4th Grade - Gateway 3

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that underlying design of the materials distinguishes between problems and exercises.

The Solve & Share, Look Back, Visual Learning Bridge, and Convince Me! Sections contain problem sets that connect prior learning and/or engage students with a problem in which new math concepts are taught. The Guided Practice, Independent Practice, and Problem Solving sections provide problem sets for students to build on their understanding of the new concept. Assessment Practice problems at the end of each lesson provide opportunities for students to apply what they have learned and can be used to determine differentiation. Additional Practice problems are found in the Additional Practice Workbook that accompanies each lesson and support students in developing mastery of the current lesson and topic concepts.

Examples include, but are not limited to:

• In Lesson 4-5, the Solve & Share states, “A playground is divided into four sections as shown in the diagram below. Find the area of the playground. Explain how you found the answer. Solve this problem using any strategy you choose." The authors state the purpose of this problem as, “Students connect to their previous understanding of finding the area of a rectangle and computing partial products to find the area of a rectangular playground that is divided into four sections. Their work shows prior and emerging understandings you can build on during the Visual Learning Bridge."
• In Lesson 3-7, Independent Practice, Question 22 states, “Maura’s dance team wants to buy costumes that cost $56 each. They have $523 saved in a fund. How much money will they have left in the fund after they buy 9 costumes?” Students independently choose an appropriate strategy to multiply 2-, 3-, and 4-digit numbers by 1-digit numbers.

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that design of assignments is not haphazard and exercises are given in intentional sequences.

Lessons are structured to build mastery. First, students are introduced to concepts and procedures with a problem-solving experience in the Solve & Share section. Next, the important mathematics are explicitly explained with visual direct instruction and connected to the problem-solving experience in the Visual Learning Bridge. Finally students are assessed at the end of each lesson so appropriate differentiation can be provided in the Assessment Practice section.

The following is an example of sequential learning from Lesson 5-4: Interpret Remainders:

• Step 1: Solve & Share: “There are 47 students taking a field trip. The students are being driven in cars to a play by adult volunteers. Each driver can take at most 4 students. How many cars are needed for the field trip? Will each car have four students? Use counters or draw pictures to solve this problem. Explain how you found your answer." The authors state the purpose of this lesson as, “Students connect to their understanding of finding quotients to find and interpret a remainder in order to solve a division problem. Their work shows prior and emerging understandings you can build on during the Visual Learning Bridge." 
• Step 2: Visual Learning Bridge: “Ned has 27 soccer cards in an album. He put 6 cards on each page. He knows 27 ÷ 6 = 4 with 3 left over, because 6 x 4 = 24 and 24 + 3 = 27." Students are explicitly taught how to analyze the relationship between the remainder and divisor.
• Step 3: Assessment Practice: “There are 39 children at a park. They want to make teams with 9 children on each team. Two of the children go home. How many complete teams can they make? Explain."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide variety in what students are asked to produce.

The instructional materials prompt students to produce answers and solutions within the Solve & Share, Guided Practice, Independent Practice, Problem Solving, and 3-Act Math sections. Students are also given opportunities to produce oral arguments and explanations during lesson discussions. Additionally students critique fictional student work. Finally, students are often prompted to solve problems “any way they choose” which provides opportunities for students to create diagrams and mathematical models. Examples include, but are not limited to:

• In Lesson 7-4, Question 10, students are shown the number 7 and then asked to, “Tell whether each number is prime of composite." 
• In Lesson 2-1, Look Back! states, “How could you use mental math to solve 1,289 + 1,566? 1,034 + 1,566? How is the thinking different?” 
• In Lesson 4-5, Question 4, students are shown 18 x 25 and asked to “Draw an area model to find each product."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

Manipulatives and other mathematical representations are aligned to expectations and concepts in the standards. Visual manipulatives are embedded within the problem sets to represent ideas and build conceptual understanding. Students and teachers have access to digital manipulatives to build conceptual understanding and solve problems.

Examples include, but are not limited to:

• Students have access to place value charts, number lines, place value blocks, $$\frac{1}{4}$$ in grid paper, 2-color counters, 2 color square counters, centimeter grid paper, fraction strips, clock faces, decimal models, decimal place value charts, hundredths grids, money, centimeter rulers, meter sticks, pattern blocks, protractors, inch rulers, and yardsticks.

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Each lesson contains an overview with discussion questions to increase classroom discourse, support for the teacher of what to look for, and ways to ensure understanding of the concept. Essential questions found at the beginning of each topic are revisited throughout the topic and teaching strategies for answering the Topic Essential Questions are provided in the Topic Assessment pages. Examples include, but are not limited to:

• Topic 6, Essential Questions, students are asked, “How is comparing with multiplication different from comparing with addition? How can you use equations to solve multi-step problems?” 
• In Lesson 2-2, Solve & Share, Discussion Questions,  students are asked, “How much does Sarah earn each week? How long has Sarah saved her weekly earnings?” 
• In Lesson 14-2, Visual Learning Bridge, Discussion Questions,  students are asked, “What is the rule for finding the number of leaflets on 4 cloverleafs? What features are in the pattern in the table?” 

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials.

Each topic contains a Topic Planner providing an overview of every lesson, which includes: Lesson Objectives, Essential Understanding, Vocabulary, Materials Needed, Technology and Activity Centers, and Math Standards. The Topic Planner also includes lesson resources such as the Digital Student Edition, Additional Practice Workbook, available print materials, Digital Lesson Courseware, and lesson support for teachers. Examples include, but are not limited to:

• Visual Learning Bridge lessons include a Visual Learning Animation Plus for each lesson. 
• Digital math tools and games, technology resources, and PDF work pages available for each lesson are noted.
• Each Lesson Overview includes an Objective and an Essential Understanding, “I can” learning target statements written in student language, CCSSM that are either being “built upon” or “addressed” for the lesson, Cross-Cluster Connections, the aspect(s) of rigor addressed, support for English Language Learners, and possible Daily Review pages with Today’s Challenge to be implemented.
• Each lesson activity contains an overview, guidance for teachers, student-facing materials, anticipated misconceptions, extensions, differentiation support based on Quick Checks, and opportunities for further practice in the online materials. 
• Annotations and suggestions on how to present the content within the lesson structure of: Step 1: Engage and Explore; Step 2: Explain, Elaborate, and Evaluate; and Step 3: Assess and Differentiate are provided. The corresponding Launch section explains how to set up the activity and what to tell students.

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials contain a teacher’s edition that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Materials explain how mathematical concepts are built from previous grade levels or topics and lessons as well as how the grade-level concepts fit into future grade-level work. Additionally, Look Back, This Lesson, Look Ahead, and Cross-Cluster Connections are found in the Coherence Section for each lesson. Examples include, but are not limited to:

• In Topic 7, Math Background: Coherence, the Look Ahead states, “Use common factors to write equivalent fractions, extend the concept of factors and multiples to fraction multiplication. In Grade 5 students will use common multiples to write fractions with a common denominator." 
• In Lesson 2-4, Lesson Overview, the Look Back states, “In Lesson 2-3, students added numbers with 3 digits. This Lesson: students add greater whole numbers using the standard algorithm. Look Ahead: In Lesson 2-5, students will subtract whole numbers with up to 3 digits."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

The Program Overview provides information about using assessments to gather information about students’ prior knowledge. The authors state, “Readiness assessments help you find out what students already know. Formative instruction in lessons inform instruction. Various summative assessments help you determine what students have learned. Rubrics are provided for assessing math practices. Auto-scored online assessments can be customized.” The Readiness Test can be printed or distributed digitally. In this assessment, prerequisite skills from the prior grade necessary for understanding the grade-level mathematics are assessed. The Daily Review is designed to engage students in thinking about the upcoming lesson and/or to revisit previous grades' concepts or skills. Review What You Know, found in the Topic Opener, gathers information about prior knowledge and provides an Item Analysis for Diagnosis and Intervention. Examples include, but are not limited to:

• In Grade, 4 Readiness Assessment, Item 4 states, “Round 341 to the nearest hundred” (3.OA.7).
• In Topic 6, Review What You Know, Question 10 states, “Find each product. 53 x 9." (4.NBT.5)

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide strategies for teachers to identify and address common student errors and misconceptions.

Lessons include an Error Intervention section identifying where students may make a mistake or have misconceptions and how to provide support. Additionally, lessons contain side matter in the Teacher Edition that identifies possible misconceptions and ways for teachers to prevent them.  Examples include, but are not limited to:

• In Lesson 14-2, Error Intervention states, “If students get factors and multiples confused, then remind them of the difference with simple numbers.  For example, 2 is a factor of 6 and 6 is a multiple of 2. Some students may find it helpful to remember that multiple sounds like multiply and you can multiply 2 by a number to get 6, so 6 is a multiple of 2."
• In Lesson 8-4, Visual Learning Bridge, teacher side matter prompts teachers to prevent misconceptions: “Some students may try to divide the numerator and denominator by two difference factors.  Explain that dividing by 1, like multiplying by 1, and does not change the value of a number. Students must divide the numerator and denominator by the same whole number so that they are dividing by 1." 

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The lesson structure, consisting of Solve & Share, Visual Learning Bridge, Guided Practice, Independent Practice, Problem Solving, and Assessment Practice, provides students with opportunities to connect prior knowledge to new learning, engage with content, and synthesize their learning. Throughout the lesson, students have opportunities to work independently, with partners, and in groups, and review, practice, and feedback are embedded into the instructional routine. In addition, practice problems for each lesson activity reinforce learning concepts and skills, and enable students to engage with the content and receive timely feedback. Discussion prompts in the Teacher Edition provide opportunities for students to engage in timely discussion on the mathematics of the lesson.

Examples include, but are not limited to:

• Each Topic includes a “Review What You Know/Concept and Skills Review” that includes a Vocabulary Review and Practice Problems. This section includes review and practice on concepts that are related to the new topic. 
• The Cumulative/Benchmark Assessments, found at the end of Topics 4, 8, 12, and 16, provide review of prior topics as an assessment. Students can take the assessment online, with differentiated intervention automatically assigned to students based on their scores. 
• Different games online at Pearson Realize support students in practice and review of procedural skills and fluency.

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials offer ongoing formative and summative assessments that clearly denote which standards are being emphasized. 

Assessments are located in the Assessment Book or online portion of the program and can be accessed at any time. For each topic there is a Practice Assessment, an End-Unit Assessment, and a Performance task. Assessments in the Teacher Edition provide a scoring guide and standards alignment for each question. Examples include, but are not limited to:

• In Topic 3, Performance Task, Question 3 states, “Miranda’s teacher is Ms. Katz. Her class wants to buy 9 laptop computers and 4 printers. Miranda said the total cost should be $7,494. Is this amount reasonable? Explain. Does the class have enough money?”  This question is noted as being DOK Level 3 and addresses 4.OA.3, 4.NBT.5, and MP1.
• In Topic 6, Assessment, Question 7 states, “Darcy ordered 18 boxes of red balloons and 12 boxes of blue balloons for a party. She ordered a total of 240 balloons. How many balloons are in each box?” This question is noted as DOK Level 2 and addresses 4.OA.3.

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The materials provide a detailed Scope and Sequence and the Topic Overview identifies prerequisite skills. Each lesson contains a Daily Review and a Solve & Share Activity that reviews prior knowledge and/or prepares students for the activities that follow. Each lesson contains explicit instructional support for sequencing and scaffolding. Lesson side matter provides guidance on discussion questions, sample student work, and look fors. Step 3: Assess and Differentiate, contains optional activities that can be used for additional practice or support before moving on to the next activity or lesson. Examples include, but are not limited to:

• In Lesson 13-4, Solve & Share, teacher prompts to check for understanding state, “In what units should you measure the marker? What are you asked to describe?” To prevent misconceptions, teacher side matter states, “Students might count all the small millimeter markings, or they might convert centimeters to millimeters. If needed, ask, ‘What do the bold markings and light markings on the ruler represent?’”
• In Topic 4, Math Background: Coherence, students use strategies and properties to multiply by 2-digit numbers. In the Look Back section, the materials note the Grade 3 standards needed: “In Topics 1, 2, 3, and 5, students learned about multiplication and developed fluency with the basic multiplication facts. In Topic 10, students used place value patterns to multiply 1-digit numbers by multiples of 10."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Additional Practice Materials include a lesson for each topic that includes specific questions for the leveled assignment for all learning ranges, Intervention, On-Level, and Advanced with verbal, visual, and symbolic representations. Response to Intervention strategies for each lesson give teachers “look fors,” suggestions to address the needs of struggling students, and discussion questions. Additional examples within the lesson help students extend their understanding of the concept being taught and include extra problems for the teacher to use. Differentiated Interventions, Reteach to Build Understanding, and Enrichment sections provide reteach scaffolding and concept extensions. Examples include, but are not limited to:

• In Lesson 3-3, Reteach to Build Understanding, students are presented with vocabulary and then guided through a series of problems that have partially completed portions: “Use the array to show 5 x 137. Use the array to find the partial products. 5 x 7 = ______, 5 x 30 = ______, and 5 x 100 = ______." 
• In Lesson 3-3, Enrichment, students match multiplication expressions with models and solutions: “Draw lines to match each expression in the first column to the expression or model that represents the same problem in the second column. Then draw a line from the second column to the product in the third column."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The Solve & Share, Visual Learning Bridge, Guided and Independent Practice, and Quick Check/Assessment Practice sections provide opportunities for students to apply mathematics from multiple entry points. Materials sometimes ask students to use a specific strategy, but questions within the lesson allow students to use a variety of strategies. Lesson and task narratives provided for teachers offer possible solution paths and presentation strategies for various levels. Examples include, but are not limited to:

• In Lesson 1-3, Solve & Share states, “A robotic submarine can dive to a depth of 26,000 feet. Which oceans can the submarine explore all the way to the bottom? Solve this problem any way you choose.” Students are presented with a table of the data needed to solve the problem. The helping character in the text states, “You can model with math. Use what you know about place value to help solve the problem.” (4.NBT.2)
• In Lesson 6-3, Independent Practice, Question 6 states, “You have $350 to buy 26 tickets for a baseball game. You need to buy some of each kind of seat. You want to spend most of the money. How many of each type of ticket can you buy? Find two different solutions to the problem. Use an equation to show each solution. Box seats are $18 each. Upper deck seats are $12 each." (4.OA.3)

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The ELL Design is highlighted in the Teacher Edition Program Overview and describes support based on the student’s level of language proficiency: emerging, expanding, or bridging, as identified in the WIDA (World-Class Instructional Design and Assessment) assessment. An ELL Toolkit provides additional support for English Language Learners. ELL suggestions are provided in Solve & Share and Visual Learning Bridge activities. Visual Learning support is also embedded in every lesson to support ELL learners. 

Support for other special populations is also provided in the Teacher Edition Program Overview. Resources and a key are provided for Ongoing Intervention during a lesson, Strategic Intervention at the end of the lesson, and Intensive Intervention as needed at anytime. The Math Diagnosis and Intervention System (MDIS) supports teachers in diagnosing students' needs and providing more effective instruction for on- or below-grade-level students. Diagnosis, Intervention Lessons, and Teacher Support are provided through teachers' notes to conduct a short lesson where vocabulary, concept development, and practice can be supported. Online Auto Design Differentiation is also included, and supports the program after a lesson, a topic, assessments, or groups of topics. Teachers can track student progress using Assignment Reports and analyzing Usage Data. Examples include, but are not limited to:

• In Lesson 8-4, Solve & Share, English Language Learner support for Expanding students states, “Have students read the first two sentences of the Solve & Share with their partners. Then have each partner draw a number line to represent what is meant by tenths."
• In Lesson 11-2, Visual Learning Bridge, English Language Learner support states, “Have students find the words scale, halves, fourths, and eighths. Explain that the scale on the line plot shows the units of the data being measured." Support for Bridging ELL students states, “Have students explain the relationship between the words scale, halves, fourths, and eighths. Have students explain how equivalent fractions help them draw the dots on the line plot correctly."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Materials provide extension activities for each Solve & Share activity. Also, Independent Practice problems contain Higher Order Thinking items. Additionally, Enrichment activities follow the Quick Check Assessment in each lesson which can be used for differentiation. STEM activities are provided in the Activity Center. Finally, Additional Practice contains Advanced problems for students. However, teacher guidance is not provided for advanced students activities.nExamples include, but are not limited to:

• In Lesson 4-2, Solve & Share states, “Erica’s class collected 4,219 bottles for the recycling center. Ana’s class collected 3,742 bottles. Leon’s class collected 4,436 bottles. How many bottles did the three classes collect? Solve this problem any way you choose.” Extension problem states, “What is the least number you can add to 7,527 that would result in regrouping all four place values?” 
• In Lesson 8-4, Higher Order Item, Question 29 states, “If the numerator and denominator of a fraction are both odd numbers, can you write an equivalent fraction with a smaller numerator and denominator? Give an example to explain."

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics.

Lessons contain tasks including various demographic and personal characteristics. All names and wording are chosen with diversity in mind, and the materials do not contain gender biases. Materials include a set number of names used throughout the problems and examples (e.g., Maria, Salvatore, Carly, Elle, Luke, Carl, Ramon, Li, Nadia, Hakeem, Jerome, Chico, and Leesa). These names are presented repeatedly and in a way that does not stereotype characters by gender, race, or ethnicity. Characters are often presented in pairs with different solution strategies and a pattern of one character using more/less sophisticated strategies does not occur. When multiple characters are involved in a scenario, they are often doing similar tasks or jobs in ways that do not express gender, race, or ethnic bias.