5th Grade - Gateway 2

​The instructional materials for enVision Florida Mathematics Grade 5 meet 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 includes several opportunities to develop conceptual understanding.

• In the Teacher’s Edition, every Topic begins with “Math Background: Rigor,” where conceptual understanding for the topic is outlined.
• Lessons are introduced with a video, "Visual Learning Animation Plus,” at PearsonRealize.com; these often build conceptual understanding.
• Links within the digital program to outside resources, such as Virtual Nerd, include videos for students that introduce concepts.
• In the student practice problems, the section “Do You Understand” reviews conceptual understanding.

Materials include problems and questions that develop conceptual understanding throughout the grade level and provide opportunities for students to demonstrate conceptual understanding independently throughout the grade.

• The Topic 1 Overview, Conceptual Understanding states, "Understand Exponents. In Lesson 1-1, students are introduced to exponents. They learn that the exponent in a power of 10 tells the number of times 10 is used as a factor. When multiplying by a power of 10, they recognize the connection between the exponent in the power of 10 and the number of zeros in the product."
• The Lesson 2-3 Lesson Overview, Rigor states, “Conceptual Understanding. Students shade grids divided into hundredths to show how parts of the whole written in decimal form can be combined. They use shading to show part of a whole and crossing out to show the parts that are taken away." Students are provided opportunities to explain and use grids and place value blocks to model and regroup.
• In Lesson 11-1, students draw or construct models to find the number of cubes that make up a rectangular prism. Students do this work by determining the number of cubes in the bottom layer of a rectangular prism as they draw or get unit cubes to create the bottom layer of a prism using measurements of the length and width. They continue adding layers to the prism and find its height.

​The instructional materials for enVision Florida Mathematics Grade 5 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

Examples of the the instructional materials developing procedural skills and fluencies throughout the grade level include:

• Procedural skills and fluencies integrate with conceptual understanding and the work students completed with operations from prior grades. Opportunities to practice procedural skills are found throughout practice problem sets that follow the units and include opportunities to use fluencies in the context of solving problems.
• The Teacher Edition Program Overview articulates, “Steps to Fluency Success.” The six steps are: Step 1: Fluency Development with Understanding, Step 2: Ongoing Assessment of Fluency Subskills, Step 3: Fluency Intervention, Step 4: Practice on Fluency Subskills, Step 5: Fluency Maintenance, and Step 6: Summative Fluency Assessment. Fluency Expectations for Grades K-5 are also listed. The Teacher Edition Topic Overview explains the six steps and foundations for fluency. In each Topic Overview, Math Background: Rigor, there is a section explaining how the material builds Procedural Skill and Fluency. The Topic 3 Overview, Procedural Skill and Fluency identifies the procedural skill for understanding of multi-digit multiplication using the standard multiplication algorithm for whole numbers.
• Within each lesson, the Visual Learning Bridge integrates conceptual understanding with procedural skills. Additional Fluency and Practice pages are in the Teacher Edition and Ancillary Books as well as online with the Practice Buddy Additional Practice. The online component also contains a game center where students continue to develop procedural skills and fluencies. Each topic ends with Fluency Practice/Assessment Worksheets.

The instructional materials provide opportunities to demonstrate procedural skill and fluency independently throughout the grade level.

• In Lesson 3-1, students evaluate two-digit whole numbers multiplied by 10.
• In Lesson 3-3, students multiply multi-digit whole numbers using the standard algorithm.
• In Lesson 3-6, students multiply three-digit numbers by two-digit numbers using the standard algorithm.
• In Lesson 8-5, Problem 15, students multiply two fractions such as $$\frac{2}{3}$$ x $$\frac{7}{8}$$.

The instructional materials provide regular opportunities for students to attend to Standard 5.NBT.2.5, multiplying multi-digit whole numbers using the standard algorithm.

​​The instructional materials for enVision Florida Mathematics Grade 5 meet expectations for being 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.

Work with applications of mathematics occurs throughout the materials. In each Topic Overview, Math Background: Rigor explains how the materials utilize applications. For example, the Topic 8 Overview, Math Background: Rigor, Application states, “Throughout Topic 8, there are real-world problems involving the computations being developed. In Lesson 8-9, students make sense and persevere in solving real-world problems involving multiplication of fractions and mixed numbers.”

Following the Topic Overview, the Topic Opener includes an enVision STEM Project where application activities are provided and can be revisited throughout the topic. In each topic, Pick A Project allows students to explore areas of interests and to complete projects that apply the mathematics of the topic. Every other topic contains 3-Act MATH where students engage in mathematical modeling.

At the end of each topic, the Performance Task provides opportunities for students to apply the content of the topic. Additional application tasks are in Additional Practice pages in the Teacher Edition, Ancillary Books, and online.

Examples of opportunities for students to engage in routine and non-routine application of mathematical skills independently and to  demonstrate the use of mathematics flexibly in a variety of contexts include:

• In the Lesson 7-12, Problem Solving Performance Task, students answer questions about camp activities where information is presented in a chart. "During the 6-hour session at day camp, Roland participated in boating, hiking, and lunch. The rest of the session was free time. How much time did Roland spend on the three activities? How much free time did he have?”
• In Lesson 8-7, Problem Solving, Question 23 states, “The city plans to extend the Wildflower Trailer 2$$\frac{1}{2}$$ times its current length in the next 5 years. How long will the Wildflower Trail be at the end of 5 years?” In this question, students solve real-world problems that involve multiplication of fractions and mixed numbers.
• In Lesson 9-5, page 404, Question 19 states, “Five friends equally share half of one large pizza and $$\frac{1}{4}$$ of another large pizza. What fraction of each pizza did each friend get? How do the two amounts compare to each other?”
• In Topic 9, Performance Task, Question 2 states, “Julie and Erin have 6$$\frac{1}{3}$$ yards of red checked cloth. After making dresses for 4 dolls, they use the remaining cloth to make bows for the dolls’ hair. They need 8 bows for 4 dolls. Part A How much cloth do Julie and Erin have for each bow? Explain. Part B. Julie wrote the equations below. What is the pattern in her equations? Explain how to use the pattern to find the quotient you found in Part A.”

​The instructional materials for enVision Florida Mathematics Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The instructional materials address specific aspects of rigor, and the materials integrate aspects of rigor.

Each lesson contains opportunities for students to build conceptual understanding, procedural skills, and fluency, and to apply their learning in real-world problems. 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 and use procedural skills and/or conceptual understanding to solve problems with multiple solutions and explain/compare their solutions.

All three aspects of rigor are present independently throughout the program materials.

• In Lessons 1-1 through 1-3, students develop conceptual understanding of place value by using place value charts and place value blocks when they transition from whole number place value to decimal place value to thousandths.
• In Lesson 8-9, students apply knowledge of multiplication of fractions and mixed numbers to solve real-world problems. Students solve problems where they determine the total cost of framing a painting given the dimensions of 10$$\frac{1}{4}$$ in. and 6$$\frac{1}{4}$$ in. and the framing cost $.040 per inch of framing. Students answer questions relating to the problem-solving steps: "What is the first step you need to do? What is the answer to the first step? Write an equation to show your work.”

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

• In Lesson 3-4, Question 16, students develop understanding of calculating partial products and practice procedural skills by multiplying and calculating each partial product. Students are given partial products in an array drawn on a grid, find the partial products, and calculate the final product. Students write the multiplication equation from the illustration to show the partial products.
• In Lessons 1-5 and 1-6, students apply their procedural skills to problems with various constraints and use their conceptual understanding of decimal order and place value to explain how their solutions represent the given situation. Students are shown models of number lines, inequality statements, and how to line up decimal points vertically, using these same models to determine how best to round numbers and to what value.
• In Lesson 3-4, students practice the procedural skill of multiplying two-digit by two-digit whole numbers using the standard algorithm while applying it in a real-world scenario. First, students solve a problem by multiplying two 2-digit numbers, using any strategy they choose in the Solve and Share section. In the Visual Learning Bridge section, students evaluate the standard algorithm of multiplying multi-digit whole numbers by answering questions such as, “A ferry carried 37 cars per trip on the weekend. If the ferry made 11 trips on Saturday and 13 on Sunday, how many cars did it carry on the weekend?”

The instructional materials reviewed for enVision Florida Mathematics Grade 5 meet expectations that the Standards for Mathematical Practice (MPs) are identified and used to enrich mathematics content within and throughout the grade level.

Examples of the MPs being identified at the topic level include:

• In Topic 1 Overview, MP.1.1 is identified. “Students persevere as they try to understand problems involving place value, plan how to solve them, and determine if their solution makes sense.”
• In Lesson 8 Topic Overview, MP.7.1 is identified. “Students use quantitative reasoning as they interpret the remainders in a division problem.”

The MPs are used to enrich the mathematical content and are not treated separately. MPs are highlighted and discussed throughout the lesson narratives, and along with the lessons, the MPs are evident in the the 3-Act Math Tasks that are included in every other chapter. The MPs are listed in the student materials, and the Math Practice Handbook is available online for teachers to make available to students.

• In Lesson 4-1, MP.7.1 is identified. “Students analyze their answers in the chart to look for a pattern they can use when multiplying numbers by powers of 10.”
• In Lesson 9-2, Problem Solving, Problem 19, MP.6.1 is identified. "Tammi has 4 pounds of gala apples and 3$$\frac{1}{2}$$ pounds of red delicious apples. If she uses 1$$\frac{3}{4}$$ pounds of gala apples in a recipe, how many pounds of apples does she have left? Ask students to think about what the numbers in the problem mean and write an expression to represent the problem.”
• In Lesson 5-6, Problem Solving, Problem 15, MP.2.1 is identified. “A delivery to the flower shop is recorded at the right. The shop owner makes centerpiece arrangements using 36 flowers that are all the same type. Will they be able to make at least 10 arrangements using each type of flower? At least 100 arrangements? Explain. Encourage students to use estimation either before or after multiplying.”

The MPs are identified within a lesson in the Lesson Overview, and lesson narratives highlight when an MP is particularly important for a concept or when a task may exemplify the identified Practice. The lessons that end each Topic specifically focus on at least one MP. For example:

• In Lesson 6-6, MP.2.1 is identified. “Students extend their understanding of how to use reasoning and Thinking Habits as they solve a multi-step problem that includes dividing a decimal by a two-digit whole number.”
• In Lesson 2-6, MP.4.1 is identified. “Students will use bar diagrams to solve multi-step problems involving the addition and subtraction of decimals.”
• In Lesson 8-9, Elaborate, MP.1.1 is identified. “Listen and look for these behaviors as evidence that students are exhibiting proficiency with this practice: Chooses a strategy or strategies to use to solve problems, identifies the quantities in a problem, the data given, and if present, the question to be answered, thinks of similar problems or uses a simpler form of the problem, if needed, organizes data or uses representations to help make sense of the problem, identifies likely strategies for solving the problem, pauses when solving problems to make sure that the work being done makes sense, and makes sure the answer makes sense before stopping work."
• In Topic 7, 3-Act Math task, MP.4.1 Model with Mathematics is connected to additional MPs. “As students carry out mathematical modeling, they will also engage in sense-making (MP.1.1), abstract and quantitative reasoning (MP.2.1), and mathematical communication and argumentation (MP.3.1). In testing and validating their models, students look for patterns in the structure of their models." (MP.7.1, MP.8.1)

The instructional materials reviewed for enVision Florida Mathematics Grade 5 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others. The Solve and Share Activities, Visual Learning Bridge Problems, Problem Sets, 3-Act Math, Problem Solving: Critique Reasoning problems, and Assessments provide opportunities throughout the year for students to construct viable arguments and analyze the arguments of others.

Examples of the instructional materials supporting students to analyze the arguments of others include:

• Lesson 2-4, Problem Solving: Critique Reasoning, Question 17: “Juan adds 3.8 + 4.6 and gets a sum of 84. Is his answer correct?”
• Topic 6, Topic Assessment, Question 20: “June says that there should be a decimal point in the quotient below after the 4. Is she correct? Use number sense to explain your answer. $$43.92\div5.2$$ = 845.”
• Lesson 11-4, Question 9: “Critique Reasoning: Does it make sense for Angelica to find the combined area of the bedroom floor and closet before finding the total volume?  Explain your thinking.” Students have to work through the previous problem in order to critique Angelica’s thinking.

Examples of the instructional materials prompting students to construct viable arguments include:

• Lesson 2-1, Question 14: “Construct Arguments: Use compensation to find each difference mentally. Explain how you found each difference. A. 67.9 - 29.9  B. 456 - 198.”
• Lesson 3-1, Question 21: “Without multiplying, tell which expression is greater, 93 x $$10^3$$ or 11 x $$10^4$$? How do you know?” Students use number sense to explain that a two-digit number multiplied by $$10^4$$ will be greater  than any two-digit number multiplied by $$10^3$$.
• In Lesson 3-5, students construct arguments by using estimation. “Is 300 x 10 a good estimate for the number of bagels sold at the bakery? Explain.”
• In Lesson 8-8, students construct a mathematical argument to compare pairs of factors to determine which is the greatest. “How is $$\frac{3}{3}$$ x 2 like 1 x 2?”
  

The instructional materials reviewed for enVision Florida Mathematics Grade 5 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in constructing viable arguments and/or analyzing the arguments of others throughout the program. Many of the activities are designed for students to work with partners or small groups where they collaborate and explain their reasoning to each other.

• In Lesson 1-5, Critique Reasoning (in the margin) provides the teacher with a question pertaining to a fictional student’s reasoning on a problem.
• In Lesson 1-7, “After Whole Class” provides teachers with opportunities to have students analyze the work of others: “Discuss Solution Strategies and Key Ideas: Based on your observations, choose which solutions to have students share and in what order. Focus on the strategies and structure they used to solve the problem. If needed, show and discuss the provided student work at the right.” There are also prompting questions to support teachers as they have the students analyze the provided student work.
• In Lesson 2-3, Critique Reasoning (in the margin), provides teachers with questions that pertain to how students can tell, without adding, that an answer does not make sense.
• In Lesson 4-8, Construct Arguments (in the margin), provides teachers with questions that pertain to students explaining if the product should be less than or greater than the decimal factors.
• In Lesson 11-1, “After Whole Class” provides teachers with opportunities to have students analyze the work of others: “Discuss Solution Strategies and Key Ideas: Based on your observations, choose which solutions to have students share and in what order. Focus on how students represented the prism and solved the problem. If needed, show and discuss the (provided student) work at the right.” There are also questions to support teachers as they have the students analyze the provided student work.
  

​The instructional materials reviewed for enVision Florida Mathematics Grade 5 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 Grade 5 Glossary is located in the Teacher Edition Program Overview, and the Glossary is also present at the back of Volume 1 of the Student Edition.
• Lesson-specific vocabulary can be found at the beginning of each lesson, under the Lesson Overview, with words highlighted in yellow used within the lesson, and a vocabulary review is provided at the end of each topic.
• There is a bilingual animated glossary available online that uses motion and sound to build understanding of math vocabulary and an online vocabulary game in the game center.
• 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.
• The Teacher Edition Program Overview, “Building Mathematical Literacy,” outlines the many ways the materials address mathematics vocabulary, including: My Word Cards, Vocabulary Activities at the Beginning of Each Topic, Vocabulary Reteach to Build Understanding, Vocabulary and Writing in Lessons (where new words introduced in a lesson are highlighted in yellow in the Visual Learning Bridge and lesson practice includes questions to reinforce understanding of the vocabulary used), Vocabulary Review at the back of each topic, an Animated Glossary where students can hear the word and the definition, and Vocabulary Games Online. There is also Build Mathematical Literacy within each Topic Overview that outlines support for English Language Learners, Mathematics Vocabulary, and Math and Reading within the topic.
• In Topic Planner, there is a vocabulary column that lists the words addressed within each lesson in the topic. For example, Lesson 16-2 lists the following words: trapezoid, parallelogram, rectangle, rhombus, and square. These same words are listed in the Lesson Overview.
• In Lesson 7-2, students interpret equivalent fractions and common denominators. Students use the context to build proper mathematical vocabulary.
• Lesson 11-1 introduces volume, cubic unit, cube, rectangular prism, and unit cube to the students. The Visual Learning Bridge provides definitions and models/diagrams using this new vocabulary. In Guided Practice, students are provided questions within the context of the lesson to answer using vocabulary. For example, Question 2 states, “Vocabulary: What is the difference between a unit cube and a cubic unit?” A sample answer is provided to support teachers using precise vocabulary language and definitions with students.

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