The materials reviewed for enVision Mathematics Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials develop conceptual understanding throughout the grade level. According to the Teacher’s Edition’s Program Overview, “conceptual understanding and problem solving are crucial aspects of the curriculum.” In the Topic Overview, Math Background: Rigor, “Conceptual Understanding Background information is provided so you can help students make sense of the fundamental concepts in the topic and understand why procedures work.” Each Topic Overview includes a description of key conceptual understandings developed throughout the topic. The 3-Act Math Task Overview indicates the conceptual understandings that students will use to complete the task. At the lesson level, Lesson Overview, Rigor, the materials indicate the Conceptual Understanding students will develop during the lesson.

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. The Visual Learning Bridge and Guided Practice consistently provide these opportunities. Examples include: 

• Topic 1, Lesson 1-3, Lesson Overview, Conceptual Understanding states, “By making a 10 to add, students deepen their understanding of addition, and they begin to develop flexibility in their ability to add mentally.” In Guided Practice, Problem 1, students use different strategies to solve addition problems. “Make a 10 to add. Use counters and ten-frames.” The materials show two ten frames: one with seven red counters and the other with four yellow counters. Three yellow counters are moved to the first ten-frame to make 10. “10 + 1 = 11 so, 7 + 4 = ___ .” Students develop conceptual understanding as they use different strategies to fluently add numbers within 20. (2.OA.2)

• Topic 7, Lesson 7-6, Lesson Overview, Conceptual Understanding states, “Students find one missing number in equations that relate two whole numbers by addition or subtraction on each side. This structure brings focus to the understanding that in order for an equation to be true, both sides must have the same value.” In Convince Me!, students find a missing number to make the equation true and explain how they know.  “What number goes in the blank to make this equation true? Explain how you know. ___ - 5 = 14 - 7.” Students develop conceptual understanding as they use place value understanding and properties of operations to add and subtract. (2.NBT.B)

• Topic 12, Lesson 12-8, Lesson Overview, Conceptual Understanding states, “Students consider lengths in relation to one another as they compare lengths and find the difference.” In the Visual Learning Bridge, the materials show three frames: A)  “Which path is longer? How much longer?” A boy suggests, “Think about both parts of the path when you estimate and measure.” A blue path and a red path are estimated to be 5 cm and 6 cm, respectively. B) shows two centimeter rulers (one horizontal and one vertical), and the boy reasoning about the need to measure the two parts of the blue and red paths. “One part of the blue path is about 2 cm. The other part is about 2 cm. Add to find the length. 2 + 2 = 4 The blue path is about 4 cm long.” In addition, “One part of the red path is about 1 cm. The other part is about 4 cm. Add to find the length. 1 + 4 = 5 The red path is about 5 cm long.” C) shows “Subtract to compare lengths. 5 - 4 = 1” and the boy states, “The red path is about 1 cm longer than the blue path.” Classroom Conversation asks students the following questions: “A) Why do you need to think about both parts of the path when you estimate and measure? B) What do you need to measure the two paths of the blue and red paths? What are the lengths of the two parts of the blue path? The red path? What is the length of each path? C) Model with Math Which path is longer? How do you know? How much longer is the red path?” Students develop conceptual understanding as they work to determine which path is longer and by how much longer. Students develop conceptual understanding as they determine how much longer one object is than another and expressing the length difference using a standard unit of length. (2.MD.4)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Practice problems consistently provide these opportunities. Examples include:

• Topic 4, Lesson 4-1, Lesson Overview, Conceptual Understanding states, “Students use and draw models to develop understanding of the strategy of breaking numbers apart in order to add them using place value. Key goals of this topic are for students to understand place value and properties of operations and to become fluent in adding 2-digit numbers.” In Independent Practice, Problems 4–7, students use place value to find sums, regrouping when needed. Teacher prompts require students (i) to explain why it is important to write a 0 in the ones column and (ii) to determine which problems require regrouping in advance in order to avoid regrouping out of routine. “4. 36 + 29 = ___  5. 27 + 23 = ___  6. 59 + 13 = ___   7.  24 + 35 = ___” Students independently demonstrate conceptual understanding by fluently adding within 100 using strategies based on place value. (2.NBT.5)

• Topic 9, Lesson 9-5, Lesson Overview, Conceptual Understanding states, “Students understand that it takes 10 of a number in one place value to make a number in the next greater place value.” In Independent Practice, Problem 2, students use place-value blocks to count the hundreds, tens, and ones and then show two other ways to make the number. The materials show a place value mat that includes two pairs of 2 hundreds blocks, one tens block, and eight ones blocks and provide space for students to write three different equivalent representations of 418. Students independently demonstrate conceptual understanding by writing numbers in expanded form. (2.NBT.3)

• Topic 13, Lesson 13-3, Lesson Overview, Conceptual Understanding states, “Students deepen their understanding of 2-dimensional shapes by drawing polygons based on descriptions of attributes, including number of sides, number of vertices, number of angles, and lengths of sides. Students also draw polygons that include right angles.” In Independent Practice, Problem 3, students draw shapes based on a description and then complete sentences based on the shape they drew. “Draw a polygon with 3 vertices and 1 right angle. The polygon also has ___ sides. The polygon is a _______.” Students independently demonstrate conceptual understanding by identifying and drawing plane shapes that have specified attributes. (2.G.1) 

The materials reviewed for enVision Mathematics Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skills and fluency throughout the grade level within various portions of lessons. The Teacher’s Edition Program Overview indicates, “Students perform better on procedural skills when the procedures make sense to them. So procedural skills are developed with conceptual understanding through careful learning progressions. … A wealth of resources is provided to ensure all students achieve success on the fluency expectations of Grades K-5.” Various portions of lessons that allow students to develop procedural skills include Solve & Share, Visual Learning Bridge, Convince Me!, Guided Practice, and 3-ACT MATH; in addition, the materials include Fluency Practice Activities. Examples include: 

• Topic 3, Lesson 3-2, Lesson Overview, Procedural Skill states, “Students add tens and ones on an open number line, which involves decomposing and composing numbers and supports the development of place-value addition strategies as well as number sense and computational fluency.” In Guided Practice, Problem 2, the materials prompt “Use an open number line to find each sum.” The materials show a number line. Students complete the equation “47 + 25 = ___.” (2.NBT.5)

• Topic 7, Lesson 7-2, Lesson Overview, Procedural Skill states, “Students use drawings and equations to make sense of word problems and to solve word problems.” In Guided Practice, Problem 1, students develop procedural skills and fluency by labeling each part of a bar diagram to help them make sense of the numbers and solve word problems. The materials ask, “Lakota has 11 fewer magnets than Jeffrey. Lakota has 25 magnets. How many magnets does Jeffrey have?” The materials show a bar diagram that consists of one whole “?” and another with two parts “25, 11.” Students complete the equation “25 + 11 = __” and indicate the number of  magnets. (2.OA.1)

• Topic 11, Lesson 11-3, Lesson Overview, Procedural Skill states, “Students use concrete representations to reinforce their understanding of regrouping.” In Guided Practice, Problem 2, students develop procedural skills and fluency as they use blocks to find the difference 363 - 127. The materials provide a place-value mat that includes hundreds, tens, and ones. (2.NBT.7)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. Independent Practice and Problem Solving consistently include these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Topic 1, Lesson 1-1, Lesson Overview, Procedural Skill and Fluency states, “Students develop fluency with addition facts to 20 by using the counting on strategy along with changing the order of addends.” In Independent Practice, Problem 7, students independently demonstrate procedural skill and fluency as they count on to find sums of single-digit numbers and repeat the task upon changing the order of the addends. The materials show 7 + 10 = ___ and ___ + ___ = ___. (2.OA.2) 

• Topic 8, Lesson 8-3, Lesson Overview, Procedural Skill states, “Students find the value of a set of bills by counting on.” In Independent Practice, Problem 5, students independently demonstrate procedural skills and fluency as they solve word problems involving dollar bills and use $ appropriately. The materials pose the word problem, “Mr. Abreu has these dollar bills. Count on to find the total value.” The materials show five bills: three $1 bills and two $10 bills. (2.MD.8).

• Topic 14, Lesson 14-3, Lesson Overview, Procedural Skill states, “Students will create and solve equations to represent situations involving measurement. In Problem Solving, Problem 6, students independently demonstrate procedural skills and fluency when they use subtraction within 100 to solve a word problem that involves length. “Make Sense The yellow boat is 15 feet shorter than the green boat. The green boat is 53 feet long. How long is the yellow boat? Think about what you are trying to find. Write an equation to solve. Show your work.” (2.MD.5)

The materials for enVision Mathematics Grade 2 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Engaging applications—which include single and multi-step, routine and non-routine applications of the mathematics—appear throughout the grade level and allow for students to work with teacher support and independently. In each Topic Overview, Math Background: Rigor provides descriptions of the concepts and skills that students will apply to real-world situations. Each Topic is introduced with a STEM Project, whose theme is revisited in activities and practice problems in the lessons. Within each lesson, Application is previewed in the Lesson Overview. Practice & Problem Solving sections provide students with opportunities to apply new learning and prior knowledge.

Examples of routine applications of the math include:

• In Topic 5, Topic Performance Task, Problem 3, students use addition within 100 to solve a routine word problem. “Chen’s sisters have toy boats. They have 21 yellow boats. They have 9 fewer red boats than yellow boats. How many boats do they have in all? Choose any strategy. Show your work. ___ boats.” (2.OA.1)

• In Topic 8, Lesson 8-4, Problem Solving, Problem 8, students independently solve a word problem that involves dollar bills. “Make Sense Lily has two $10 bills, three $5 bills, and one $1 bill. She gives Grace $11. How much money does Lily have left? $___” The materials include the image of a boy saying, “How much is two $10 bills?” (2.MD.8)

• In Topic 13, Lesson 13-2, Problem Solving, Problem 12, students independently recognize and identify pentagons and hexagons. “Be Precise Which plane shapes are sewn together in the soccer ball?” The materials show the image of a soccer ball. (2.G.1)

Examples of non-routine applications of the math include:

• In Topic 2, Lesson 2-1, Problem Solving, Problem 11, students independently determine whether a group of objects (up to 20) has an odd or even number of members and write an equation to model the sum. “Model Tyrone puts 4 marbles in one jar. He puts 3 marbles in another jar. Does Tyrone have an odd or even number of marbles? Draw a picture to solve. Then write an equation.” Students fill in the blanks ___ + ___ = ____ and “Tyrone has an ___ number of marbles.” (2.OA.3)

• In Topic 7, Lesson 7-5, Problem Solving, Problem 7, students use the relationship between addition and subtraction within 100 to write a number story. “ Higher Order Thinking Write a two-step number story using the numbers 36, 65, and 16. Then solve the problem. Write equations to show each step.” (2.OA.1 and 2.NBT.5)

• In Topic 15, Lesson 15-1, Enrichment Activity, students independently generate measurement data by measuring lengths of several objects to the nearest whole unit and then showing the measurements on a line plot. “Choose 10 objects in the classroom that measure from 5 to 15 inches in length. Measure each object. Then record each length on the line plot.” A horizontal line plot marked off in whole-inch units from 5 to 15 is provided. Students respond to questions such as “2. How many objects measured more than 10 inches?” and  “4. What is the length of the longest object? ___inches” (2.MD.9)

​The materials for enVision Mathematics Grade 2 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

Each Topic Overview contains Math Background: Rigor, where the components of Rigor are addressed. Every lesson within a topic contains opportunities for students to build conceptual understanding, procedural skills and fluency, and/or application. During Solve and Share and Guided Practice, students explore alternative solution pathways to master procedural fluency and develop conceptual understanding. During Independent Practice, students apply the content in real-world applications, use procedural skills and/or conceptual understanding to solve problems with multiple solutions, and explain/compare their solutions.

The three aspects of rigor are present independently throughout the grade. For example:

• Topic 4, Lesson 4-6, Independent Practice, Problem 8, students attend to procedural skills and fluency as they recall and apply place-value concepts and properties of operations to add up to four two-digit numbers. “Add. Use any strategy. Show your work. 25 + 17 + 24 + 15 = ___ .” Students may reorder addends, break apart, make a ten, or draw tens and ones for each addend. (2.NBT.6)

• Topic 7, Lesson 7-5, Independent Practice, Problem 3, students attend to application as they use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. “Solve each problem any way you choose. Show your work. Sandy has 12 balloons. Tom has 11 more balloons than Sandy. Some of Tom’s balloons popped and now he has 14 balloons. How many balloons popped?” Students fill in the blank, “____ balloons popped.” (2.OA.1)

• Topic 13, Lesson 13-2, Problem Solving, Problem 14, students attend to conceptual understanding as they recognize and draw shapes having special attributes. “Higher Order Thinking Draw a polygon shape that has 7 angles. How many sides does the polygon have? How many vertices does it have?” (2.G.1)

Multiple aspects of Rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:

• Topic 5, Lesson 5-1, Problem Solving, Problem 17, students attend to conceptual understanding and procedural skills and fluency as they develop their ability to subtract mentally using place value. “Use Tools Use the hundred chart to solve the problems below. Higher Order Thinking Chris wants to subtract 76 - 42. Write the steps he can take to subtract 42 from 76 on the hundred chart.” (2.NBT.5)

• Topic 8, Lesson 8-4, Independent Practice, Problem 4, students attend to application and procedural skills and fluency as they solve word problems involving dollar bills. “Solve each problem. Show your work. Roberto buys a baseball mitt. He pays for it with $100 bill and receives $29 in change. How much does the baseball mitt cost?” Students apply skills with subtracting dollar amounts to solve word problems about money. (2.MD.8)

• Topic 15, Lesson 15-1, Solve & Share, students attend to conceptual understanding, procedural skills, and application as they generate measurement data by measuring lengths of objects and representing the measurements on a line plot. “Find four objects that are each shorter than 9 inches. Measure the length of each object to the nearest inch. Record the measurements in the table. Then plot the data on the number line. Which object is longest? Which is shortest?” The materials show a table with columns labeled “Object” and “Length in Inches” and a number line extending from 0 to 9 with whole-unit increments. (2.MD.9)

The materials reviewed for enVision Mathematics Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 4, Lesson 4-2, Independent Practice, Problem 8, students make sense of problems and persevere in solving them as they draw the second addend to get the total sum. “Add. Use place value. Draw blocks or use another way. Higher Order Thinking Draw the second addend.” The materials show the First Addend as two vertical lines and three dots and the Sum as 5 vertical lines and 8 dots. Students determine the Second Addend and report the number of tens and the number of ones. 

• Topic 11, Lesson 11-6, Independent Practice, Problem 2, students make sense of problems and persevere in solving them as they use place value to add and subtract numbers. “Use the table to solve each problem. Show your work. How much heavier is a grizzly bear than an arctic wolf and a black bear together?” The materials show a table, “Weights of Wild Animals (in pounds),” that lists the weights of an arctic wolf, a black bear, a grizzly bear, a mule deer, and a polar bear. Students identify the weight of the indicated animals: 990, 176, and 270, respectively. 

• Topic 14, Lesson 14-2, Problem Solving, Problem 8, students make sense of problems and persevere in solving them as they determine what a problem is asking, what information they know, and how they can use addition and/or subtraction within 100 to solve the problem. “Higher Order Thinking Jack jumped 15 inches. Tyler jumped 1 inch less than Jack and 2 inches more than Randy. Who jumped the farthest? How far did each person jump?”  

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the Topics. Examples include:

• In Topic 1, Lesson 1-1, Convince Me!, students reason abstractly and quantitatively as they determine if changing the order of the addends changes the sum. “Does $5+2=2+5$? How do you know?” 

• In Topic 7, Lesson 7-6, Independent Practice, Problem 11, students reason abstractly and quantitatively as they reason about how to use place-value understanding and properties of operations to find an unknown value that makes an equation true. “Write the missing number that makes each equation true. Show your work. ” 

• In Topic 12, Lesson 12-1, Convince Me!, students reason abstractly and quantitatively as they estimate the lengths of objects using units of feet and yards by relating the lengths of the objects to measurements they know. “Is your height closer to 4 feet or 4 yards? How do you know?”

The materials reviewed for enVision Mathematics Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning as well as corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 1, Lesson 1-7, Independent Practice, Problem 15, students construct viable arguments and critique the reasoning of others as they explain why they can add or subtract to make a 10 to solve subtraction problems. “Higher Order Thinking Carol subtracts 6 from 15. First, she adds to get to 10. Then she adds again to find her answer. Her answer is 10. Is Carol correct? Explain.” The materials suggest to teachers, “Remind students to provide reasons why Carol was correct or incorrect. Suggest that they make a 10 to find 15 - 6 on their own to check Carol’s work and find possible reasons she may be incorrect.”

• Topic 4, Lesson 4-4, Solve & Share, students construct viable arguments and critique the reasoning of others as they check if a suggested sum is correct and use partial sums to show their work as part of their argument. “Add 46 + 26. Explain how you solved the problem.” Teachers say, “What did you do first? Next? How did this help you find the sum?” and share student solutions that focus on how different strategies give the same sum. Teachers are prompted to use questions and additional work to help students construct viable arguments and critique the reasoning of others such as: “Based on your [teacher] observations, choose which solutions to have students share and in what order…If needed, show and discuss the student work at the right.” There are two pieces of work displayed at the right one is labeled Eli’s Work and the other is labeled Jay’s Work. The following questions are asked: “What strategies did Eli use to solve the problem? What mistake did Jay make?”

• Topic 10, Lesson 10-6, Solve & Share, students construct viable arguments and critique the reasoning of others as they use addition strategies to solve problems and explain why their and other strategies work. “Find 375 + 235. Explain your strategy. Then explain why your strategy works.” Teachers are prompted to use questions and additional work to help students construct viable arguments and critique the reasoning of others such as: “Based on teacher observations, choose which solutions to have students share and in what order…If needed, show and discuss the student work at the right.” There are two pieces of work displayed at the right one is labeled Isaac’s Work and the other is labeled Pippa’s Work. The following questions are asked: “Isaac used an open number line. He started at 375 and mentally added on 2 hundreds, 3 tens, and 5 ones to find the correct sum, 610. Why does Isaac’s strategy work? Pippa used partial sums to find the total sum. She added the hundreds, then the tens, and then the ones. Pippa then added all of the partial sums to find the correct sum, 610. Why does Pippa’s strategy work?”

• Topic 13, Lesson 13-1, Solve & Share, students construct viable arguments and critique the reasoning of others as they name and classify plane shapes by the number of sides and vertices they have. “Look at the picture. How many triangles can you find? Trace each triangle. Be ready to explain how you know you have found them all.” The materials show a large triangle that consists of three rows of triangles. Teachers are prompted to use questions and additional work to help students construct viable arguments and critique the reasoning of others such as: “Based on teacher observations, choose which solutions to have students share and in what order…If needed, show and discuss the student work at the right.” There are two pieces of work displayed at the right one is labeled Rebecca’s Work and the other is labeled Shawn’s Work. The following questions are asked: “Rebecca knows that a triangle has 3 sides. She correctly traced and counted all the triangles. How did she do this? Shawn found 9 triangles. He did not correctly find all the triangles. What mistake did he make?”

The materials reviewed for enVision Mathematics Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP4 and MP5 across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level.  The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 2, Lesson 2-5, Independent Practice, Problem 2, students model with mathematics as they use visuals, numbers, and symbols to solve real-world problems to find the total number of objects. “Draw a picture and write an equation to show each problem. Then solve. Mika has 4 rows of playing cards. If there are 4 playing cards in each row, how many cards does Mika have in all?” Students will model the problem, write an equation, and fill in the blank ___ cards. 

• Topic 8, Topic Performance Task, Problem 4, students model with mathematics as they represent time on an analog clock and write time in two different ways. “Ted walks to the toy store in the afternoon. Part A He starts walking at the time shown on the digital clock. Draw hands on the second clock to show the same time. Is the time on the clocks above 3:35 a.m. or 3:35 p.m.? Explain how you know. Part B Write the time on the clocks in two different ways.”

• Topic 15, Lesson 15-4, Problem Solving, Problem 11, students model with mathematics as they use picture graphs to represent and compare data. “Use the tally chart to complete the picture graph. Use the picture graph to solve the problems. Model Bob makes a tally chart to show the trees in a park.” The materials show a tally chart, “Trees in the Park,” that identifies four types of trees and a blank picture graph (key included). A boy states, “You can model data using a picture graph.”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students use appropriate tools strategically as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 3, Lesson 3-1, Problem Solving, Problem 13, students use appropriate tools strategically as they use a hundreds chart to solve problems. “Use Tools Use the hundreds chart to solve the problems. Sara has 48 buttons. Luis has 32 buttons. How many buttons do they have in all?” The materials show a portion of a hundreds chart that begins at 31 and ends at 100. Students fill in the blank, “___ buttons.”

• Topic 10, Lesson 10-3, Solve & Share, students use appropriate tools strategically as they use place-value blocks as a tool to find sums of 3-digit numbers: adding hundreds to hundreds, tens to tens, and ones to ones. “Use place-value blocks to find 243 + 354. Tell which place value you added first and why. Then draw a picture to show your work.” The materials show “243 + 354 = ___ .”  Teacher guidance: “How do students use the place value blocks to represent the addends? Students might only represent the first addend. If needed, ask When adding two numbers together, do you need to represent both numbers? Why?” 

• Topic 12, Lesson 12-2, Independent Practice, Problem 6, students use appropriate tools strategically as they estimate the measures of objects in inches and then use a ruler to measure the objects to the nearest inch to check their estimates.“Estimate the height or length of each real object. Then use a ruler to measure. Compare your estimate and measurement.” The materials show a standard box of 64 crayons, indicate that students should measure the “length of a crayon box,” and provide the incomplete statement, “about ___ inches” for both the estimate and the measurement.

The materials reviewed for enVision Mathematics Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP6 across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson-level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews. 

Students attend to precision in mathematics in connection to grade-level content as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 2, Lesson 2-1, Convince Me!, students attend to precision as they communicate their understanding of odd and even numbers using clear definitions in their discussion and reasoning. “You break apart a tower of cubes to make two equal parts, but there is one cube left over. Is the number of cubes even or odd? Explain.” Teacher guidance: “Be Precise Have students show two cube towers with 4 cubes in each tower. Have them add 1 cube to one of the towers. Ask students to discuss why the total number of cubes in all is an odd number.”

• Topic 6, Lesson 6-5, Independent Practice, Problem 9, students attend to precision as then choose and use strategies (e.g., regroup place-value blocks and partial differences) to accurately subtract 2-digit numbers. “Use any strategy to subtract. Show your work. Be ready to explain why your strategy works.” Students complete the equation “86 - 19 = ___” Teacher guidance: “Guide students … to think about which strategy would be good to use to solve that particular problem.”

• Topic 12, Lesson 12-4, Visual Learning Bridge and Guided Practice, Problem 2, students attend to precision when they estimate and  measure the length and height of objects in inches, and feet. Visual Learning Bridge (B), “Measure the bookcase in feet.” The materials show a bookcase with three one-foot rulers spanning its top surface and a boy who states, “It is about 3 feet long.” Teacher guidance: “Attend to Precision Explain to students that when they measure an object that is longer than one ruler, they have to position the next ruler exactly where the last ruler ends with no gaps or overlapping.” Problem 2, “Measure each real object using different units. Circle the unit you use more of to measure each object.” The materials show a red chair. Students fill in the blanks and circle the appropriate unit: “about ___ inches about ___ feet I use more of: inches feet.”

Students attend to the specialized language of mathematics in connection to grade-level content as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 1, Lesson 1-2, Convince Me!, students use specialized language when they use math terms such as “doubles” and “near doubles” to communicate precisely and solve problems. “How could you use the doubles fact 7 + 7 to find 7 + 9?” Teacher guidance: “Be Precise Encourage students to give a clear, correct, and precise explanation. Suggest that they use math terms … to communicate precisely.”

• Topic 8, Lesson 8-7, Problem Solving, Problem 7, students use specialized language as they learn different ways to say and write the same times. “Higher Order Thinking Look at the clock to solve each problem. What time will it be in 50 minutes? Write this time in two different ways.” The materials show an analog clock that shows the time 9:40.

• Topic 13, Lesson 13-2, Problem Solving, Problem 15, students use specialized language when they identify shapes represented by real-world objects. “Assessment Practice Name the shape of the sign below. Write 3 things that describe the shape.”

The materials reviewed for enVision Mathematics Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are explicitly identified for teachers within the Program Overview within the Topic Contents at the Lesson Level. The Math Practices and Problem Solving Handbook introduces each of the Math Practices with specific emphasis on making connections among representations to develop meaning and corresponding Thinking Habits. The Teacher’s Edition provides support for developing, connecting, and assessing each math practice. Topic Planners include the Math Practices at the lesson level; relevant practices are specified in Lesson Overviews.

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 1, Lesson 1-2, Independent Practice, Problem 13, students use and look for structure as they discern a pattern and use their knowledge of the doubles fact pattern to solve near doubles problems. “Complete the doubles facts. Use the doubles facts to solve the near doubles. Use cubes if needed.” Students complete the doubles fact “5  + 5 = ___” to solve the near doubles problem “5 + 7 = ___.”

• Topic 9, Lesson 9-6, Problem Solving, Problem 12, students use and look for structure when they use place-value patterns to count by 10s. “Look for Patterns Yoshi sees a pattern in these numbers. Describe the pattern.” The materials show the numbers “341, 351, 361, 371, 381, 391.” Teacher guidance: “What would the next number in the pattern be? Discuss how both the hundreds digit and the ten digit change.”

• Topic 13, Lesson 13-2, Independent Practice, Problem 5, students use and look for structure when they name polygons by counting their number of angles. “Write the number of angles and then name the shape.” The materials show a convex pentagon. Students fill in the blanks: ___ angles Shape: ___ ”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with the support of the teacher and independently throughout the Topics. Examples include:

• Topic 5, Lesson 5-5, Independent Practice, Problem 11, students look for and express regularity in repeated reasoning when they use compensation to make numbers that are easier to subtract. “Higher Order Thinking Yoshi says that to find 91 - 32, he can subtract 2 from both numbers. Then he can subtract using mental math. He says the answer is 59. Do you agree? Explain. 

• Topic 9, Lesson 9-8, Independent Practice, Problem 12, students look for and express regularity in repeated reasoning when they use their understanding of place value to find a number that makes all comparisons true.  “Higher Order Thinking Find one number that will make all three comparisons true.” Students complete the following comparisons: ___ < 111, ___ > 109, and ___ = 110. 

• Topic 13, Lesson 13-8, Solve & Share, students look for and express regularity in repeated reasoning to create designs in equal shares. “Design two different flags. Draw 15 equal size squares in each flag. Use rows and columns. Make three equal shares of different colors in each flag. Then write an equation for each flag to show the total number of squares.” The materials show two rectangles: one each with a horizontal and vertical orientation, Students show two different ways to divide a design and use repeated addition to write equations for their designs.

The materials reviewed for enVision Mathematics Grade 2 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

The Teacher’s Edition Program Overview provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: Overview of enVision Mathematics, User’s Guide, Correlation and Content Guide.

• The Overview provides the table of contents for the course as well as a pacing guide for a traditional year long course as well as block/half year course. The authors provide the Program Goal and Organization, in addition to information about their attention to Focus, Coherence, Rigor, the Math Practices, and Assessment.

• The User’s Guide introduces the components of the program and then proceeds to illustrate how to use a ‘lesson’: Lesson Overview, Problem-Based Learning, Visual Learning, and Assess and Differentiate. In this section, there is additional information that addresses more specific areas such as STEM, Building Mathematical Literacy, Routines, and Supporting English Language Learners.

• The Correlation provides the correlation for the grade.

• The Content Guide portion directs teachers to resources such as the Big Ideas in Mathematics, Scope and Sequence, Glossary, and Index.

Within the Teacher’s Edition, each Lesson is presented in a consistent format that opens with a  Lesson Overview, followed by probing questions to provide multiple entry points to the content, error intervention, supports for English Language Learners and ends with multiple Response to Intervention (RtI) differentiated instruction.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Teacher’s Edition includes numerous brief annotations and suggestions at the topic and lesson level organized around multiple mathematics education strategies and initiatives, including the CCSSM Shifts in Instructional Practice (i.e., focus, coherence, rigor), CCSSM practices, STEM projects, and 3-ACT Math Tasks, and Problem-Based Learning. Examples of these annotations and suggestions from the Teacher’s Edition include:

• Topic 1, Lesson 1-6, Independent Practice, Problem 5, “Subtract. Complete the addition fact that can help you.” Students fill in the blanks.  8-1 = ___ ; 1 + ___ = 8. Teacher guidance: “Encourage students to think about how subtraction is related to addition as they work through the problems. If needed, have them practice writing related addition and subtraction equations.”

• Topic 3, Lesson 3-1, Visual Learning Bridge, Essential Question, “Ask How can you use patterns on a hundred chart to help you add numbers mentally?” Teachers begin the Classroom Conversation by saying the following, “Use Appropriate Tools Strategically What tool do you see here? [Part of a hundred chart] Why do we start at 54? [It is the greater of the two addends.] How many tens are in 18? [One] How can you add 1 ten on the hundreds chart? [Move down one row] How can you add 8 ones? [Move right 6 spaces to 70 and then move 2 more to 72.]”

• Topic 6, Lesson 6-3, Problem Solving, Problem 13, “Higher Order Thinking Write a subtraction story using two 2-digit numbers. Then solve the problem in your story.” Teacher guidance: “Higher Order Thinking After students have written their stories, ask them to explain if their stories compare two numbers, show a take-away situation, or describe a part-part-whole problem. Showcase each of those different types of subtraction problems as students share their stories.” 

The materials reviewed for enVision Mathematics Grade 2 meet expectations for containing adult-level explanations and examples of the more complex grade concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

The materials provide professional development videos at two levels to help teachers improve their knowledge of the grade they are teaching.

• “Professional development topic videos are at SavvasRealize.com. In these Topic Overview Videos, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The video is a quick, focused ‘Watch me first’ experience as you start your planning for the topic.

• Professional development lesson videos are at SavvasRealize.com. These Listen and Look for Lesson Videos provide important information about the lesson.

An example of the content of a Professional development video:

• Topic 2: Professional Development (topic) Video, “The Understanding of equal groups prepares students for the concept of multiplication and division by understanding the concept of even and odd and arrays … Students should build on their experiences with doubles by decomposing or breaking apart a number into two equal addends into equal groups. … When students have an understanding of equal groups it supports their development of multiplication and division.”

The Math Background: Coherence, Look Ahead section, provides adult-level explanations and examples of concepts beyond the current grade as it relates what students are learning currently to future learning.

An example of how the materials support teachers to develop their own knowledge beyond the current grade:

• Topic 14, Math Background: Coherence, Look Ahead, the materials state, “Grade 3 One-and Two-Step Word Problems In Topic 11, students will use letters to represent unknown quantities when solving one- and two-step word problems. Solve Word Problems Involving Time, Mass, and Liquid Volume In Topic 14, students will use number lines and bar diagrams to represent word problems involving other measurable attributes: time, liquid volume, and mass.”  An example word problem with the solution is shown. “Perimeter In Topic 16, students will solve problems involving perimeter.” An example word problem is shown, with a diagram of a floor of a room, teachers are tasked with writing an equation to find the missing length of the room. 

The materials reviewed for enVision Mathematics Grade 2 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Standards correlation information is indicated in the Teacher’s Edition Program Overview, the Topic Planner, the Lesson Overview, and throughout each lesson. Examples include:

• The Teacher’s Edition Program Overview, Grade 2 Correlation to Standards For Mathematical Content organizes standards by their Domain and  Major Cluster and indicates those lessons and activities within the Student’s Edition and Teacher’s Edition that align with the standard. Lessons and activities with the most in-depth coverage of a standard are distinguished by boldface. The Correlation document also includes the Mathematical Practices. Although the application of the mathematical practices can be found throughout the program, the document indicates examples of lessons and activities within the Student’s Edition and Teacher’s Edition that align with each math practice.

• The Teacher’s Edition Program Overview, Scope & Sequence organizes standards by their Domain, Major Cluster, and specific component. The document indicates those topics that align with the specific component of the standard.

• The Teacher’s Edition, Topic Planner indicates the standards and Mathematical Practices that align to each lesson.

The Teacher’s Edition, Math Background: Coherence provides information that summarizes the content connections across grades. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:

• Topic 3, Math Background: Coherence, the materials highlight two of the learnings within the topics: “Decompose Tens and Ones, and Mental Math” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 3 connect to what students will learn later?” and provides a Grade 3 connection, “Add and Subtract Within 1,000 In Topic 8, students will use place-value understanding, properties of operations, and the relationship between addition and subtraction to add and subtract within 1,000. In Topic 9, students will fluently add and subtract within 1,000.”

• Topic 8, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Solve Problems with Coins and Bills, Tell Time, and Count Money and Tell Time” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 8 connect to what students will learn later?” and provides a Grade 3 connection, “Time to the Minute and Elasped Time In Topic 14, students will learn to tell time to the nearest minute. They will also learn to measure elapsed time and to solve word problems involving time. To find time to the nearest minute, students count by 5s and then 1s. The clock shows 10:27.”

• Topic 12, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Estimate and Measure Length, Use Different Length Units, and Addition, Subtraction, and Length” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 12 connect to what students will learn later?” and provides a Grade 3 connection, “Measure to the Nearest Fourth Inch and Half Inch In Topic 12, students will extend their understanding of measuring lengths in customary units to measuring to the nearest fourth inch or half inch.”

The materials reviewed for enVision Mathematics Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Teacher’s Edition Program Overview provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted all examples are found in the Teacher’s Edition Program Overview.

Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

• The Program Goal section states the following: “The major goal in developing enVision Mathematics was to create a program for which we can promise student success and higher achievement. We have achieved this goal. We know this for two reasons. 1. EFFICACY RESEARCH First, the development of enVision Mathematics started with a curriculum that research has shown to be highly effective: the original enVisionMATH program (PRES Associates, 2009; What Works Clearinghouse, 2013). 2. RESEARCH PRINCIPLES FOR TEACHING WITH UNDERSTANDING The second reason we can promise success is that enVision Mathematics fully 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.”

• The Instructional Model section states the following: “There has been more research in the past fifteen years showing the effectiveness of problem-based teaching and learning, part of the core instructional approach used in enVision Mathematics, than any other area of teaching and learning mathematics (see e.g., Lester and Charles, 2003). Furthermore, rigor in mathematics curriculum and instruction begins with problem-based teaching and learning. … there are two key steps to the core instructional model in enVision Mathematics. STEP 1 PROBLEM-BASED LEARNING Introduce concepts and procedures with a problem-solving experience. Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014)... STEP 2 VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The important mathematics is the new concept or procedure students should understand (Hiebert, 2003; Rasmussen, Yackel, and King, 2003). Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions from the Solve and Share task…”

• Other research includes the following:

  • Hiebert, J.; T. Carpenter; E. Fennema; K. Fuson; D. Wearne; H. Murray; A. Olivier; and P.Human. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann, 1997.

  • Hiebert, J. (2003). Signposts for teaching mathematics through problem solving. In F. Lester, Jr. and R. Charles, eds. Teaching mathematics through problem solving: Grades Pre-K–6 (pp. 53–61). Reston, VA: National Council of Teachers of Mathematics.

• Throughout the Teacher’s Edition Program Overview references to research-based

The materials reviewed for enVision Mathematics Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the online Teacher Resources for each grade, a Materials List is provided in table format identifying the required materials and the topic(s) where they will be used. Additionally, the materials needed for each lesson can be found in the Topic Planner and the Lesson Overview. Example includes:

• Topic 1, Topic Planner, Lesson 1-2, Materials, “Counters (or Teaching Tool 6), Connecting cubes (or Teaching Tool 5)”

• Topic 6, Lesson 6-1, Lesson Resources, Materials, “Place-value blocks (or Teaching Tool 19), Place-Value Mat A (or Teaching Tool 26)”

• Teacher Resources, Grade 2: Materials List, the table indicates that Topic 12 will require the following materials: “1-inch squares (or Teaching Tool 39), Buttons or counters (or Teaching Tool 6), Centimeter rulers and meter sticks (or Teaching Tool 40), ...”

The materials reviewed for enVision Mathematics Grade 2 meet expectations for including an assessment system that provides multiple opportunities throughout the grade  to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

The assessment system provides multiple opportunities to determine student’s learning throughout the lessons and topics. Answer keys and scoring guides are provided. In addition, teachers are given recommendations for Math Diagnosis and Intervention System (MDIS) lessons based on student scores. If assessments are given on the digital platform, students are automatically placed into intervention based on their responses.

Examples include:

• Topic 2, Lesson 2-4, Independent Practice, Evaluate, Quick Check, Problem 4, “A check mark indicates items for prescribing differentiation on the next page. Items 4 and 9: up to 2 points each; Item 10: 1 point.” For example, Directions: “Draw an array to show each problem. Use repeated addition to solve. 4. Kristin makes an array with her books. She places them in 5 rows with 4 books in each row. How many books does Kristin have in all?” Students complete the equation ___ + ___ + ___ + ___ + ___  = ___ books. The following page, Step 3: Assess and Differentiate states,  “Use the Quick Check on the previous page to prescribe differentiated instruction. I Intervention 0-3 points, O On-Level 4 points, A Advanced 5 points.” The materials provide follow-up activities—to be assigned at the teacher’s discretion—to students at each indicated level: Intervention Activity I, Technology Center I O A, Reteach to Build Understanding I,  Build Mathematical Literacy I O, Enrichment O A, Activity Centers I O A, and Additional Practice Leveled Assignment I Items 1-3, 5, O Items 1-5, and A Items 1-5.

• Topic 6, Topic Assessment Masters, Problem 4, “Dan has 58 shells. Kelsey has 6 fewer shells than Dan. Kelsey gives 8 shells to her friend. Part A Which pair of equations should be used to find how many shells Kelsey has now? (A) 58 - 6 = 52, 52 - 8 = 44 (B) 58 - 6 = 52, 52 + 8 = 60 (C) 58 + 6 = 64, 64 - 8 = 56 (D) 58 + 6 = 64, 64 + 8 = 72. Part B How many shells does Kelsey have now? ___ shells” Item Analysis for Diagnosis and Intervention indicates: DOK 2; MDIS C26, E2, and E3; Standard 2.OA.A.1.” Scoring Guide indicates: “4A 1 point, Correct choice selected; 4B 1 point, Correct solution.”

• Topic 9, Topic Performance Task, Problem 5, “Reading Record These students love to read! These books show the number of pages each student has read so far this year.” The materials show Ken’s book 512 pages, Luisa’s book 493 pages, Ruth’s book 427 pages, and Tim’s book 378 pages. “Diane reads the same number of minutes each week. How many minutes does she read after 4 weeks? After 5 weeks? Skip count on the number line above to find the answers.” Students fill in the blanks: “After 4 weeks: ___ minutes; After 5 weeks: ___ minutes.” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS A40, Standard 2.NBT.A.2, 2,NBT.B.8, MP7. Scoring Guide indicates: 2 points “Two correct answers” and 1 point “One correct answer.”

• Topics 1-12, Cumulative/Benchmark Assessment, Problem 12, “What is the value of the 2 in the number 283? What is the expanded form of 283?” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS A33 and A34, Standard 2.NBT.A.1, 2.NBT.A.3.1. Scoring Guide indicates: “2 points “Correct value of digit in number AND correct expanded form” and 1 point “Correct value of digit in number OR correct expanded form.”

The materials reviewed for enVision Mathematics Grade 2 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The materials provide formative and summative assessments throughout the grade as print and digital resources. As detailed in the Assessment Sourcebook, the formative assessments—observational tools, Convince Me!, Guided Practice, and Quick Checks—occur during and/or at the end of a lesson. The summative assessments—Topic Assessment, Topic Performance Task, and Cumulative/Benchmark Assessments—occur at the end of a topic, group of topics, and at the end of the year.  The four Cumulative/Benchmark Assessments address Topics 1-4, 1-8, 1-11, and 1-15. 

• Observational Assessment Tools “Use Realize Scout Observational Assessment and/or the Solve & Share Observation Tool blackline master.”

• Convince Me! “Assess students’ understanding of concepts and skills presented in each example; results can be used to modify instruction as needed.”

• Guided Practice “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.”

• Quick Check “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to prescribe differentiated instruction.”

• Topic Assessment “Assess students’ conceptual understanding and procedural fluency with topic content.” Additional Topic Assessments are available with ExamView.

• Topic Performance Task “Assess students’ ability to apply concepts learned and proficiency with math practices.

• Cumulative/Benchmark Assessments “Assess students’ understanding of and proficiency with concepts and skills taught throughout the school year.”

The formative and summative assessments allow students to demonstrate their conceptual understanding, procedural fluency, and ability to make application through a variety of item types. Examples include: 

• Order; Categorize

• Matching

• Graphing

• Yes or No; True or False

• Number line

• True or False

• Multiple choice

• Fill-in-the-blank

• Technology-enhanced responses (e.g., drag and drop)

• Constructed response (i.e., short and extended responses)