2nd Grade - Gateway 2

​The instructional materials for enVision Florida Mathematics 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.

• In the Teacher Edition, every Topic begins with Math Background: Rigor, where conceptual understanding for the topic is outlined.
• Lessons are introduced via video, Visual Learning Animation Plus, at PearsonRealize.com building on conceptual understanding.
• Each Lesson Overview includes Rigor highlighting how conceptual understanding is incorporated into the lesson.
• Each lesson includes Solve and Share where students are able to build and demonstrate conceptual understanding.

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. The conceptual understanding incorporated into each lesson is stated in the Rigor section of the Lesson Overview:

• In Lesson 1-5, students “explore different ways to subtract using a number line. Students deepen their understanding of how counting is connected to adding and subtracting, and of how addition and subtraction are related.” In Solve and Share, students solve, “How can counting help you find 12 - 4? Use a number line to show your work.” (2.OA.2)
• In Lesson 3-3, students “deepen their conceptual understanding of addition when they break apart numbers to add.” The Visual Learning Bridge shows students ways to break apart 57 and 13 to add the two numbers. (2.NBT.2.5 and 2.NBT.2.9)
• In Lesson 3-5, students use strategies to “demonstrate an understanding of place value, properties of operations, and the relationship between addition and subtraction.” During Independent Practice, students use different strategies to solve given problems. (2.NBT.2.5, 2.NBT.2.6, and 2.NBT.2.9)
• In Lesson 4-1, students “use and draw models to develop understanding of the strategy of breaking numbers apart in order to add them using place value.” In Solve and Share, “Leslie collects 36 rocks. Her brother collects 27 rocks. How many rocks do they collect in all? Use place-value blocks to help you solve.”  (2.NBT.2.5 and 2.NBT.2.9)
• In Lesson 6-2, students “use concrete and symbolic representations to reinforce understanding of subtracting two-digit numbers.” In Solve and Share, students solve, “You have 42 pipe cleaners. You use 19 of the pipe cleaners. How many pipe cleaners do you have now? Use place-value blocks to help you solve. Draw place-value blocks to show your work.” Possible student work includes both standard and symbolic representation. (2.NBT.2.5 and 2.NBT.2.9)
• In Lesson 8-3, “A key goal of this topic is for students to be able to count money and solve word problems about money. Conceptual work focuses on understanding what the value of each bill is and how to identify each bill.” In Solve and Share, “What is one way you can show 100¢ with coins?  Use coins to model. Draw and label the coins you use.” (2.MD.3.8.c, 2.MD.3.8.a, and 2.NBT.1.2)
• In Lesson 9-1, 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.” In Solve and Share, students solve, “What is another way to show 100? Draw a picture and explain.” (2.NBT.1.1.a and 2.NBT.1.1.b)
• In Lesson 9-2, students “use place-value blocks to reinforce understanding of place-value concepts and 3-digit numbers.” In Solve and Share, “How can you use place-value blocks to show 125? Explain. Draw your blocks to show what you did.” (2.NBT.1.1 and 2.NBT.1.3)
• In Lesson 9-5, students “understand that it takes 10 of a number in one place value to make a number in the next greater place value.” In Solve and Share, “Use place value blocks. Show two ways to make 213. Then draw each way. Tell how your ways are alike and different.” (2.NBT.1.3 and 2.NBT.1.1.a)

​The instructional materials for enVision Florida Mathematics 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:

• Activity Centers
• Reteach to Build Understanding
• Build Mathematical Literacy
• 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 problems 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.

• In Lesson 1-1, students “develop fluency with addition facts to 20 by using the counting on strategy along with changing the order of addends.” In Guided Practice, students “Count on to find the sum. Then change the order of the addends.” Convince Me!: “Does 5+2=2+5? How do you know?” (2.OA.2.2)
• In Lesson 1-2, students’ “proficiency and fluency with addition facts build as they develop the habit of looking for and thinking about doubles and near doubles.” In Independent Practice, students “Complete the doubles facts. Use the doubles facts to solve the near doubles. Use cubes if needed.” (2.OA.2.2)
• In Lesson 3-3, 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, which is a goal of Grade 2. Developing flexibility with mental math strategies is a critical component of fluency.” Activity Centers, students solve, “Most African lions are found in national parks or on game reserves. Scientists try to keep track of how many lions there are. Break apart numbers to solve each problem. Use blocks to help, if needed.” (2.NBT.2.5)
• In Lesson 4-1, Solve and Share, students solve, “Leslie collects 36 rocks. Her brother collects 27 rocks. How many rocks do they collect in all? Use place-value blocks to help you solve. Show your place-value blocks.” There is additional practice in Independent Practice and Problem Solving sections of the materials, as well as online games identified on PearsonRealize.com. (2.NBT.2.5)
• In Lesson 4-2, Guided Practice, students “Add. Use place value. Draw blocks or use another way.” Additional practice is provided in the Independent Practice and Problem Solving sections, as well as online games identified on PearsonRealize.com. (2.NBT.2.5)
• In Topic 10, the Lesson Fluency Practice Activity contains two worksheets. Add and Subtract Within 20 has 19 fluency practice problems (2.OA.2.2). Add and Subtract Within 100 has 17 problems that “Use place value understanding and properties of operations to add and subtract.” (2.NBT.2.5)

​The instructional materials reviewed for enVision Florida Mathematics Grade 2 meet expectations for teachers and students spending 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. Each Topic also includes:

• 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 & Share, students draw a picture and write an equation to model and solve a word problem: “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.3.4)
• In Lesson 5-8, Solve and Share, students solve, “Bill collects and sells seashells. He has 45 shells, finds 29 shells, and sells 20 shells. How many seashells does Bill have now? Tara says you have to subtract 45-29 and then add 20 to solve the problem. Do you agree with Tara’s thinking? Circle your answer. Use pictures, words, or equations to explain.” (2.NBT.2.9)
• In Lesson 7-7, Independent Practice, Number 14, “Reasoning: Jill and Tim have the same number of toy cars. Tim has 10 red card and 20 blue cars.  Jill has 8 red cars, 15 blue cars, and some yellow cars. How many yellow cars does Jill have?” Students then, “Write an equation to show each problem. Then solve. Show your work.” (2.OA.1.a)
• In Lesson 11-6, Solve & Share, students solve a hidden question in order to solve a two-step word problem: “Jody wants to bake 350 muffins. She bakes one batch of 160 muffins and one batch of 145 muffins. How many more muffins does Jody need to bake? Solve any way you choose. Show your work. Be prepared to explain why your way works.” (2.NBT.2.7)

​The instructional materials for enVision Florida 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 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:

• Lesson 3-6 emphasizes Application, “Students apply their knowledge and understanding of addition and subtraction within 100 as they solve word problems.” In the Solve and Share, “students solve a one-step word problem, and they explain their answer using counters, drawings, or equations.” (2.OA.1.1 and 2.NBT.2.5)
• Lesson 5-1 emphasizes Procedural Skill, “By subtracting tens and ones on a hundred chart, students develop their ability to subtract mentally. Students practice subtracting a 1-digit or 2-digit number from a 2-digit number, to begin working toward becoming fluent in finding differences (and sums) within 100 by using strategies based on place value.” In the Solve and Share, “students explain how they use a hundred chart to find the difference of two 2-digit numbers.” (2.NBT.2.5 and 2.NBT.2.9)
• Lesson 12-5 emphasizes Conceptual Understanding in Solve and Share, "The green cube is 1 centimeter long. How can you use 1 centimeter cubes to find the length of the line in centimeters. Measure the line and explain." (1.MD.1.2)

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 Lesson 4-3, Lesson Overview, Conceptual Understanding and Procedural Skill and Fluency are the focus of the lesson. “Conceptual Understanding: Using partial sums supports students’ understanding of place value and properties of operations. Procedural Skill and Fluency: Using partial sums to add helps students develop fluency in adding within 100.” Students demonstrate both aspects of Rigor in the Solve and Share, where they “draw cubes from a bag, sort by color, and make drawings to compare groups.” (2.NBT.2.5, 2.OA.1.1, and 2.NBT.2.9)
• In Lesson 5-7, Lesson Overview, Application and Procedural Skill are the focus of the lesson. “Application: Students apply their understanding of addition and subtraction to solve one- and two-step word problems. Procedural Skill: Students practice adding and subtracting within 100 to achieve fluency by the end of Grade 2.” Students demonstrate both aspects of Rigor in the Solve and Share, when they “solve an add-to, start-unknown word problem.” (2.OA.1.1 and 2.NBT.2.5)
• In Lesson 14-2, Lesson Overview, Conceptual Understanding, Procedural Skill, and Application are emphasized. “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 these aspects of Rigor in the Solve and Share, when they “measure objects and add lengths to find a given total length.” (2.MD.2.5 and 2.OA.1.1)

​The instructional materials reviewed for enVision Florida Mathematics 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.

The MPs are used to enrich the mathematical content and are not treated separately. MPs are highlighted and discussed throughout Topic Planners, Topic Overviews, 3-Act Math Tasks, and identified within each lesson of every topic. Additionally, the Math Practice and Problem Solving Handbook includes a list of the MPs and real-world scenarios modeled through questions and answers. The online tools offer a Math Practices Animation video that explains the MPs and offers problems that demonstrate each one.

Examples of the MPs identified within the materials include:

• In Topic 9, Topic Overview, the “math practices are highlighted in all lessons and are given special emphasis in lessons that focus on problems solving.” MP.7.1: Look for and make use of structure. Students are required to analyze the structure of three-digit numbers when looking at a place-value chart or place-value blocks. (e.g., [Lesson 9-1], Visual Learning Bridge).”
• In Topic 13, 3-Act Math Task (Straw Shaped), “As students carry out mathematical modeling, they engage in sense-making (MP.1.1), abstract and quantitative reasoning (MP.2.1), and mathematical communication and argumentation (MP.3.1). They use appropriate tools to develop their models (MP.5.1). In testing and validating their models, students attend to precision (MP.6.1) and look for patterns in the structure of their models (MP.7.1 and MP.8.1)."

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

• In Lesson 2-3, MP.7.1, “Students will closely analyze each array. They will see that adding the rows or adding the columns result in the same sum.” MP.7.1 is used to enrich the content in the Solve and Share as students analyze an array and show their work to find a total number in the array.
• In Lesson 5-6, MP.5.1, “Students used the strategies learned in previous lessons as tools to solve. They use the information given in the problem to choose the best strategy to solve.” MP.5.1 is used to enrich the content in the Solve and Share as students use any strategy to solve a take-apart, total-unknown word problem and to show and explain their work.
• In Lesson 8-1, MP.2.1, “Students use their knowledge of place value and addition to solve problems involving coins.” MP.2.1 is used to enrich the content in the Solve and Share as students solve and express values in cents and they add using different strategies. Students can draw or do numerical expressions to solve.

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

Several structures exist within the Grade 2 materials prompting students to construct viable arguments such as:

• In Convince Me!, students answer open-ended questions to demonstrate how they know the answer.
• In the 3-Act Math activities, Critique Reasoning, students share solutions and analyze the work of others allowing students to construct viable arguments.
• In most lessons, Construct Arguments, students answer open-ended questions to construct viable arguments.
• In Solve and Share, there is an opportunity for students to critique the reasoning of others and construct viable arguments.
• In Visual Learning Bridge, there is an opportunity for students to construct viable arguments.

Student materials consistently prompt students to both construct viable arguments and analyze the arguments of others.

• In Lesson 5-8, students “use their knowledge of math to support their arguments about what they and others do when solving problems.”
• In Lesson 8-2, Convince Me!, Construct Arguments, “Students describe how doing computations with money is like adding and subtracting whole numbers. Some students may also note that you can use coins to show the computations with money.”
• In Lesson 11-1, Convince Me!, Construct Arguments, students evaluate 447-100. Students share their reasoning and approaches to these problems with a partner and compare answers.
• In Lesson 15-5, Convince Me!, students determine “how many tickets Kim and Neil sell in all, and then they explain their work.”

​The instructional materials reviewed for enVision Florida Mathematics Grade 2 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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

• At the beginning of each lesson the Solve and Share contains After: Discuss Solution strategies and Key Ideas.
• Each lesson has Convince Me! in the Visual Learning Bridge where teachers are provided with prompts to assist students in constructing viable arguments.
• The 3-Act Math Activity has Construct Arguments with prompts for the teacher to use during the activity.
• The Math Practices and Problem Solving Handbook for each grade level identifies the lessons for each grade level focusing on constructing viable arguments and critiquing the reasoning of others (Lessons 1-10 and 5-4).

Teacher materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others frequently throughout the program:

• In Lesson 4-1, Convince Me!, Construct Arguments, “Encourage students to share their thinking and to give an example to support their opinion. As a class, discuss how the ones digits can be used to decide if regrouping is needed.”
• In Lesson 6-2, Convince Me!, Construct Arguments, “Students explain how the two methods are similar and different. Ensure that students are thorough in their explanations. Have students use place-value blocks to act out each method to help them see how they are different.”
• In Lesson 11-4, Convince Me!, “Find 254 - 174. Jason says he can subtract 100 and then 4 and then 70 to find the difference. Do you agree?”

​The instructional materials reviewed for enVision Florida Mathematics Grade 2 meet expectations for explicitly attending to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• The Teacher Edition Topic Overview: Build Mathematical Literacy outlines multiple ways the materials address mathematics vocabulary. These components can be found in every topic under Build Mathematical Literacy:
• Build math vocabulary: “using the vocabulary cards, vocabulary activities, vocabulary review, and glossary plus the online glossary and vocabulary game.”
• My Word Cards: “Vocabulary cards for a topic are provided online at PearsonRealize.com. Students use the example on the front of the card to complete the definition on the back.”
• Vocabulary Activities: “The Teacher Edition provides vocabulary activities at the start of topics. These include activities for vocabulary in My Word Cards or activities for vocabulary in Review What You Know.”
• Vocabulary Review: “A page of vocabulary review is provided at the end of each topic. It reviews vocabulary used in the topic.”
• Glossary: “A glossary is provided at the back of Volume 1 of the Students’ Edition.”
• Animated Glossary: “An online, bilingual, animated glossary uses motion and sound to build understanding of math vocabulary.”
• Online Vocabulary Game: “An online vocabulary game is available in the Game Center.”
• Lesson-specific vocabulary can be found in each Topic Planner. For example, in the Teacher Edition, Topic 8, Lesson 8-1: dime, nickel, penny, quarter, half-dollar, cents; Lesson 8-3: dollar, dollar sign, dollar bill; and so on. The same vocabulary words are listed in the Lesson Overview under Lesson Resources.
• A Glossary is provided in the back of the Student Edition.
• Both the topic and the lesson narratives contain specific guidance for the teacher to support students to communicate mathematically. Within the lesson narratives, new terms are highlighted in yellow and explained as related to the context of the material.
• Topic 13, Vocabulary Review, the words identified for the topic, “angle, cube, edge, equal shares, face, fourths, halves, hexagon, pentagon, polygon, quadrilateral, right angle, vertex.” Problem 8: “Tell how many you can divide a square into two equal shares. Then tell how you can divide that same square into 3 equal shares. Use terms from the word list.”
• Topic 15, Vocabulary Review, students are given a word list including, “bar graph, data, line plot, picture graph, and symbol,” students are then instructed to “Label each data display. Write line plot, bar graph, or picture graph.” Number 4: “Look at the graph in Item 2. Use words to tell how to find which ball game is the most popular. Use terms from the Word List.” Throughout the lesson, the vocabulary words are implemented.

No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.