4th Grade - Gateway 2

​The instructional materials for enVision Florida Mathematics Grade 4 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 6 Overview, Rigor: Conceptual Understanding states, “Comparison Problems. In Lessons 6-1 and 6-2, students’ understanding of multiplicative comparison goes beyond the basic situation where the product is unknown. They also learn about multiplicative comparison situations in which the lesser quantity is unknown or the 'times as many' number is unknown. In these two latter types of multiplicative comparison situations, division is the operation used for finding the answer.”
• The Lesson 7-1 Visual Learning Animation Plus states, “How Can You Use Arrays to Find the Factor Pairs of a Number?” The scenario begins by having students work with the music director to find the best way to arrange the chairs for a performance. Students use grids to show all the ways the chairs can be arranged.
• The Lesson 8-5 Lesson Overview, Conceptual Understanding states, “Students extend their understanding of comparing fractions to include those with unlike numerators and denominators.” Students are provided opportunities to explain and use benchmarks, area models, and number lines to compare fractions.

​The instructional materials for enVision Florida Mathematics Grade 4 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 11 Overview, Procedural Skill and Fluency identifies the procedural skill for creating line plots and identifying data points on a line plot.
• 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 2-3, Independent Practice, students find the sum to addition problems such as 389 + 461.
• In Lesson 2-6, Independent Practice, students find differences of greater numbers using the standard algorithm.
• In Lesson 12-4, Independent Practice, students add fractions such as $$\frac{28}{10}$$ + $$\frac{72}{10}$$ + $$\frac{84}{100}$$ .
• In Lesson 15-5, Independent Practice, Problem 6 states, “Use the angle measures you know to write an equation to find the angle measure of angle EGH. What kind of angle is EGH?” Students develop fluency in adding multi-digit numbers.

The instructional materials provide regular opportunities for students to attend to Standard 4.NBT.2.4, adding and subtracting multi-digit whole numbers using the standard algorithm.

​The instructional materials for enVision Florida Mathematics Grade 4 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 5 Overview, Math Background: Rigor, Applications states, “Throughout Topic 5, there are real-world problems involving division of whole 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 Lesson 15-6, Problem Solving Performance Task, students solve real-world problems involving angles. The problem shows a picture of a mural that needs to be painted and the angles where certain items should appear in the painting. Students find the angle measures so that the mural meets specifications.
• In Lesson 2-6, Problem Solving, Question 19, “On Monday, from the peak of Mount Kilimanjaro, a group of mountain climbers descended 3,499 feet. On Tuesday, they descended another 5,262 feet. How many feet did the mountain climbers descend after 2 days? How many more feet do they have to descend to reach the bottom?”
• In Lesson 10-4, Problem Solving, Question 11, students find the difference in lengths of times for a boat trip that takes 2$$\frac{2}{4}$$ hours and a canoe trip that takes 3$$\frac{1}{4}$$ hours.
• In Lesson 9-8, Problem Solving, Question 24 states, “Joe biked 1$$\frac{9}{12}$$ miles from home to the lake, then went some miles around the lake, and then back home. Joe biked a total of 4$$\frac{9}{12}$$ miles. How many miles did Joe bike around the lake?”
• In Lesson 10-4 Problem Solving Performance Task, students answer questions about the amount of paint they would need to mix in order to have different amounts of orange paint for different projects. The task involves mixing red and yellow paints to make orange paint ($$\frac{5}{8}$$ gallon of red paint and $$\frac{3}{8}$$ gallon of yellow paint are need to make the correct shade of orange).
• In Topic 10 Performance Task, students multiply whole numbers by fractions to determine how long it will take certain people to paint their portion of the mural.
• In Lesson 10-3, Problem Solving, Question 20, “It takes Mario ¼ hour to mow Mr. Harris’ lawn. It takes him 3 times as long to mow Mrs. Carter’s lawn. How long does it take Mario to mow Mrs. Carter’s lawn? Write your answer as a fraction of an hour, then as minutes.”

The instructional materials for enVision Florida Mathematics Grade 4 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 Topic 1, students develop conceptual understanding of place value by using place value charts, place value blocks, and number lines to develop understanding of the structure of our number system by writing numbers in base-ten numerals, number names, and expanded form. Students develop their conceptual understanding when they transition to working with whole number place value to comparing and rounding whole numbers in relationship to their place value.
• In Lesson 6-2, students develop procedural skill when writing and solving equations involving multiplicative comparison.
• In Lesson 1-5, students apply knowledge of place value to solve problems related to land areas that are 10 times greater than Georgia.

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

• In Lesson 3-1, Question 15, students develop conceptual understanding of the operations, work on procedural fluency skills, and apply their understanding by multiplying products and finding the difference between the products. “How much money did they save on two children’s tickets for Plan C instead of buying separate tickets for Plan A and Plan B?” Students must find two products before calculating how much money was saved.
• In Lesson 1-3, students apply their procedural skills to problems with various constraints and use their conceptual understanding of place value to compare whole numbers and explain how their solutions represent the given situation. Students are shown models of place value charts, drawings, inequality symbols, and lining up numbers vertically based on place value.
  

​The instructional materials reviewed for enVision Florida Mathematics Grade 4 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 lesson and topic level include:

• In Topic 1, Topic Overview states, “Students persevere as they try to understand problems involving place value, plan how to solve them, and consider whether their answers make sense.” (MP.1.1)
• In Lesson 3-1, MP.7.1 is identified: “Students use basic facts and patterns to multiply a one-digit number by a multiple of 10, 100, or 1,000.”
• In Topic 5, Topic Overview identifies MP.6.1: “Students attend to precision when they use symbols, numbers, or drawings to solve problems involving division of whole numbers.”

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 Topic 5, 3-Act Math task, MP.4.1 is identified and linked 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)
• In Lesson 15-3, Problem Solving, Problem 14, MP.5.1 Use Appropriate Tools: “What is the measure of the angle of the yellow hexagon pattern block?” The Guidance for the Teacher states: “Have students put three pattern-block hexagons together so that there is no space between the figures. How many angles make up the center of the figure? How can this help you find the measure of one angle?”
• In Lesson 9-8, Problem 25, Reasoning, identifies MP.2.1: “The bus took 4$$\frac{3}{5}$$ hours to get from Jim’s home station to Portland and 3$$\frac{4}{5}$$ hours to get from Portland to Seattle. How long did the bus take to get from Jim’s home station to Seattle? Teacher guidance includes: “Students can use a bar diagram to decide what computation is needed to solve this problem. They also may want to write an equation to show how quantities are related in the problem.”
• In Lesson 11-1, MP.4.1 is identified: “Students consider how the data shown in a line plot models the real-world data that it represents”

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 7-4, MP.2.1 is identified. “Students will use abstract reasoning when connecting factors of a number to the possible number of rectangular arrays that can be made to represent that number.”
• In Lesson 9-10, MP.4.1 is identified. “All of the problems in this lesson elicit the use of multiple mathematical practices. For example, making sense of problems and persevering are required to solve all problems. Any mathematical practices that come into play in the work on this lesson should be made explicit. However, the classroom conversation should focus on the meaning and use the Thinking Habits shown on the Solve and Share task for mathematical modeling.”
• In Lesson 6-6, Guided Practice, the teaching notes identify MP.1.1. “Listen and look for these behaviors as evidence that students are exhibiting proficiency with this practice. Give a good explanation of the problem, thinking about a plan before jumping into the solution, think of similar problems, try special cases, or use a simpler form of the problem. If needed, organize data or use representations to help make sense of the problem, identify likely strategies for solving the problem, pause when solving problems to make sure that the work being done makes sense, make sure the answer makes sense before stopping work, do not give up when stuck, and look for alternative ways to solve the problem when stuck."

​The instructional materials reviewed for enVision Florida Mathematics Grade 4 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:

• In Lesson 2-2, Problem Solving, Question 16: “Elle says, when rounding to the nearest thousand, 928,674 rounds to 930,000. Do you agree? Explain.”
• In Lesson 3-6, Problem Solving, Question 14 directs students to “Critique Reasoning: Quinn used compensation to find the product of 4 x 307.  First, she found 4 x 300 = 1200. Then she adjusted the product by subtracting 4 groups of 7 to get her final answer of 1,172.  Explain Quinn’s mistake and find the correct answer.”
• In Topic 3, Topic Assessment, Question 17A:, “Mr. Luca would like to purchase a digital keyboard for each of his 3 nieces and 1 nephew. The keyboard costs $105. Mr. Luca thinks the total cost should be about $200. Is this amount reasonable? Explain.”
• In Lesson 13-4, Problem Solving, Question 12 directs students to  “Critique Reasoning: Milo thinks 8 hours is greater than 520 minutes. Is Milo correct? Remember 1 hour is equal to 60 minutes.”

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

• Lesson 1-5, Independent Practice, Question 5: “Construct a math argument that explains why Gerald did not write the population of his city correctly;” Question 6: “Correct Gerald’s argument. Explain how to compare the populations of Gerald's and Emily’s cities.”
• In Lesson 2-4, Visual Learning Bridge, Convince Me!, students use mathematical understanding and procedural fluency to construct an argument of how the standard algorithm can be used to regroup more than 10 tens. “When using the standard algorithm to add 24,595 + 19,255, how do you regroup 1 ten + 9 tens + 5 tens?”
• In Lesson 8-7, Solve and Share Activity, students construct a mathematical argument to compare fractions. “Sherry and Karl both started their hike with a small bottle filled with water. Tia started her hike with a larger bottle that was $$\frac{1}{2}$$ full. At the end of the hike, Sherry and Tia’s bottles were each half filled with water. Karl’s bottle was $$\frac{1}{3}$$ filled with water. Who has the most water left? Construct a math argument to support your answer.”

​The instructional materials reviewed for enVision Florida Mathematics Grade 4 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 1-4, Critique Reasoning (in the margin), the materials provide questions for teachers that pertain to how answers could be different when numbers are rounded to different places.
• In Lesson 2-2, Construct Arguments (in the margin), the materials provide questions for teachers that pertain to which of the two versions of the problem on the page results in the correct answer.
• In Lesson 2-6, “After Whole Class” provides teachers opportunities to engage students in analyzing 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 subtract. Some students may break the numbers apart and others may use the standard algorithm. If needed, show and discuss the provided student work at the right.” There are also prompting questions to support teachers if they have the students analyze the provided student work.
• In Lesson 5-4, Critique Reasoning (in the margin) provides teachers with questions that pertain to why a calculation where the remainder is greater than the divisor is incorrect.
• In Lesson 13-4, “After Whole Class” provides teachers with opportunities to engage students in analyzing 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 determined and described the relationship between the two measurements. 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 4 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 4 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 the Topic Planner, there is a vocabulary column that lists the words addressed within each lesson in the topic. For example, Lesson 10-1 lists the following word: unit fraction. These same words are listed in the Lesson Overview.
• Lesson 1-1 introduces numbers through one million and expanded form. Within the Visual Learning Bridge, students write numbers in expanded form. The definition of expanded form is developed as students write a number in expanded form to show the sum of each digit. For example, “Each digit in 356,039 is written in its place on the chart. Expanded form shows the sum of the values of each digit.” Within the Guided Practice Activity, students write numbers in expanded form, and during further practice, there are discussion questions for the teacher to help solidify the concept of expanded form. “What is the same about the three expanded form equations that are shown? How is the first equation different from the other two? What is another way you could write 21,125 in expanded form?”
• In Lesson 7-4, students interpret prime and composite numbers. Students use the context to build proper mathematical vocabulary.
• Lesson 14-1 introduces the mathematical idea of a rule to students. The Visual Learning Bridge provides definitions and models/diagrams using this new vocabulary. Within the Guided Practice Section, students are provided questions within the context of the lesson to answer using vocabulary. For example, Question 17 states, “Vocabulary Define rule. Create a number pattern using the rule, “Subtract 7.” 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.