Kindergarten - Gateway 2

​The instructional materials for enVision Florida Mathematics Kindergarten 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.

• In Lesson 2-1, students “understand what it means to describe quantities as being equal as they compare groups of objects.” In Solve and Share, students “explain how they know if two given groups of objects are the same in number.” (K.CC.3.6)
• In Lesson 5-3, students “analyze data in a new way as they compare numbers in each category.” In Solve and Share, students “sort objects into two categories, then count and compare the numbers in each category.” Students determine, “What do you need to know first before you can compare? How can you find this? How can you compare the numbers in each category? How have you compared numbers before?” (K.MD.2.3)
• In Lesson 6-1, students utilize multiple representations to “conceptualize addition problems in order to solve them.” In Solve and Share, students “determine the number of flowers in all and represent the total in different ways,” using counters, showing with their fingers, or counting on. Multiple representations are also present in Visual Learning Animation Plus, where students are presented with blocks, fingers, and a student’s thinking process in order to show how to solve a problem. (K.OA.1.1)  
• In Topic 6, Lesson 6-5, students “build upon their understanding of addition as they work more closely with interpreting, representing, and solving addition word problems.” In Solve and Share, students “show ways to add to solve a word problem.” (K.OA.1.2)
• In Lesson 7-2, students “continue to explore subtraction with the idea that a whole can be separated into parts.” In Visual Learning Animation Plus, students are provided with the visual of pears, apples, and peaches to take apart. (K.OA.1.1)
• In Lesson 11-2, students further their “understanding of counting with both tens and ones... The continued use of a number chart allows students to see the relationships between the numbers they count and encourages the application of previously learned patterns, including consideration of tens and ones digits.” (K.CC.1.1)
• In Lesson 11-3, students “use the decade numbers to count by tens to 100. Counting to 100 in this way gives students an understanding of the structure of numbers greater than those they have encountered so far. Recognizing patterns within this sequence and having experience with the hundred chart will allow students to generalize about counting later as they apply what they know to other numbers. This lays the foundation for counting within 100 in different ways.” In Solve and Share, students “identify decade numbers and count by tens to 100.” (K.CC.1.1)

​The instructional materials for enVision Florida Mathematics Kindergarten 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 6-2, students “count on from a number to find the total.” In Visual Learning Animation Plus, students “use connecting cubes to model adding to the group when more boats come, and then write and addition sentence to hell how many in all.” Independent Practice and Additional Practice sections provide opportunities for students to independently demonstrate procedural skill and fluency. (K.OA.1.1, K.CC.1.3, and K.OA.1.2)
• In Lesson 6-7, students “solve addition equations without corresponding pictures or number stories (and) follow the procedure of filling in parts of an addition equation...The strategy of using patterns to solve addition problems should help students increase both the speed in which they solve the problems and the accuracy of their answers.” Independent Practice, Build Mathematical Literacy, Additional Practice, and Enrichment include opportunities for students to develop procedural skill and fluency in addition by using patterns to complete number sentences, fact families, and equations. (K.OA.1.5 and K.OA.1.1)
• In Lesson 8-1, students “work on the procedural skill of showing parts of a number and representing those parts in an equation as they solve word problems.”  Visual Animation Plus, Guided Practice, Independent Practice, Reteach to Build Understanding, Build Mathematical Literacy, and Enrichment provide opportunities for students to solve word problems in different ways, with both addends unknown. (K.OA.1)
• In Lesson 8-4, students “find different ways to solve equations within 5.” In Independent Practice, “students solve the equation any way they choose, and then tell how they solved the problem.” Reteach to Build Understanding, Build Mathematical Literacy, and Enrichment sections provide additional opportunities for students to solve equations within 5. (K.OA.1.5 and K.OA.1.1)

​The instructional materials reviewed for enVision Florida Mathematics Kindergarten 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 Topic 2, Topic Centers, students apply comparing numbers 0-5, including writing numbers sequences in the correct order, numbering plants, counting the number of sprouts and seeds in each cup, and walking on a taped line taking no more than 5 steps.
• In Topic 2, Performance Task, students “count the large puzzles and cars that David can see in the toy chest, and then write the numbers to tell how many of each toy. Then have them draw a circle around the number that is greater than the other number and mark an X on the number that is less than the other number.” (K.CC.1.3 and K.CC.3.6).
• In Topic 5, STEM Project, students classify animals by attributes, such as color, count the animals by each color, and create a poster after researching the animal. (K.MD.2, K.CC.2)
• In Topic 5, 3-Act Math, students create mathematical models representing a situation by tallying, drawing tens frames, counting in order to classify, or counting data. (K.MD.2.3, K.OA.1.1, K.CC.1.2, and K.CC.2.5)
• In Lesson 14-6, Lesson Overview, Rigor, the materials state, “mathematics instruction calls for the selection, use, and management of problem-solving methods. Use the Thinking Habits shown in the Solve & Share task to help focus thinking on precision in the lesson.” Students are asked to compare the length of a piece of ribbon to a 3 cubed train. (K.MD.1.2)

​The instructional materials for enVision Florida Mathematics Kindergarten 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 3-7 emphasizes Conceptual Understanding, “Students deepen their basic understanding of the counting sequence as they count to find a quantity that is 1 less than or 1 greater than a given number.” (K.CC.1.2)
• Lesson 9-7 emphasizes Application, “mathematics instruction calls for the selection, use, and management of multiple problem-solving methods. Use the Thinking Habits shown in the Solve & Share task to help focus thinking in the lesson on explaining why answers work with the criteria and identifying why there is more than one correct answer.” (K.CC.1.2 and K.CC.2.5)
• Topic 12, Fluency Practice Activity, students “practice fluently adding and subtracting within 5 during a partner activity.” (K.OA.1.5)

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 4-1, Lesson Overview, Conceptual Understanding and Procedural Skill are the focus of the lesson. “Conceptual Understanding: Students further their understanding of comparison as they compare larger groups to determine which is greater or less in number. Procedural Skill: Students use a 1-to-1 matching strategy to compare groups.” 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.” (K.CC.3.6 and K.CC.2.5)
• In Lesson 8-8, Lesson Overview, Conceptual Understanding and Application are emphasized. “Conceptual Understanding: Students continue to develop their understanding of addition situations as they work with problems involving sums of 10. Application: Students show parts of a number and represent those parts in an equation as they solve real-life word problems.” Students demonstrate both aspects of rigor in the Solve and Share, when “students use counters and an equation to model a way to solve a word problem where only the total is known.” (K.OA.1.a and K.OA.1.1)

​The instructional materials reviewed for enVision Florida Mathematics Kindergarten 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 individual lessons:

• In Lesson 3-8, MP.7.1, “Students use a pattern to show all the ways to make a number. Students make the pattern both with colors and numbers to help them visualize and keep track of the ways they have found.”
• In Lesson 5-1, MP.6.1, “Students classify objects into two categories, analyzing to find those which have a given attribute and those which do not.”
• In Lesson 9-7, MP.2.1, “Students identify possible answers to word problems that have more than one potential answer.”
• In Lesson 14-4, MP.5.1, “Students see that objects can be described by more than one measurable attribute, and consider the appropriate tool to measure.”

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

• In Topic 6, the Topic Overview states, “math practices are highlighted in all lessons and are given special emphasis in lessons that focus on problem solving.” MP.1.1: Make sense of problems and persevere in solving them. Students make sense of word problems with two addends, and they use drawings and addition equations to represent the problems (e.g., [Lesson 6-5], Solve and Share).”
• In Topic 7, 3-Act Math Task (Fruit Salad), “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)."

​The instructional materials reviewed for enVision Florida Mathematics Kindergarten 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 Kindergarten 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 that allows 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 1-10, students “focus on explaining solutions to problems where they have to count. They draw on the concepts learned previously in this topic to clearly communicate their understanding in different ways.”
• In Lesson 5-4, students “tell whether a given statement makes sense and provide reasons for their choice using numbers, pictures, or words to explain.”
• In Lesson 8-4, Critique Reasoning, students answer, “Daniel says 2 - 1= 3 because he starts with 2 counters, add 1 more, and has 3 in all. Is he correct? Explain.”
• In Lesson 11-3, Solve and Share, students answer, “Alan says he knows which numbers to count because they are the only numbers that have a zero in them. Is Alan correct? Why does Alan not put an X on 10 or 20?”

​The instructional materials reviewed for enVision Florida Mathematics Kindergarten 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 1-10, Solve and Share, Discuss Solution Strategies and Key Ideas, “Based on your observations, choose which solutions to have students share and in what order. Focus on the explanations of ways to tell there are 5. If needed, show and discuss the student work at the right.”
• In Lesson 3-3, Visual Learning Bridge, Classroom Conversation, Construct Arguments, “How can you find out how many counters there are? Does it matter which counter you count first? Why are there 9 counters? How many counters should you color?”
• In Lesson 5-4, Visual Learning Bridge, Classroom Conversation, Critique Reasoning, “What do you see? Tucker and Olivia counted 6 cars and 5 vehicles that are not cars. How can you check whether their answers make sense?” In Convince Me, Critique Reasoning, “Emily says the number of cars is equal to the number of vehicles that are not cars. Does this make sense?”
• In Lesson 10-1, Solve and Share, “Based on your observations, choose which solutions to have students share and in what order.  Focus on how students knew how the equation should be written. If needed, show and discuss the students work at the right.”

​The instructional materials reviewed for enVision Florida Mathematics Kindergarten 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 5, Lesson 5-1: category, classify; Lesson 5-2: chart, tally mark; 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 1, Vocabulary Review, “Have students 1. draw a circle around the number, 2. write the number that means none, 3. Draw a circle around the number four, 4. mark an X on the one red cube and draw a circle around all five cubes in the group.” The vocabulary listed for this topic include: number, none, four, one, and five.
• Topic 11, Vocabulary Review, “Understand vocabulary: Have students 1-draw a circle around the part of the number in the orange column that is 3 ones 2-draw a circle around the part of the number in the blue column that shows the pattern of 8 ones 3-color the decade numbers red.” The vocabulary listed for the topic includes: ones, pattern, and decade.

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