6th Grade - Gateway 2

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 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.”