practice & Problem Solving, Item 33, students analyze the arguments of others as they divide whole numbers and decimals and apply these skills to solve mathematical problems, “Henrieta divided 0.80 by 20 as shown. Is her work correct? If not, explain why and give a correct response.”
• Lesson 2-1, Understand Integers, Explain It!, students analyze the arguments of others as they explain differences between integers, “Sal recorded the outdoor temperature as -4℉ at 7:30 A.M. At noon, it was 22℉. Sal said the temperature changed by 18℉ because 22 - 4 = 18. Problem A. “Is Sal right or wrong? Explain.”
• Lesson 3-1, Understand and Represent Exponents, Practice & Problem Solving, Item 35, students analyze the arguments of others using their understanding of exponents. “Kristen was asked to write each of the numbers in the expression 80,000 x 25 using exponents. Her response was (8 x 10$$^3$$ ) x 5$$^2$$ . Was Kristen’s response correct? Explain.”
• Lesson 6-1, Understand Percent, Practice & Problem Solving, Item 22, students analyze the arguments of others as they represent and find the percent of a whole, “Kyle solved 18 of 24 puzzles in a puzzle book. He says that he can use an equivalent fraction to find the percent of puzzles in the book that he solved. How can he do that? What is the percent?” 
• Lesson 7-6, Find Surface Area of Prisms, Practice & Problem Solving, Item 14, students analyze the arguments of others as they explain how to find the surface area of a cube, “Jacob says that the surface area of the cube is less than 1,000 cm$$^2$$. Do you agree with Jacob? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in constructing viable arguments frequently throughout the program. Examples include:

• Lesson 1-3, Multiply Fractions, Visual Learning, Example 1, Try It!, “Find $$\frac{1}{4}$$ x $$\frac{1}{5}$$ using the area model. Explain.” ETP (Effective Teaching Practices) teacher prompts state, “Could you use the rows in the area model to represent $$\frac{1}{5}$$? Explain. How does the model support your answer?”
• Lesson 2-6, Represent Polygons on the Coordinate Plane, Solve & Discuss It!, “Draw a polygon with vertices at A(-1,6), B(-7,6), C(-7,-3), and D(-1,-3). Then find the perimeter of the polygon.” ETP Discuss Solution Strategies and Key Ideas for teachers states, “Have students present their solutions. Have groups discuss how they knew what distances they needed to find to determine the side lengths. Have students discuss how they knew when to add absolute values to find the distance and when they had to subtract; add when points are on opposite sides of an axis and subtract when points are on the same side. Have students who used different perimeter formulas present how they calculated the perimeter. (P = l + w + l + w versus P = 2l + 2w).”
• Lesson 8-5, Summarize Data Using Measures of Variability, Visual Learning, Example 1, Try It!, “Ann’s vocabulary quiz scores are 75, 81, and 90. The mean score is 82. What is the mean absolute deviation?” The ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “Does the fact that the MAD is a decimal mean that Ann could score a decimal value on her quiz? Explain. Suppose Ann’s friend also found the mean and the MAD for his three quiz scores. If his MAD was 10.5 points, how do his quiz scores compare to Ann’s? Can the absolute value of a number ever be negative? Explain.”

Teacher materials assist teachers in engaging students in analyzing the arguments of others frequently throughout the program. Examples include:

• Lesson 3-6, Generate Equivalent Expressions, Explain It!, “Juwon says all three expressions are equivalent. 8n + 6, 2(4n + 3), and 14n. Do you agree with Juwon that all three expressions are equivalent? Explain.” ETP Discuss Solution Strategies and Key ideas teacher prompt states, “Have students who used substitution to describe why Juwon is incorrect share first, followed by students who simplified each expression. Have students discuss why each method is valid; equivalent expressions simplify to expressions with the same terms and have equivalent results for all values of the variable. Have students discuss which method they think is better, and why; simplifying the expressions is the better method since expressions with the same terms are definitely equivalent, but substitution may lead to choosing a value that coincidentally yields the same result for all expressions even though the expressions are not equivalent for all values.”
• Lesson 4-8, Understand Dependent and Independent Variables, Do You Understand?, Item 2, “Viola says the number of calories, c, they burn is the dependent variable. Do you agree? Explain.” The ETP Item 2 Critique Reasoning teacher prompt states, “Does the number of miles Jake and Viola bike depend on the number of calories they burn? What affects the number of calories they burn?”
• Lesson 5-10, Relate Customary and Metric Units, Explain It!, “Gianna and her friends are in a relay race. They have a pail that holds 1 liter of water. They need to fill the 1-liter pail, run 50 yards, and dump the water into the large bucket until it overflows. Gianna says that as long as they do not spill any of the water, they will need 7 trips with the 1-liter pail before the large bucket overflows. Which conversion factor could you use to determine whether Gianna is correct? Explain.” ETP Observe Students at Work teacher prompt states, “How do students critique Linus’s reasoning? Students might say approximate values are very close to actual values, so they can be used to solve problems. If needed, have students look up more precise conversion factors to compare.”

Teacher materials assist teachers in engaging students in both the construction of viable arguments and analyzing the arguments or reasoning of others frequently throughout the program. Each Topic Overview highlights specific Math Practices and suggests look fors in student behavior and provides questioning strategies. Examples include:

• Topic 2, Integers and Rational Numbers, Math Practices, look fors, “Mathematically proficient students: Critique the strategies of others as they compare and order rational numbers. Construct arguments to defend the application of formulas to new situations. Construct arguments using accurate definitions and terminology related to rational numbers and coordinate planes. Ask questions to clarify others’ reasoning to decide whether arguments make sense or to improve the arguments.”
• Topic 2, Integers and Rational Numbers, Math Practices, questioning strategies, “How can you justify your answer? What mathematical language, models, or examples will help you support your answer? How could you improve this argument? How could you use counterexamples to disprove this argument? What do you think about this explanation? What questions would you ask about the reasoning used?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. Each Topic Overview provides a chart in the Topic Planner that lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow. Lesson practice includes questions to reinforce vocabulary comprehension and students write using math language to explain their thinking. Each Topic Review contains a Vocabulary Review section for students to review vocabulary taught in the Topic. Students have access to an Animated Glossary online in both English and Spanish. Examples include:

• Topic 1, Use Positive Rational Numbers, Use Vocabulary in Writing, “Explain how to use multiplication to find the value of $$\frac{1}{3}$$ ÷ $$\frac{9}{5}$$. Use the words multiplication, divisor, quotient, and reciprocal in your explanation.”
• Lesson 1-4, Understand Division with Fractions, Visual Learning, Example 3, “Two numbers whose product is 1 are called reciprocals of each other. If a nonzero number is named as a fraction, $$\frac{a}{b}$$, then its reciprocal is $$\frac{b}{a}$$.”
• Lesson 5-5, Understand Rates and Unit Rates, Visual Learning, Example 1, “A rate is a special type of ratio that compares quantities with unlike units of measure.”
• Lesson 7-5, Represent Solid Figures Using Nets, Visual Learning, Example 1, “A polyhedron is a three-dimensional solid figure made of flat polygon-shaped surfaces called faces. The line segment where two faces intersect is called an edge. The point where several edges meet is called a vertex.”
• Topic 8, Solve Area, Surface Area, and Volume Problems, Mid-Topic Checkpoint, Question 1, “How many pairs of opposite sides are parallel in a trapezoid? How is this different from a parallelogram?” 

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. A Vocabulary Glossary is provided in the back of Volume 1 and lists all the vocabulary terms and examples. Teacher side notes, Elicit and Use Evidence of Student Thinking and Pose Purposeful Questions, provide specific information about the use of vocabulary and math language. Examples include:

• Lesson 2-2, Understanding Integers, Visual Learning, Example 3, Pose Purposeful Questions, “What do distances of opposite numbers have in common? Is 0 positive or negative? Explain.”
• Lesson 4-8, Understand Dependent and Independent Variables, Visual Learning, Example 1, Elicit and Use Evidence of Student Thinking, “What does the number of pancakes that the baker can make depend on? In this situation, does the number of cups of batter depend on the number on the number of pancakes that the baker can make? Explain.”
• Lesson 7-3, Find Areas of Trapezoids and Kites, Visual Learning, Example 3, Try It!, Elicit and Use Evidence of Student Thinking, “When you decompose the trapezoid in Part a of the Try It! into two triangles and a rectangle, are the triangles identical? Explain. What is the height of the two large, identical triangles that compose the kite in Part b of the Try It!?”
• Lesson 8-1, Recognize a Statistical Question, Visual Learning, Example 2, Pose Purposeful Questions, “How does the dot plot show numerical data? Which parts of the dot plot help you determine the statistical question?”
• Student Edition, Glossary, “dependent variable: A dependent variable is a variable whose value changes in response to another (independent) variable.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas, in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

Materials engage students in both problems and exercises through the grade level. Problems where students learn new mathematics are typically found in the Lesson’s Visual Learning Bridge. This portion of the lessons consists of visual examples that formalize the mathematics of the lesson by providing guided instruction of the math concepts with one example stepped-out. Examples from the Teacher Resource include:

• Lesson 1-3, Multiply Fractions, Visual Learning, Example 1, students model multiplication of fractions using two methods, “There was $$\frac{1}{4}$$ of a pan of lasagna left. Tom ate $$\frac{1}{3}$$ of this amount. What fraction of a whole pan of lasagna did Tom eat? One Way: Divide one half into fourths. Another way: Shade 1 of the 3 rows yellow to represent $$\frac{1}{3}$$. Shade 1 of the 4 columns red to represent $$\frac{1}{4}$$.”
• Lesson 4-6, Understand and Write Inequalities, Visual Learning, Example 1 students model inequalities to present more than one value, “How can you write an inequality to describe the ages of the children who must be accompanied by an adult at this sledding hill? One model: You can show some of the ages on a number line. Second model: Let a represent the ages of children who must be accompanied by an adult. Use the less than symbol (<) to write the inequality.”
• Lesson 7-3, Find Areas of Trapezoids and Kites, Visual Learning, Example 1, students use properties of a trapezoid to find its area, “The pasture is in the shape of a trapezoid. What is the area of the pasture? One strategy: Decompose the trapezoid into a rectangle and two right triangles. Second strategy: Find the area of each shape and then add the areas. The triangles are identical.”

Exercises, where students apply learning to build mastery, are typically found in the Practice and Problem Solving section. These exercises build independent proficiency, challenge higher-order thinking, and simulate high-stakes testing questions. Examples from the Teacher Resource include:

• Lesson 2-2, Represent Rational Numbers on the Number Line, Practice & Problem Solving, Item 30, students explain how to represent rational numbers using a number line, “Suppose $$\frac{a}{b}$$, $$\frac{c}{d}$$, and $$\frac{e}{f}$$ represent three rational numbers. If$$\frac{a}{b}$$ is less than$$\frac{c}{d}$$ and$$\frac{c}{d}$$ is less than$$\frac{e}{f}$$, compare$$\frac{a}{b}$$ and$$\frac{e}{f}$$. Explain.”
• Lesson 3-3, Write and Evaluate Numerical Expressions, Practice & Problem Solving, Item 27, students evaluate numerical expressions, “Frederick evaluates the numerical expression [(53.7 + 37.2) - (3$$^3$$+ 3.8)]- 8.6 and records the answer as 51.5. Lana evaluates the numerical expression 53.7 + 37.2 - 3$$^3$$+ 3.8- 8.6 and records the answer as 59.1. The expressions have the same numbers and operations. Explain how Frederick and Lana can both be correct.”
• Lesson 5-4, Represent and Graph Ratios, Practice & Problem Solving, Item 12, students explain how to use a table or graph to represent equivalent fractions, “Ishwar can read 5 pages in 15 minutes. Anne can read 15 pages in 1 hour. Explain How you could use a table or graph to find how much longer it would take Anne to read a 300-page book than Ishwar.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that the design of assignments is not haphazard: exercises are given in intentional sequences.

Lesson activities within each Topic are intentionally sequenced developing student understanding and leading towards mastery of the content. Students are introduced to concepts and procedures with a problem-solving experience, Solve & Discuss it. The Visual Learning Bridge provides direct instruction that makes the important mathematics explicit through class discussion of student thinking and solutions. Examples from the Teacher Resource include:

• Lesson 6-5, Find the Percent of a Number, Solve & Discuss It!, Example 2, students use models to find the percent of a number in order to find the part in a percent problem, “D’wayne plans to wallpaper 72.5% of a 60-square foot wall. How many square feet of the wall does D’wayne plan to wallpaper? Find 72.5% of 60.”
• Lesson 8-3, Display Data in Box Plots, Solve & Discuss It!, Example 1, students describe data by finding the minimum value, maximum value, median and connect it to making a box plot, “Helen wants to display the lengths of 15 fish she caught this year to compare to the lengths of fish she caught last year. How can she use the data to make a box plot? Find the minimum, median, and maximum values of the data. Find the median for each half. Draw the box plot. Show a number line with an appropriate scale, a box between the first and third quartiles, and a vertical segment that show the median.”
• Lesson 8-6, Choose Appropriate Statistical Measures, Solve & Discuss It!, Example 1, students determine whether the mean, median, or mode best describes the data in a set and select the best measure of center to describe a data set, “Gary reviews the scores on his weekly quizzes. What measure should Gary use to get the best sense of how well he is doing on his weekly quizzes? Display Gary's scores on a dot plot. Describe the shape of the data. Then find the mean, median, and mode of the data set.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that there is variety in what students are asked to produce. For example, students are asked to produce answers and solutions; but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc. Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Do You Understand?, Item 4, students use integers in real-world contexts, “Which amount represents a debt of two hundred fifty dollars, $250 or -$250? Explain.”
• Lesson 4-1, Understand Equations and Solutions, Practice & Problem Solving, Item 25, students determine whether a given number makes an equation true, “Alisa’s family planted 7 palm trees in their yard. The park down the street has 147 palm trees. Alisa guessed that the park has either 11 or 31 times as many palm trees as her yard has. Is either of Alisa’s guesses correct? Use the equation 7n = 147 to justify your answer.” 
• Lesson 8-1, Recognize Statistical Questions, Practice & Problem Solving, Item 11 students use data to make a frequency table to display answers to statistical questions, “Sergei asked his classmates, Will you take Spanish or French next year? He collected these responses: 15 classmates chose Spanish and 13 chose French. Make a frequency table to display the data.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods. 

Students have access to Anytime Math Tools powered by Desmos to build understanding and are accessible from the Tools panel online. Desmos tools include a graphing calculator, a scientific calculator, and a geometry construction tool. In addition, students have access to digital math tools such as algebra tiles, integer chips, area models, and bar diagrams. Students see an icon with a wrench when tools are suggested for use during examples and questions. Examples from the Teacher Resource include:

• Lesson 4-2, Apply Properties of Equality, Visual Learning, Example 1, students use a balance to define properties of equality, “What do the Addition and Subtraction Properties of Equality state? What does the Multiplication Property of Equality state? How does the Division Property of Equality differ from the other three properties of equality? Explain.”
• Lesson 6-2, Relate Fractions, Decimals, and Percents, Solve and Discuss It!, students use grid paper or colored tiles to model how to represent values out of 100 using fractions, decimals, and percents and make connections when converting between these forms, “The grid is shaded with blue, orange, and yellow? What part of the grid is shaded blue? What part is shaded orange? What part of the grid is shaded?”
• Lesson 7-7, Find Surface Areas of Pyramids, Visual Learning, Example 2, Try It!, students use a net to visualize the faces of square and triangular pyramids when finding the surface area, “Draw a net and find the surface area of the triangular pyramid. Find the Area (T) of each equilateral triangle. Find the surface area (SA) of the triangular pyramid.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 have a visual design (whether in print or online) that is not distracting or chaotic, and supports students in engaging thoughtfully with the subject.

The font size, graphics, amount of directions, and language used on student pages and in Digital Lessons is appropriate for students. Graphics promote understanding of the mathematics being learned. The digital format is easy to navigate and is engaging. There is ample “white space” for students to calculate and write answers in the student materials.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Effective Mathematics Teaching Practices (ETP) side notes provide quality questions that are designed to promote reasoning and problem solving, support productive struggle, and engage students in mathematical discourse. Establish the Mathematical Goal provides questions related to the Essential Question. Use and Connect Mathematical Representations and Pose Purposeful Questions provide probing questions to enrich the mathematics. Elicit Student Thinking is an opportunity to formatively assess students to determine their understanding of concepts learned. Examples from the Teacher Resource include:

• Lesson 1-4, Understand Division with Fractions, Visual Learning, Key Concept, Pose Purposeful Questions, “How is dividing a whole number by a fraction similar to dividing a fraction by a whole number? Sample answer: You write the whole number as a fraction. You multiply by the reciprocal of the divisor.”
• Lesson 5-10, Relate Customary and Metric Units, Visual Learning, Do You Understand? Essential Question, “How can you use ratios to convert customary and metric units of measure? Students should know that to convert between customary and metric units they can find the conversion rate that relates the appropriate units and then find an equivalent rate or use dimensional analysis.” 
• Lesson 7-4, Find Areas of Polygons, Visual Learning, Example 3, Pose Purposeful Questions, “Why do you need to find the area of the polygon to find the amount of bamboo flooring needed? The polygon is decomposed into which shapes? How do you find the dimensions of the shapes? Why was addition used to find the area of the polygon instead of subtraction?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student materials and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning. 

Effective Mathematics Teaching Practices (ETP) side notes provide Before, During, and After suggestions regarding lesson implementation. Examples from the Teacher Resource include:

• Lesson 3-1, Understand and Represent Exponents, Solve & Discuss It!, ETP: Before, “1. Introduce the Problem: Provide sheets of paper as needed. 2. Check for Understanding of the Problem: Engage students by asking them to fold a piece of paper in half in as many different ways as possible.”
• Lesson 7-2, Solve Triangle Area Problems, Solve & Discuss It!, ETP: During, “3. Observe Students at Work: Students might notice that both triangles have the same base and height, so the triangles have the same area. Students might calculate the area of the parallelogram and divide it by 2 to find the area of each triangle. If needed, ask, is there anything you learned in the previous lesson that can help you solve this problem? For Early Finishers, pose the question, How would the problem change if points B and C were moved 2 units up?”
• Lesson 8-5, Summarize Data Using Measures of Variability, Solve & Discuss It!, ETP: After, “4. Discuss Solution Strategies and Key Ideas: Have students share their work. Have them discuss the connections between the dot plot that varied a little; all values on the dot plots are close to the median. Have students discuss the connections between the dot plots that varied a lot; the values on the dot plot are spread out from the median. Have students discuss how their points are distributed and whether they need to be symmetrically distributed around 6; they do not, as long as they are the same number of points on either side of 6 the median will remain the same. 5. Consider Instructional Implications: After presenting Example 2, have students find the interquartile range for both of their dot plots from the Solve & Discuss It. Have students articulate what the interquartile ranges mean in the context of the problem; the interquartile ranges show the range of numbers of fruits eaten by the middle 50% of the students.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Each Topic contains a Topic Opener, Math Background: Focus section that provides a discussion of the math content in the topic along with sample work and strategies that illustrate the underlying concepts to help teachers anticipate the works students will do. The Topic Opener also contains Advanced Concepts for the Teacher that provides examples and adult-level explanations of more advanced mathematical concepts related to the topic with explanations and examples to support teacher understanding of the underlying mathematical progressions. Examples from the Teacher Resource include:

• Topic 1, Use Positive Rational Numbers, Math Background, “Multiplying Decimals: The standard algorithm for multiplying decimals extends the algorithm for multiplying multi-digit whole numbers by combining it with properties of multiplication and exponents. However, the manner in which decimal multiplication is typically represented presents some apparent contradictions. Consider the product of 4.32 and 1.8. The partial products, in red, are integers. These results appear to be false. Likewise, the sum of the 3456 and 4300 appears to be 7.776, which is also false. These apparent contradictions arise because the original factors, 4.32 and 1.8, were each multiplied by the smallest integral power of 10 required to turn them into integers. This step is hidden in the standard algorithm above. The question has actually been changed so that integers are being multiplied. Now the partial products 3456 and 4320 make sense. But their sum 7776, is not the answer to the original problem.” Visual representations are provided.
• Topic 5, Understand and Use Ratio and Rate, Math Background, “Ratios as a Multiplicative Relationship: A ratio describes a multiplicative relationship between values. The comparison between the two values is not achieved by finding a difference, but rather by scaling. A ratio of 5 red to 1 blue does not describe relationships where there are 4 more red than blue, but rather that there are 5 times as many red as blue, or 5 red for every 1 blue. As the relationship is multiplicative, increasing just one quantity that is described by the ratio changes the ratio of the two quantities in a way that may be unintuitive. For example, you have 10 marbles with a ratio of 4 red to 1 blue. That means 8 marbles are red and 2 marbles are blue in the batch of 10. If 1 more red marble is added, the new ratio is not 5 red to 1 blue, but rather 9 red to 2 blue. 
• Topic 8, Display, Describe, and Summarize Data, Math Background, “Choosing a Data Displays. When choosing a data display, a number of key questions need to be considered: Is displaying individual data values important? Plots like dot plots and scatter plots include a point for every data value. In contrast, histograms show the data in given intervals and box-and-whisker plots show the range and interquartile range of data. Graphs such as dot plots quickly show clusters and gaps in the data, which can be hidden in a box-and-whisker plot. Is displaying the data as a part of a whole important? Graphs like pie charts and stacked bar charts show the proportion of a set of data that falls within a given category, unlike a regular bar graph which displays absolute but not relative amounts. Is displaying the distribution of the data important? Graphs such as histograms and dot plots help to show how the data is distributed. It makes the data range, outliers, and data skew more apparent. A box-and-whisker plot clearly shows the spread of the data and the median, but does not show the distribution as clearly as a histogram. Is identifying the relationship between sets of data important? Graphs like line plots and scatter plots may reveal how one variable relates to another. Scatterplots can be used to identify trend lines, which in turn can be used to estimate how the change in one variable affects a change in another.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Each Topic Opener contains a section Math Background: Coherence that summarizes the content connections through the materials to prior and future grades. Look Back illustrates connections to previously taught concepts and skills include those within the grade, across content, or across grades. Look Ahead illustrates connections within or across grades. Examples from the Teacher Resource include:

• Topic 2, Integers and Rational Numbers, Math Background, Look Back, “Grade 5: Rational Numbers - In Grade 5, students extended their understanding of decimal place value to the thousandths place. They graphed decimals on a number line to help them compare and round decimals. They also extended their ability to do computations with rational numbers to include adding, subtracting, multiplying, and dividing decimals and fractions. Graph Points on the Coordinate Plane- In Grade 5, students learned about the coordinate plane and graphed points in the first quadrant to solve real-world and mathematical problems.”
• Topic 4, Represent and Solve Equations and Inequalities, Math Background, Look Back, “Earlier in Grade 6: Numerical and Algebraic Expressions- In Topic 3, students developed understanding of algebraic expressions, including evaluating algebraic expressions, generating equivalent expressions, and simplifying algebraic expressions. Graph Rational numbers on the Coordinate Place-In Topic 2, students graphed points with rational c\coordinates on a coordinate plane.”
• Topic 7, Solve Area, Surface Area, and Volume Problems, Math Background, Look Ahead, “Grade 7: Solve Measurement Problems -In Grade 7, students will solve real-world and mathematical problems involving area, surface, area, and volumen of two-and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prism.Scale Drawings, students will solve problems involving scale drawings, including finding the actual area of a figure given a scale drawing. Three-dimensional Figures, students will describe the two-dimensional figures that result from slicing three-dimensional figures.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 provides a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

Each Topic Opener contains a Topic Planner that provides an overview of the Learning Objective, Essential Understanding, and Standards. The Content Overview Introduction also contains a breakdown of each Topic into lessons, objectives, and standards. Finally, the Teacher Resource Program Overview contains a Pacing Guide with Topic titles and number of instruction days required, “Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 to 2 days for the enVisionSTEM project and Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessment.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 contain some strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The online Teacher’s Resource Masters have Home School Connection Letters, in English and Spanish, for each Topic. The letters include information on the mathematical content and activities parents can do with their child to support the mathematical content. For example, Grade 6, Topic 6, Understand and Use Percents, “Dear Family, In this topic, your child will learn about percents. Your child will use models as he or she learns, including hundreds grids, number lines, and circle graphs. He or she will develop an understanding of the concept of percents as parts of 100, find percents of a number and find the whole given a part and the percent. For example, if the teacher states that 33% of the 24 students in her class are boys, a number line could be used to help find the number of boys. The equation 0.33 x 24 ≅ 8 could also be used to conclude that there are 8 boys in the class. You can help your child strengthen these skills by using the following activity.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 contain explanations of the instructional approaches of the program and identification of the research-based strategies.

enVision is based on research-based strategies. According to the Teacher Resource Program Overview, “enVision Mathematics embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.” Additionally, the core instructional model is based in research, “Over the past twenty years, there have been numerous research studies measuring the effectiveness of problem-based learning, a key part of the core instructional approach used in enVision Mathematics. These studies have found that students taught partly or fully through problem-based learning showed greater gains in learning. However, the interaction of problem-based learning, which fosters informal mathematical learning, and more explicit visual instruction that formalizes mathematical concepts with visual representations leads to the greatest gains for students. The enVision Mathematics instructional model is built on the interaction between these two instructional approaches.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Materials provide strategies for gathering students’ prior knowledge. Examples include:

• Grade Level Readiness Test diagnoses students’ readiness for learning by assessing prerequisite content. This assessment is also available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource. 
• Topic Readiness Assessment diagnoses students’ proficiency with Topic prerequisite concepts and skills. This assessment is available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource. 
• Review What You Know, found at the beginning of each Topic, checks for understanding of key math concepts previously learned. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide strategies for teachers to identify common student errors and misconceptions. 

Materials provide strategies to identity student errors. Prevent Misconceptions are found in the Teacher Resource sidenotes for the Visual Learning portion of the lesson and Error Interventions are found in the Practice & Problem Solving Section. Examples from the Teacher Resource include:

• Lesson 3-1, Understand and Represent Exponents, Practice & Problem Solving, Error Intervention, Item 32, “Students may not realize that numbers can be written using exponents in more than one way. Q: What rule do you use to evalue 10$$^0$$ and 1 1.0$$^0$$? [Any nonzero number raised to an exponent of 0 has a value of 1.] Q: Write 16 in two different ways using exponents.”
• Lesson 5-3, Compare Ratios, Practice & Problem Solving, Error Intervention, Item 9, “Students may not know how to answer because the question does not include a second ratio table. Q: What is the ratio of minutes of news to minutes of music for radio station WILM? [4:25] Q: Do you need to find the number of minutes of news that each station plays in one hour to answer the question? Explain. [No; Whichever station has a greater ratio of news to music will broadcast more news each hour.]”
• Lesson 8-5, Summarize Data Using Measures of Variability, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 3, “Explain that the mean absolute deviation shows the average distance each data value is from the mean. Q: What does MAD of 2 indicate? [On average, the data values are 2 units from the mean.] Q: What does MAD of 4 indicate? [On average, the data values are 4 units from the mean.] Q: Why does a greater MAD show greater variability? [It means that the data are more spread out from the mean.] ”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review of concepts and skills. Examples Include:

• Each Topic includes Review What You Know to activate prior knowledge and and review prerequisite skills needed for the Topic. Both vocabulary and practice problems are provided.
• The Cumulative/Benchmark Assessments are found at the end of Topics 2, 4, 6 and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the year. An item analysis is provided for diagnosis and intervention. Students can take the assessment online, with differentiated intervention automatically assigned to students based on their scores. 
• The Math Diagnosis and Intervention System has practice pages which are specific to a skill or strategy (i.e. Markups and Markdowns and Mental Math). 
• There are multiple pages of extra practice available at Pearson Realize online that give students extra opportunities to review skills assigned by the teacher. Each of these pages is able to be customized by the teacher or used as is.
• Different games online at Pearson Realize support students in practice and review of skills, as well procedural fluency.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials offer ongoing formative and summative assessments, clearly denoting which standards are being emphasized. 

Formative and summative assessments clearly denote standards being assessed. Examples include:

• Try It! and Convince Me! are found following the Visual Learning Examples and assess students’s understanding of concepts and skills presented in each Example and results can be used to modify instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Do You Understand? And Do You Know How? are found after the Visual Learning instruction and assess students’ conceptual understanding and procedural fluency and results can be used to review content. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Following each lesson is a Lesson Quiz that assesses students’ conceptual understanding and procedural fluency with the lesson content. Results can be used to determine differentiated instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• At the end of each Topic there is a Topic Assessment with 2 forms, Form A and Form B, that assesses students’ conceptual understanding and procedural fluency with the topic content. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• At the end of each Topic there is a Performance Task with 2 forms, Form A and Form B, that assess students’ ability to apply concepts learned and proficiency with math practices. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• Cumulative/Benchmark Assessments found at the end of Topics 2, 4, 5, and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the school year; results can be used to determine intervention. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials offer ongoing formative and summative assessments, which include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Following Lesson Quizzes, Topic Assessments, Topic Performance Task and Cumulative/Benchmark Assessments Scoring Guides are provided. Teachers can also assign these assessments online where they are auto-scored and differentiated intervention is automatically assigned to students based on their scores. Examples from the Teacher Resource include:

• Lesson 1-7, Solve Problems with Rational Numbers, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments. Intervention 0-3 Points. On-Level 4 Points. Advanced 5 Points. You may opt to have students take the Lesson Quiz online. The Lesson Quiz will be automatically scored and appropriate remediation, practice, or enrichment will be assigned based on student performance.”
• Topic 3, Numeric and Algebraic Expressions, Topic Assessment, Form A, “Greater Than 85%: Assign the corresponding MDIS for items answered incorrectly. Use Enrichment activities with the student. 70% - 85%: Assign the corresponding MDIS for items answered incorrectly. You may also assign Reteach to Build Understanding and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Less Than 70%: Assign the corresponding MDIS items answered incorrectly. Assign appropriate intervention lessons available online. You may also assign Reteach to Build Understanding, Additional Vocabulary Support, Build Mathematical Literacy, and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly.”
• Topic 7, Solve Area, Surface Area, and Volume Problems, Performance Task, Form A, Item 3, “Rafael needs to find the area of each section of the design to determine how much paint he needs. Part A. Describe a strategy to find the total area of the sections painted with each color. Part B. Find the total area of the sections painted with each color. Complete the table. Show your work.” Two charts are provided for the teacher, Item Analysis for Diagnosis and Intervention and Scoring Rubric for forms A and B. The Item Analysis for Diagnosis and Intervention Chart contains information to help the teacher with RTI such as DOK, MDIS, and standard. The scoring rubric provides the teacher with solutions and scoring explanations. “Item 3, Form A 2 Points: Correct response. 1 Point: Partially correct response.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 encourage students to monitor their own progress. 

Each Topic contains a Mid-Topic Checkpoint for students to monitor their understanding of concepts and skills taught in the first lessons of the Topic. Following the assessment students are asked, “How well did you do on the mid-topic checkpoint? Fill in the stars.” Three stars are provided.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The Topic Overview in the Teacher Resource provides a coherence section which enhances the opportunity to scaffold instruction by identifying prerequisite skills needed. All lessons include instructional notes and classroom strategies in the side matter labeled ETP, Effective Teaching Practices. ETP notes provide teachers with sample questions, differentiation strategies, discussion questions, possible misconceptions, and student “look fors” to assist in making content accessible to all learners. Additionally, the Solve and Discuss It! section provides teachers with Before, During, and After instruction notes to help scaffold learning for students. Examples from the Teacher Resource include:

• Lesson 4-2, Apply Properties of Equality, Example 3, ETP: Pose Purposeful Questions, “Q: Do you think adding 12 to one side and adding 12 to the other side would maintain equality? [Yes]. Q: What happens to the left side of the equation y - 12 + 12 = 30 + 12 when the expression is simplified? [The only thing left is y.] Q: What happens to the right side when the expression is true? Explain. [You get 42.] Q: How can you show that the equation is true? Explain. [Sample answer: Substitute 42 in for y in the original equation and then simplify. The equation is true because 30 = 30.]”
• Lesson 6-6, Find the Whole Given a Part and the Percent, Solve and Discuss It!, ETP: Before, “1. Introduce the Problem. Provide blank number lines, as needed. 2. Check for Understanding of the Problem. Ask students: How many games did your favorite team win last year?”
• Lesson 7-2, Solve Triangle Area Problems, Example 1, ETP: Use and Connect Mathematical Representations, “Q: How can you find the area of a parallelogram? [Multiply the base by the height]. Q: How do you know that the two triangles are identical? [Sample answer: One triangle can be rotated onto the other triangle. The heights, bases, and angles of the two triangles are the same.] Q: If the two different triangles have the same base and height, what do know about their areas? [The areas are the same.] Q: What relationship does the height of a triangle have to its base? [The height is perpendicular to the base.]”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Each lesson contains Response to Intervention and Enrichment strategies in each lesson. Additional Examples and Additional Practice are provided if students need more support. At the end of each lesson Differentiated Intervention is provided for Intervention, On-Level, and Advanced learners. Examples from the Teacher Resource include:

• Lesson 3-4, Write Algebraic Expressions, Response to Intervention, “Use with Example 3: Some students may need additional help identifying parts of an expression. Provide students with strips of paper and scissors. Have them write expressions with at least three terms on the strips and then trade the strips with a partner. Q: How many terms does the expression on your strip have? Q: Make a cut in your strip after each operation to separate the terms. How many pieces do you have? Students should check this answer against their answer to the first question. Discuss any discrepancies.”
• Lesson 5-7, Solve Unit Rate Problems, Enrichment, “Use with Example 3 Try It!, Challenge advanced students to use the distance equation to solve the following problems. Ask them to show their work. Q: At the same rate, how far would the submarine travel in 5 hours? Q: At the same rate, how long would it take the submarine to travel 285 miles? Q: What would the submarine’s rate be if it traveled 60 miles in 2$$\frac{1}{2}$$ hours?”
• Lesson 7-7, Find Surface Area of Pyramids, Differentiate Intervention, Reteach to Build Understanding, Problem 1, “Find the surface area of the square pyramid. First, complete the net below with four identical ____faces and one____ face.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Each lesson begins with a Problem-Based Learning activity, Solve & Discuss It, Explore It or Explain it! that offer multiple entry-points. 3-Act Mathematical Modeling tasks and Performance Tasks also include questions with multiple entry points that can be solved using a variety of representations. Examples from the Teacher Resource include:

• Topic 3, Numeric and Algebraic Expressions, Performance Task Form A, Item 1, “Ali is renting tables for the cookout and wants to seat an equal number of people at each table. She needs to decide how many tables to get. How could she arrange the seating so that a reasonable and equal number of people sit at each table? Explain.”
• Topic 5, Understand and Use Ratio and Rate, 3-Act Mathematical Modeling, Get in Line. Students are shown a video and then encouraged to consider the situation and ask any questions that come to mind. Teachers pose the Main Question, “How long will it take to get past the traffic light?” Teachers are given questions and tips to facilitate discussion about the 3-Act Mathematical Modeling activities. “Why do you think your prediction is the answer to the Main Question? Who had a similar prediction? How many agree with that prediction? Who has a different prediction?” 
• Lesson 7-6, Find Surface Area of Prisms, Solve & Discuss It!, “Marianne orders boxes to pack gifts. When they arrive, she finds flat pieces of cardboard as shown below. Marianne needs to cover each face of the boxes with green paper. What is the least amount of paper needed to cover the box? Explain. Suppose Melanie has only one large sheet of green paper that is 15 inches by 30 inches. Is the area of this sheet of paper great enough to cover all of the faces of one box? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Each lesson contains instructional strategies for Emerging, Developing, and Expanding English Language Learners. Additionally, the Language Support Handbook provides Topic and Lesson instructional support and online academic vocabulary activities. Examples from the Teacher’s Resource include:

• Lesson 2-3, Absolute Values of Rational Numbers, English Language Learners, “Emerging: As students work through Examples 1-3, ask: Q: What words do you not know? Circle them. Write or display the words with simple definitions. Refer to the terms as you discuss the examples. Q: What words in your language have the same meaning as order? Write or display these words as appropriate. Be sure to point out that order is a homonym. Elicit only words that have the same meaning as the word used in the lesson.”
• Lesson 5-4, Represent and Graph Ratios, English Language Learners, “Developing: Have students complete the following sentences stems for each equivalent ratio in Example 1. The ratio ____ is equivalent to 3:2. It can be written as the ordered pair ______. [6:4; (6,4)] The point ____ shows that the cost of _____ balloons is ____. [(6,4); 6; $4]”
• Lesson 7-2, Solve Triangle Area Problems, English Language Learners, “Expanding: Have students read Example 1 with partners. Write parallelogram, decompose, area, triangle, base and height on the board. Ask each pair of students to write the vocabulary terms on index cards. Partners should take turns drawing a card and using the word in context.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Each lesson provides an Enrichment side note with instructional strategies for advanced learners. The Problem-Based Learning activity provides instructional strategies During the lesson for Early Finishers. A Challenge question is presented in the teacher side notes for Practice & Problem Solving. Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Enrichment, “Use with Example 1 Try It!, Challenge advanced students to apply their answer to another situation. State the following question or write it on the board. May measures the temperatures at 5 A.M. Its is -4°F. At 9 A.M., the temperature is the opposite of its value at 5 A.M. What is the temperature at 9 A.M.? Q: What is the opposite of -4°F? Q: How much did the temperature rise from 5 A.M. to 9 A.M.?”
• Lesson 4-6, Understand and Write Inequalities, Solve & Discuss It!, ETP: During, “Early Finishers, Find some possible times Camilla would have to swim to break a record of 24.02 seconds.”
• Lesson 8-6 Choose Appropriate Statistical Measures, Practice & Problem Solving, Item 15, “Challenge: Reasoning, You can use this item to extend students’ understanding of mean, median, and mode. Q: If there were 7 data points could 12,500 occur three times?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics.

Different cultural names and situations are represented. Role names are used instead of pronouns referencing gender. Objects, animals, and cartoon drawings are used in place of actual people. Examples from the Teacher Resource include:

• Lesson 2-1, Understand Integers, Additional Practice, Item 30, students compare integers, “Roberto and Jeanne played a difficult computer game. Roberto’s final score was -60 points, and Jeanne’s final score was -160 points. Use <, >, or = to compare the scores, then find the player who had the higher final score.” 
• Lesson 4-1, Understand Equations and Solutions, Visual Learning, Example 2, three students are shown guessing the correct number of marbles: one caucasion female, one caucasion male, and one African American female.
• Lesson 7-3, Find Areas of Trapezoids and Kites, Practice & Problem Solving, Item 15, students decompose a shape into a trapezoid and then find the area, “A craftsman wants to build this symmetrical fiddle. He needs to know the area of the face of the fiddle. How could he use the measurement shown to find the area? Use your strategy to find the area of the face of the fiddle.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 provide opportunities for teachers to use a variety of grouping strategies.

Each lesson begins with a Problem-Based Learning activity which is introduced to the whole class. Then students break into small groups to work on the activity and come back together to discuss solutions and strategies as a whole class. Independent practice is found in the Problem & Practice Solving portion of the lesson. Icons in the Teacher’s Edition indicate whether the activity should be completed with Whole Class or Small Group.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 encourage teachers to draw upon home language and culture to facilitate learning. 

The Language Support Handbook provides research-based support strategies for English Language Learners, Academic Vocabulary Activities, a list of key vocabulary in 6 languages, and specific language support for each Topic Lesson. Digital and Student Edition Glossaries are in both English and Spanish. Assessments in Spanish can be accessed online. Each Topic’s Home-School Connection Letter explains the content of the Topic in English or Spanish.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 integrate technology including interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Examples include:

• The Math Practices and Problem Solving Handbook is an online reference available for students.
• Digital Desmos Activities provide embedded technology with engaging instruction of real-world content. 
• Visual Learning Animation Plus provides scaffold animations of learning with real aloud options to support English learners.
• Animated Glossary in digital resources provides math terms with support in English and Spanish.
• Math Practice Animations are online videos explaining the Practices and sample problems supporting the Practices.
• A variety of Interactive Math Tools are available online for students and teachers.
• Topic Readiness Tests and Lesson Quizzes taken online are automatically graded and remediation and enrichment activities are automatically assigned to students.

The digital instructional materials reviewed for enVision Mathematics Common Core Grade 6 are web-­based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are “platform neutral” (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The instructional materials reviewed for enVision Mathematics Common Core Grade 6 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. Examples include:

• Digital games that enhance fluency and provide opportunities for students to use procedural skills to solve problems are available online.  
• Virtual Nerd offers tutorials on a variety of math concepts with procedural skill emphasised.
• The online Readiness Assessment tab for each topic includes a Remediation link that has tutorials and opportunities for students to practice procedural skills using technology. 
• Fluency Practice Pages for each Topic are available online.

The digital materials reviewed for enVision Mathematics Common Core Grade 6 include opportunities for teachers to personalize learning for all students. Adaptive technology is not provided by digital materials. 

Digital materials include opportunities for teachers to personalize learning for all students. Examples include:   

• Teachers can select and assign individual practice items for student remediation based on the Topic Readiness assessment. If students take the test online it is automatically scored and students are automatically assigned enrichment or remediation activities.
• Teachers can create online classes and assignments for students.  
• Interactive Student Edition is accessible online and can be assigned to students.

The digital materials reviewed for enVision Mathematics Common Core Grade 6 can easily be customized for local use. Digital materials provide online materials for teachers to assign to students. Examples include:

• Interactive media lessons are accessible that cover all learning standards
• Lesson plans can be customized by day, week, or month or resequenced to match the district curriculum map.
• Outside content can be uploaded and Teacher Resource Masters can be customized.

The materials reviewed for enVision Mathematics Common Core Grade 6 include technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.). 

Teachers can create Online Discussion Boards and monitor student participation.