2nd Grade - Gateway 2

​The instructional materials for enVision Mathematics Common Core Grade 2 meet the expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

The structure of the lessons include several opportunities that address conceptual understanding. For example:

• Math Background: Rigor page contains information about where conceptual understanding is built within the topic
• The Lesson Overview includes a narrative on how conceptual understanding is included in the lesson.
• Solve & Share activity whose purpose is “to elicit productive struggle that builds understanding by connecting prior knowledge to new ideas.”
• Lessons are introduced via video, Visual Learning Animation Plus, at PearsonRealize.com building on conceptual understanding.
• Students have the opportunity to independently demonstrate conceptual understanding through Independent Practice and Problem Solving pages within lessons.

Materials include problems and questions developing conceptual understanding throughout the grade-level and provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. For example:

• In Lesson 1-6, Solve and Share, students answer the question, “How can you use an addition fact to solve 14 - 6?” and “use counters to show how.” (2.OA.2)
• In Lesson 5-4, Visual Learning Bridge, features a demonstration of how to break apart one of the addends (6 into 3+3) in order to make finding 33 - 6 easier. Convince Me!, students “look at the problem above. Why wasn’t the 6 broken apart into 1+5 to find 33 - 6?” (2.NBT.7, 2.NBT.9)
• In Lesson 9-8, Solve & Share, students solve, “Jay and Zach flipped three number cards. Then they each made a number. Jay made 501. Zach made 510. Who made the greater number? How do you know? Use place-value blocks to help you solve.”  (2.NBT.4)
• In Lesson 10-5, Guided Practice, “Add. Use partial sums. Show your work. Use place-value blocks if needed. Item 1, 425 + 148 = ___. “ There is a place value chart with room for partial sums with answers dotted for students to trace. (2.NBT.7)

​The instructional materials for enVision Mathematics Common Core Grade 2 meet the 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. 

In the Teacher Edition, each Topic begins with Math Background: Rigor, where procedural skill and fluency for the topic is outlined for teachers. The structure of the lessons include several opportunities to develop procedural skill and fluency, including:

• Math Background: Rigor page contains information about where procedural skill and fluency is built within the topic
• The Lesson Overview includes a narrative on how procedural skills are addressed in the lesson, when applicable. 
• A Steps to Fluency Success chart details steps to move students to fluency and provides resources to use for practice, intervention, and enrichment. 

Later Topics include Additional Practice and Fluency worksheets, Math Diagnosis and Intervention Systems, and My Fluency Progress Forms. Additional practice is located online at PearsonRealize.com. 

Materials include items and questions intended to develop procedural skill and fluency throughout the grade-level and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. For example:

• In Lesson 1-2, Solve & Share, students “use counters. Show 6+6=12. Then show and explain how knowing that fact can help you find 6 + 7.”  (2.OA.2) 
• In Lesson 1-8, Guided Practice, Items 1-12,  “Add or subtract. Use any Strategy.” Item 1, “14 - 9 = ”.  (2.OA.2)
• In Lesson 3-5, Independent practice, Items 1-11, “Find each sum. Use any strategy. Show your work.” Item 1, “33 + 52 = __”, Item 2, “27 + 6 = __” (2.NBT.B.5)  
• In Lesson 6-5, Guided Practice, “Use any strategy to subtract. Show your work.  Draw blocks if needed. Explain why the strategy works.” Item 1, “67 - 39 = ” (2.NBT.5)
• In Lesson 12-9, Guided Practice, students look at a picture of a crayon lined up with a centimeter ruler that has space before the 0. Item 1. “Bev measures the crayon and says it is 5 centimeters long. Is her work precise? Explain.” The answer is “the work is not precise, because the crayon is not lined up with the 0 mark, it is lined up with the end of the ruler.” (2.MD.1)

The instructional materials for enVision Mathematics Common Core 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 include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

In the Teacher Edition, each Topic begins with Math Background: Rigor, where applications for the topic are outlined for teachers. Math Background: Rigor for Topic 7, Applications states, “Real-World Contexts In Lessons 7-1 to 7-5, students use addition and subtraction within 100 to solve one- and two-step real-world word problems.” Each Topic also includes a variety of application tasks, for example:

• Topic Opener, containing a contextual STEM problem designed to spark interest in the content of the topic,
• Topic Centers with application problems, 
• 3-Act Math activities where students engage in application problems, and
• Performance Tasks, where students apply mathematics of the topic in multi-step, real-world situations. 

The structure of the lessons includes several opportunities for students to engage in routine and non-routine application problems. Practice & Problem Solving sections provide students with a variety of problem types to apply what they have learned. The way in which application is incorporated into specific lessons is stated in the Rigor section of the Lesson Overview of those lessons. 

Examples of opportunities for students to engage in routine and non-routine application problems include:

• In Lesson 2-5, Solve and Share, students solve “There are 4 rows in a classroom. Two rows have 3 tables in each row. Two rows have 4 tables in each row.  How many tables are there in all? Draw a picture and write an equation to model and solve the problem.” (2.OA.4)
• In the Topic 5 Performance Task, Item 3, students solve, “Nina sees 34 red roses. She sees 9 fewer yellow roses than red roses. How many roses does she see in all?” (2.OA.1)
• In Lesson 7-4, Solve and Share, students solve, “3 bees land on some flowers. 10 more bees join them. Then 4 bees fly away. How many bees are left? Solve the problem any way you choose. Write equations to show how you solved each part of the problem.” (2.OA.1)
• In Lesson 14-5, Visual Learning Bridge, students solve, “Sara plays soccer. She is 56 feet away from the goal. Then she runs 24 feet straight toward the goal. How many feet from the goal is Sara now?” (2.MD.5)
• In Lesson 14-2, Solve and Share, students apply addition and measurement skills to determine which two pieces of yarn have a total length of 12 cm. “Julie and Steve each cut a piece of yarn. The total length of both pieces is 12 cm. Measure each piece of yarn. Circle Julie and Steve’s pieces. Then explain your thinking.” (2.MD.5)
• In Topic 7, Pick a Project - The Chrysler Building, students solve, “Find the total number of floors in the Chrysler Building. In 1930, there was a viewing deck. Do research to find what floor the deck was on. Then find the number of floors from the viewing deck to the top floor. Write two equations you could use to find the number.” (2.OA.1, 2.NBT.5)
• In Topic 14, Performance Task, Item 6, students see a picture of a car pulling a boat and the measurements of each component: car - 7 feet, space between car and boat - 2 feet, and boat - 21 feet. Students solve, “Jim’s family meets a man with a big boat. A parking spot at the dock is 32 feet long. Will the man’s car and boat fit in the parking spot?” (2.MD.5, 2.MD.6, 2.OA.1)

Examples of where instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts include:

• In Lesson 2-5, Independent Practice, Item 4, students solve, “Tina drew an array to show 9 shells. The same number of shells are in each row and each column. How many shells are in each row and each column? Explain how you know.” (2.OA.4)
• In Lesson 7-2, Solve and Share, students solve, “Aiden has 27 fewer crayons this week than last week. Last week he had 56 crayons. How many crayons does Aiden have this week?” (2.OA.1)
• In Lesson 7-5, Problem Solving, Item 7, students “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,  2.NBT.5)
• In Lesson 8-2, Independent Practice, Item 4, students solve, “Trina buys a ring. She pays for it with 9 dimes. She receives 8 pennies in change. How much did the ring cost?” (2.MD.8)

​The instructional materials for enVision Mathematics Common Core 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 skill 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.

In some instances, the three aspects of Rigor are present independently throughout the instructional materials. For example:

• In Lesson 9-5, Solve and Share, students build conceptual understanding as they, “Use place-value blocks. Show two ways to make 213. Then draw each way. Tell how your ways are alike and different.” (2.NBT.3)
• In Lesson 14-4, students represent addition and subtraction with whole numbers on a number line. In Independent Practice, Items 3 and 4, students “use the number lines to add or subtract 80 - 35 and 19 + 63.”  (2.MD.6)
• In Lesson 4-4, Lesson Overview, “Using partial sums helps students develop fluency in adding within 100.” In Independent Practice, Item 4, students “Write the addition problem. Use partial sums. Add any way you choose”  as they solve 15 + 28. (2.NBT.5)
• In Lesson 10-5, Solve and Share, engages students with application: “On Monday, 248 people visit the museum. On Tuesday, 325 people visit the museum. How many people visit the museum on Monday and Tuesday? Solve any way you choose. Be prepared to explain your thinking.” (2.NBT.7)

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

• In Topic 3, Lesson 3-3, Lesson Overview, “Conceptual Understanding: Students deepen their conceptual understanding of addition when they break apart numbers to add. Procedural Skill: Students break apart the second addend into tens and ones to add to the first addend. This helps students develop fluency with addition within 100.” This is illustrated in Convince Me!, when students “explain how you can break apart 28 to find 33+28.” (1.NBT.7)
• In Lesson 2-5, Solve and Share, students apply their understanding of equal groups to solve the word problem, “There are 4 rows in a classroom. Two rows have 4 tables in each row. How many tables are there in all? Draw a picture and write an equation to model and solve the problem.” (2.OA.4)
• In Lesson 9-1, Lesson Overview, “Conceptual Understanding: Students understand that 100 can be thought of as a group of 10 tens. They learn that a number such as 300 can be expressed as 3 hundreds, 0 tens, and 0 ones. Procedural Skill: Students use drawings of place-value blocks to demonstrate understanding of composing and decomposing hundreds.” Students demonstrate both aspects of rigor in Problem Solving, Item 9, as they complete two sentences using the vocabulary words hundred, tens, and ones. “There are 100  ____ in 100. There are 10 ____ in one ____.” (2.NBT.1a, 2.NBT.1b)
• In Lesson 14-2, Lesson Overview, “Conceptual Understanding: Students further develop their understanding of measurement by solving word problems involving length. Procedural Skill: Students use equations and drawings, such as a number line, to solve problems about unknown measurements. Application: Students apply reasoning when determining whether to use addition or subtraction to solve problems about unknown measurements.” Students demonstrate all three aspects of rigor in the Independent Practice, Item 3, as they solve, “Filipe’s pencil box is 24 centimeters long. Joe’s pencil box is 3 centimeters shorter than Filipe’s. How long is Joe’s pencil box?”  (2.MD.5, 2.OA.1)
• In Lesson 15-3, Guided Practice, Item 2, students are given a Favorite Pet table and a bar graph to complete. The directions state, “Use the table to complete the bar graph and use the bar graph to solve the Item. How many students chose a bird or dog?” (2.MD.10)

The instructional materials reviewed for enVision Mathematics Common Core Grade 2 meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade level, and are not treated separately.

The math practices are identified throughout the materials. For example: 

• Every Topic includes a Math Practices and ETP (Effective Teaching Practices) page with an explanation of how students engage with the MPs throughout the topic.
• Every lesson includes a Lesson Overview where an explanation of how students engage with the math practice during the lesson.  
• Special Item Solving lessons in each topic focus on specific math practices. 
• Specifically flagged comments and Items in all lessons focus on specific math practices. 
• Math Practice Animation videos for each MP provide a student-friendly explanation with demonstration Items. These can be found in the Digital Resources in Pearson Realize. 
• The Math Practices and Item Solving Handbook contains a detailed explanation for each MP, identifies “Thinking Habits” unique to each MP, connections to content and other MPs, and student behavior look-fors to monitor progress toward proficiency.

Examples of the MPs identified within individual lessons:

• MP1: Topic 7, Math Practices and ETP, “Students use information in a world Item to determine whether they should add or subtract to solve the Item. (e.g., p. 286, Visual Learning Bridge.)”
• MP2: Lesson 4-5, Lesson Overview, “Students reason about why breaking apart the second addend in a two-digit addition Item is a helpful mental math strategy for addition.”
• MP4:  Topic 7, Math Practices and ETP, “Students understand when and why to use subtraction to represent a real-world Item. (e.g., p. 284, Item 7.)”
• MP5: Topic 12, Math Practices and ETP, Teacher Edition, page 505H, “Students measure objects using units and different tools..  (e.g., p.511, Items 3-4.)”
• MP6: Lesson 2-1, Lesson Overview, “Students will communicate their understanding of odd and even numbers using clear definitions in their discussions and reasoning.”
• MP7: Lesson 13-4, Lesson Overview, states, “Students describe and draw cubes, looking for information about faces, edges, and vertices.” 
• MP8: Topic 7, Math Practices and ETP, states, “Students recognize that the same Item can be represented by different equations.  (e.g., p. 282, Convince Me!)”

Examples of where MPs are identified and used to enrich the content:

• In Lesson 10-1,Lesson Overview, students engage with MP1 as they “recognize that when adding 10 or 100 to a 3-digit number, only the tens or hundreds digit will increase by 1.” In Solve and Share, students use dollar bills, place value blocks, or mental math to solve, “Forest Park Nursery sells trees. Sal buys a maple tree for $125. A spruce tree costs $10 more than a maple tree. An elm tree costs $100 more than a maple tree. What is the cost of a spruce tree? An elm tree?” (2.NBT.8)
• In Lesson 10-6, Problem Solving, Item 7, students engage with MP2 as they solve each Item any way they choose and show their work. “There are 229 people at the football game. 108 more people arrive at the game. How many people are at the football game now?” (2.NBT.9)
• In Lesson 2-5, Lesson Overview, students engage in MP4 as they “draw and use arrays to write repeated addition equations to help them solve real-world Items.”  In Independent Practice, Item 2, “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 “draw a picture and write an equation to show each Item. Then solve.”  (2.OA.4)
• In Lesson 14-5, Lesson Overview, students engage in MP5 as they “consider the available tools when solving a  mathematical Item.” In Solve and Share, students are shown a picture of a straight line and a squiggly line and “Choose a tool to solve each part of the Item.  Be ready to explain which tools you used and why. Which line is longer? How much longer? Draw a line that is that length.” (2.MD.5)
• In Lesson 6-5, Lesson Overview, students engage in MP6 as they “attend to precision as they choose and use strategies to accurately subtract 2-digit numbers.” In Solve and Share, students “Find 82 - 56. Use any strategy you have learned or your own strategy. Show your work. Explain why your strategy works.” (2.NBT.5)
• In Lesson 9-10, Independent Practice, Item 4, “The blue team wants to sort their jersey numbers from greatest to least. After they sort the numbers, what number would come next? Look for a pattern in the sorted jersey numbers. What is the pattern rule?” (2.NBT.2)
• In Lesson 13-8, Lesson Overview, students engage with MP8, as they “use repeated reasoning to create designs in equal shares.” In Solve and Share, students “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.” (2.G.2)

​The instructional materials reviewed for enVision Mathematics Common Core Grade 2 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Specific features of the materials engage students in constructing viable arguments and/or analyzing the mathematical reasoning of others. Examples include:

• Convince Me! prompts provide the opportunity for students to share their thinking and to analyze the reasoning of others.
• In Three-Act Math, students critique other’s reasoning as solution methods for the task are shared with the class. 
• In Solve and Share, students share and justify solutions with the class, and they critique the reasoning of others as teachers select which solutions to share.
• In the Visual Learning Bridge, there are opportunities for students to construct viable arguments.
• “I Can” bubbles prompt students: 
• Construct Arguments: Lesson 3-7 Solve & Share, “I Can...use pictures, numbers, and words to explain why my thinking and work are correct.”  
• Critique Reasoning of Others: Lesson 5-8, Solve & Share, “I Can...critique the thinking of others by using what I know about addition and subtraction.” 
• Thinking Habits thought bubbles  prompt students:
• Construct Arguments: Lesson 3-7, Solve & Share, “How can I use math to explain why my work is correct? Am I using numbers and symbols correctly?  Is my explanation clear?”   
• Critique Reasoning of Others: Lesson 5-8, Solve and Share, “What questions can I ask to understand other people’s thinking?  Are there mistakes in other people’s thinking?”  
• Math Practices and Problem Solving Handbook.

The materials consistently provide opportunities for students to construct viable arguments. Examples include:

• Lesson 1-10, Independent Practice, Item 3, “The Lions scored 11 runs in a baseball game. The Tigers scored 7 runs. Did the Tigers score 3 fewer runs than the Lions? Explain.” (2.OA.1)
• Lesson 3-7, Guided Practice, Item 1, students “Solve. Use pictures, words, or numbers to make a math argument. Show your work.”  “There are 16 chickens in the yard. There are 19 chickens in the barn. There are 30 nesting boxes. Will all of the chickens have a nest? Explain.” (2.NBT.9)
• Lesson 8-2, Convince Me!, “Why is subtracting 75¢ - 68¢ like subtracting 75 - 68? Explain.” (2.MD.8, 2.NBT.2)

The materials consistently provide opportunities for students to analyze the reasoning of others. Examples include:

• Lesson 3-1, Convince Me!, “Max says that to find 54 + 18 on a hundred chart, you can start at 54, move down 2 rows, and move back 2 spaces. Do you agree? Explain.” (2.NBT.5)
• Lesson 4-6, Solve & Share, “12 + 34 + 28 = ? Tom says he can find the sum by adding 28 and 12 first. He says he can add 34 to that sum to find the total. Do you agree? Use pictures, words and numbers to make a math argument. Then solve the problem. Show your work.” (2.NBT.6)

The instructional materials reviewed for enVision Common Core Grade 2 meet the expectations of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. 

There are multiple locations in the materials where teachers are provided with prompts to elicit student thinking. For example:

• The Math Practices and Problem Solving Handbook provides guidance on implementing MP3 and questions that students might ask themselves as they reflect on MP3. The Problem Solving Lessons which focus on MP3 are identified, for example, in Lessons 1-10, 3-7, and 5-8.
• In the teacher’s notes for each lesson, MP3 is identified in red print as “Construct Arguments” or “Analyze Reasoning”. Questions to elicit student thinking are included below the prompts. 
• In the teacher notes for Solve & Share activities, questions to prompt students thinking are included in Share Solution Strategies and Key Ideas.
• The Convince Me! activity, when connected to MP3, provides prompts to assist students in constructing arguments and analyzing the reasoning of others.
• Three Act Math Tasks includes Construct Arguments which provides prompts for the teacher to help students construct arguments.

The materials provide guidance to support teachers in engaging students in constructing viable arguments. Examples include:

• Lesson 1-10, Problem Solving Performance Task, Item 7, the teacher narrative prompts “After students solve the problem, have them share their solution strategies. Elicit a variety of solution strategies. As they share their solutions and their strategies, look for opportunities to point out how their work has the qualities of a good math argument.” (2.OA.1)
• Lesson 4-2, Convince Me!, “Ken adds 43 + 27. His sum is 60. Is he correct? Explain.” The teacher is prompted to “remind students that as well as deciding if Ken is correct, they need to explain how they reached that decision. Some students may disagree on the answer or may agree on the answer but have different explanations. Discuss these differences, telling different ways to correctly make an argument.” (2.NBT.5)
• Lesson 10-5, Convince Me!, “Can the problem above (257 + 384) be solved by adding the ones first, then the tens, and then the hundreds? Explain.” The teacher is prompted to “have students explain why the order they add the partial sums will not change the sum.” (2.NBT.7)

The materials provide guidance to support teachers in engaging students in analyzing the reasoning of others. Examples include:

• Lesson 2-5, Problem Solving, Item 6, students are given the information that there are 3 rows of posters and 5 posters in each row. “Mr. Miller says that he will add 3 + 5 to find the total number of posters in the posters display. Do you agree with his plan? Explain.” The teacher is prompted to ask guiding questions if students have difficulty getting started. “How can you find the total number of posters in the poster display? What equation can you write to show the number of posters in the poster display?” (2.OA.4)
• Lesson 5-8, Convince Me!, “Have students share the questions they would ask Kelly to help her check her reasoning, so that students understand there is more than one way to help Kelly check her thinking.” (2.NBT.9)
• Lesson 9-5, Problem Solving,Item 8, students are presented with a problem: “Neha wants to make the same number in different ways. She says 300 + 130 + 9 equals the same number as 500 + 30 + 9. Do you agree with Neha? Explain.” In the lesson narrative, teachers are prompted to “Encourage students to compare the ways shown. What is the sum of the numbers in the first way? Compare the first and second ways. How are they different? What does that tell you about the sum of the  numbers in the second way?” (2.NBT.3)

The instructional materials reviewed for enVision Mathematics Common Core Grade 2 meet the expectations of attending to the specialized language of mathematics.

The materials provide explicit instruction on the use of mathematical language including words, diagrams, symbols, and conventions. Each topic includes:

• My Word Cards are available online. In the Teacher Edition, page 1J Build Mathematical Literacy, explains how My Word Cards are used: “Students use the example on the front of the card to write the definition on the back.” 
• Vocabulary Activities at the beginning of Topics
• Vocabulary Review at the end of each Topic
• Glossary in the Student Edition
• Animated glossary is available online.
• Online vocabulary game in the Games Center

For each topic, the Topic Planner includes a list of the new vocabulary words for each lesson. The vocabulary words are also included in the Teacher Edition, Lesson Overview page for each lesson. For example in Topic 3:

• Lesson 3-1, identifies tens and ones
• Lesson 3-2: identifies open number line
• Lesson 3-3: identifies break apart

In the Vocabulary Review at the end of each Topic, teachers are provided several activities to help students review the vocabulary: For example, in Topic 12, Teacher Edition, page 546,

• “Have students define the terms in their own words”
• “Have students say math sentences or math questions that use the words.”
• Play a “What’s My Word?” guessing game in which you or a student thinks about one of the words and says a clue that others listen to before they guess the word.
• Play a “Right or Wrong?” game in which you or a student says a sentence that uses one of the words correctly or incorrectly. Then others say “right” or “wrong.”

The materials use precise and accurate terminology and definitions when describing mathematics, and they provide support for students to use them correctly. Examples include:

• Lesson 2-3, Visual Learning Bridge, states, “You can model repeated addition with an array. Arrays have equal rows. Each row has 3 strawberries. Arrays have equal columns. Each column has 2 strawberries. Write two equations that match the array. By Rows 3+3=6  By Columns 2+2+2=6”
• Topic 6, Vocabulary Review, Items 4 - 6, students are given three terms: equation, regroup, difference and three examples: a place-value model illustrating regrouping,  “the answer to 75 - 23”, and “72 + 25 = 97”. They are asked to “Draw a line from each term to its example.”
• Topic 12, Vocabulary Review, Item 8, Use Vocabulary in Writing, states “Use words to tell how to find the height of a table. Use terms from the Word List.” (centimeter, estimate, foot, height, inch, length, meter, nearest centimeter, nearest inch, yard)