The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that they assess grade-level content, and if applicable, content from earlier grades. In instances where above-level content is assessed, questions could easily be omitted or modified by the teacher. Probability, statistical distributions, similarities, transformations, and congruence do not appear in the assessments.

Assessments are found in the Teacher Guide and the Assessment Sourcebook. Topic Assessment and Performance Tasks are provided at the end of every unit to assess student understanding of standards taught in the Topic. Cumulative/Benchmark Assessments are given after a group of topics have been taught. Customizable Digital Assessments allow teachers to edit, add questions, and build tests from scratch.

Questions assessing grade-level content include, but are not limited to:

• Topic 6, Assessment, Question 1, states, “Jessica and her 2 sisters want to take a camping trip. They have $225 saved. Each of them will save $21 a week until they have at least $512 to pay for the trip. How much money will they save after 4 weeks? Will they have enough money to pay for the trip?” Students solve multi-step word problems with whole numbers and multiple operations (4.OA.3).
• Topic 16, Performance Task, states, “The Ottoman Empire lasted from 1299 until 1922. Much of the art from this period contained geometric shapes. The enlarged part of the figure shows 4 triangles that are all the same type. Classify these triangles by their sides and by their angles. Explain.” Students classify the triangles shown by properties of their lines and angles (4.G.2).
• Topics 1-4, Cumulative/Benchmark Assessment, Question 14, “Which of the following shows how to find 4 x 567? Which property was used?” Students use understanding of place value and properties of operations to perform multi-digit multiplication (4.NBT.5).
• Topic 12, Assessment, Question 3, states, “Lucy buys a puzzle for $3.89, a model airplane for $12.75, and a stuffed animal for $2.50. How much money did she spend in all? Draw or use bills and coins to solve.” Students perform operations with decimals to hundredths (4.MD.2).
• Topic 12, Assessment, Question 5, states, “Catalina takes the money shown to the bookstore. Does Catalina have enough for all three books? If not, how much more money does Catalina need? Explain. Draw or use bills and coins to solve. Catalina chooses to buy only 2 of the books. Choose two books for Catalina to buy, and then find how much money she will have left. Draw or use bills and coins to solve.” Students perform operations with decimals to hundredths (4.MD.2). 
• Topic 12, Performance Task, Question 2, states, “Analyze the amount of money that the students raised. How much more money did Yuna raise than Ali? Draw bills and coins to show your work.” Students perform operations with decimals to hundredths (4.MD.2).

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet  expectations for spending a majority of instructional time on major work of the grade. 

Evidence includes, but is not limited to:

• The approximate number of Topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 13 out of 16, which is approximately 81%.
• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 91 out of 104, which is approximately 88%.
• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 120 out of 144, which is approximately 83%. 

A lesson level analysis is most representative of the instructional materials since the lessons include major work, supporting work connected to major work, and assessments embedded within each topic. As a result, approximately 88% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Supporting standards/clusters are used to support major work of the grade and are connected to the major standards/clusters of the grade.

Examples of connections between supporting and major work of the grade include, but are not limited to:

• In Lesson 7-2, students use multiplication knowledge (4.NBT.5) to find factors and factor pairs of whole numbers (4.OA.4). Question 25 states, “Any number that has 9 as a factor also has 3 as a factor. Why is this?”
• In Lesson 10-4, students solve word problems involving units of time (4.MD.2) using fractions and mixed numbers (4.NF.3d and 4.NF.4c). Question 11 states, “A boat ride at the lake lasts $$2\frac{2}{4}$$ hours. A canoe trip down the river lasts $$3\frac{1}{4}$$ hours. Show each time on the number line. How much longer is the canoe trip than the boat ride in hours? minutes?”
• In Lessons 11-1 and 11-3, students solve problems using data from line plots (4.MD.4) to compare, add, and subtract fractions (4.NF.2 and 4.NF.3d). Lesson 11-1, Question 12 provides a line plot with time in fraction form and states, “How much longer was the greatest amount of time spent completing the project than the least amount of time?”
• In Lessons 13-4 and 13-5, students convert metric units of length (4.MD.1) to solve problems involving operations with whole numbers (4.OA.3). Lesson 13-4, Question 14 states, “Signs are placed at the beginning and at the end of a 3-kilometer hiking trail. Signs are also placed every 500 meters along the trail. How many signs are along the trail? Explain.”
• In Lesson 14-3, students generate shape patterns following a rule (4.OA.5) and solve problems dividing whole numbers (4.NBT.6). Question 13 states, “Josie puts beads on a string in a repeating pattern. The rule is ‘Blue, Green, Yellow, Orange.’ There are 88 beads on her string. How many times did Josie repeat her pattern?”
• In Lesson 15-2, students find angle measures (4.MD.5) using fractional parts of a given whole (4.NF.3b). Question 11 states, “A mirror can be used to reflect a beam of light at an angle. What fraction of a circle would the angle shown turn through?”

Instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the amount of content designated for one grade-level is viable for one year. 

As designed, the instructional materials can be completed in 144 days. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications. 

• There are 104 daily content-focused lessons. According to the Pacing Guide, “Each core lesson including differentiation, takes 45-75 minutes.” 
• There is a Topic/Vocabulary Review and Assessment for each of the 16 topics, which are suggested to take two days per topic.
• There are eight 3-Act Math activities where students solve problems using mathematical modeling, which are found in odd-numbered topics and are allotted one day each.

According to the Pacing Guide, additional time can be spent on the following resources (TE 23A):

• Lesson Resources: More days can be spent on some lessons for conceptual understanding, skill-development, and differentiation.
• Additional Resources: More days can be spent on the Math Diagnosis and Intervention System and the 10 Step-Up Lessons used after Topic 16.
• Assessments: More days can be spent on the Readiness Test, Review What You Know, Cumulative/Benchmark Assessments, and Progress Monitoring Assessments (Forms A, B, and C). 

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations for the materials being consistent with the progressions in the Standards. Content from prior grades is identified and connected to grade-level work, and students are given extensive work with grade-level problems. All grade-level standards are present in the Teacher Edition Program Overview “Grade 4 Common Core Standards.”

The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. The Teacher Edition contains a Topic Overview Coherence: Look Back, which identifies connections to content taught in previous grades or earlier in the grade, indicating the relevant topics and/or lessons. In addition, Overview Coherence: Look Ahead includes connections to content taught later in the grade and in future grades, topics, or lessons. For example, the Teacher Edition, Topic 4 Overview, Math Background: Coherence, includes:  

• “Look Back, Grade 3: 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. Earlier in Grade 4, Topic 3 students used arrays, area models, the distributive property, and partial products to multiply multi-digit numbers by 1-digit numbers.”
• “Connections within Topic 4 include: Students use rounding to estimate products and estimation to check reasonableness of answers. Students use arrays, area models, and the distributive property throughout the topic as they use partial products to find the product of two 2-digit numbers. Students apply strategies for whole-number multiplication to solve real-world problems.”
• “Look Ahead: In Topic 5, students use their understanding of multiplication and their skill in multiplying with patterns, models, and partial quotients to divide by 1-digit numbers. In Topic 6, students use multiplication to compare. In Grade 5, Topic 3 students use the standard algorithm to multiply multi-digit whole numbers. In Topic 4, students use models and strategies to multiply decimals to hundredths.”

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. All topics include a topic project, and every other topic incorporates a 3-Act Mathematical Modeling Task. During the Solve & Share, Visual Learning Bridge, and Convince Me! sections, students explore ways to solve problems using multiple representations and prompts to reason and explain their thinking. Guided Practice provides students the opportunity to solve problems and check for understanding before moving on to the Independent Practice. During Independent Practice, students work with problems in a variety of formats to integrate and extend concepts and skills. The Problem Solving section includes additional practice problems for each of the lessons. For example, students engage in extensive work with Standard 4.NBT.5 grade-level problems in Topic 3: Use Strategies and Properties to Multiply by 1-Digit Numbers, including:

• In Topic 3, 3-Act Math, students watch a video of a boy covering the sides of a box with square stickers. Students make predictions to determine “How many stickers do you need to cover the box?” Students discuss what information is needed to solve the problem and are given additional information in order to model the solution.
• In Lesson 3, Convince Me!, students use the distributive property to break apart 13 as 10 + 3 and multiply each addend by 5. The question states, “How are the partial products represented with the place-value blocks?”
• In Lesson 7, Problem Solving, Question 23 states, “Elaine rents a car for 5 days. It costs $44 each day to rent the car and $7 each day for insurance. At the end of the trip she spends $35 to fill the car with gas. What is the total cost for Elaine to use the car?”

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. In the Math Background: Coherence section for each topic, the Teacher Edition provides explicit connections to prior learning, but standards are not provided. Additionally, some lesson Look Back sections detail connections to previous grades.

Connections to prior grade-level learning include, but are not limited to:

• In Topic 2, Math Background Coherence: Fluently Add and Subtract Multi-Digit Whole Numbers the Look Back states, “In Grade 3, Topics 8 and 9, students developed fluency in adding and subtracting whole numbers within 1,000. They estimated sums and differences, added and subtracted mentally, and used strategies such as partial sums and partial differences.”
• In Topic 6, Math Background Coherence: Use Operations with Whole Numbers to Solve Problems the Look Back states, “In Grade 3, Topics 1-5, students solved word problems involving basic facts and the foundational understandings of multiplication and division. In Topics 8 and 9, students solved word problems as they developed fluency with addition and subtraction through 1,000. In Topic 10, students solved word problems involving multiplying by multiples of 10.” 
• In Lesson 9-1, the Look Back states, “In earlier grades, students developed an understanding of the meaning of addition of whole numbers as joining, and they developed an understanding of the meaning of a fraction $$\frac{a}{b}$$ as a number of unit fractions $$\frac{1}{b}$$.” In this lesson, students use fraction strips and number lines to add fractions.
• In Lesson 12-2, the Look Back states, “In Grade 3, students represented fractions on a number line. In Grade 4, Lesson 8-2, they represented equivalent fractions on a number line.” In this lesson students locate and describe fractions and decimals on number lines.

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Examples include, but are not limited to:

• The Topic Planner states Topic 2: “Focuses on developing fluency with standard algorithms for addition and subtraction (4.NBT.B)”. For example, in Lesson 2-2, “Students use rounding to estimate sums and differences.”
• In Lesson 6-1, the Mathematics Objective states, “Interpret comparisons as multiplication or addition equations.” This is shaped by 4.OA.A: “Use the four operations with whole numbers to solve problems.”  
• In Lesson 12-2, the Mathematics Objective states, “Locate and describe fractions and decimals on number lines.” This is shaped by 4.NF.C: “Understand decimal notation for fractions, and compare decimal fractions.” 

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples include, but are not limited to:  

• Lesson 2-2 connects 4.NBT.B to 4.OA.A when students use strategies for estimating sums and differences presented in the context of solving a multi-step word problem.  
• Lesson 5-3 connects 4.OA.A to 4.NBT.B when students solve multi-step word problems involving the four operations.
• Lesson 8-3 connects 4.NF.A to 4.NBT.B when students find equivalent fractions using multiplication.
• Lesson 14-2 connects 4.OA.5 to 4.NBT.2 when students use rules to develop shape patterns and solve problems involving multiplication and division. In Independent Practice, Problem 5: Rule Divide by 5, students complete a chart with the number of fingers listed as 5, 10, 15, 20; and the number of hands as 1, 2, _, _. 

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each lesson is structured to include background information for the teacher and problems and questions that develop conceptual understanding. Examples include, but are not limited to:

• Conceptual understanding for each topic is outlined in the Teacher Edition’s section Math Background: Rigor. For example, the Topic 3 Overview explains using number sense to estimate, partial products, and the distributive property to multiply multi-digit numbers by 1-digit. The Conceptual Understanding section states, “Arrays and area models are used to foster understanding of partial products. The partial products can be added in any order and the result is the same. The distributive property is the basis for breaking apart a multi-digit factor by place value and generating partial product.” 
• The Teacher Edition contains a Rigor section for each lesson explaining how conceptual understanding is developed in the lesson. For example, Lesson 8-1, the Rigor section states, “Students use an area model to demonstrate that two fractions are equivalent when they name the same part of the same whole.”
• Each lesson is introduced with a video: Visual Learning Animation Plus, to promote conceptual understanding.  For example, the Lesson 8-1 video states, “What are some ways to name the same part of a whole?” The scenario begins by saying, “James ate part of the pizza shown in the picture. He said $$\frac{5}{6}$$ of the pizza is left. Cardell said $$\frac{10}{12}$$ of the pizza is left. Who is correct?” Students should see that $$\frac{5}{6}$$ and $$\frac{10}{12}$$ are equal.
• Each lesson begins with a Visual Learning Bridge activity that provides the opportunity for a classroom conversation to build conceptual understanding for students. In Lesson 8-1, teachers ask, “How much of the pizza is left according to James? According to Cardell?  What do you need to do? What does the denominator of the fraction tell you? What does the numerator tell you? How can you tell from the picture that $$\frac{5}{6}$$ of the pizza is left? Why is the first area model labeled $$\frac{5}{6}$$? Why is the second area model labeled $$\frac{10}{12}$$? Why are $$\frac{5}{6}$$ and $$\frac{10}{12}$$ equivalent?  Does it matter what shape is used to show each of the two fractions?”
• Each lesson contains a Convince Me! section that provides opportunities for conceptual understanding. In Lesson 6-4, students use bar diagrams to model the question, “Chef Angela needs 8 cartons of eggs to make the cakes that are ordered. She has 2 cartons of eggs and 4 single eggs in the refrigerator. How many more eggs does she need to make all of the cakes?”
• Each lesson contains a Do You Understand? section which makes a connection to previous learning and provides opportunities for conceptual understanding. In Lesson 8-2,  “Students use number lines to find equivalent fractions that represent the same point on the number line.” Students also explain why two fractions are equivalent.

Practice problems provide students opportunities to independently develop conceptual understanding. Examples include, but are not limited to:

• In Lesson 5-7, Questions 5-8, students use drawings of place-value models and the concept of division as equal shares to find 2-digit quotients with and without remainders. For example, Question 6 states, “___ = 176 ÷ ____.” (4.OA.3)
• In Lesson 9-7, Questions 4-11, students use fraction strips and number lines to extend work with fractions to include mixed numbers. Question 5 shows a number line which students use to find the sum of $$1\frac{2}{3}+2\frac{2}{3}$$. (4.NF.3c)
• In Topic 8, Performance Task, students are told, “During gym class, the fourth-grade students climbed on a rope hanging from the ceiling.  The Rope Climbing table shows what part of the rope several students climbed.” The table displays students and the part of the rope that they climbed using the fractions: $$\frac{4}{6}$$, $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{2}{3}$$, $$\frac{5}{6}$$, $$\frac{4}{3}$$. Problem 1, Part A, states, “Compare how high the students climbed.  Who climbed a greater part of the rope, Gia or Jim? Use benchmark fractions to compare. Explain.” Part B, states, “Who climbed a greater part of the rope, Gia or Jason? Use the number line to compare.”  Part C states, “Who climbed a greater part of the rope, Rachel or Russ? Justify your comparison using fraction strips.” (4.NF.2)

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

Problem sets provide opportunities to practice procedural fluency. Regular opportunities for students to attend to Standard 4.NBT.4: add and subtract fluently within 1,000,000, are provided.

The instructional materials develop procedural skill and fluency throughout the grade-level. Examples include, but are not limited to:

• Each topic contains a Math Background: Rigor page with a section entitled “Procedural Skill and Fluency.”  In Topic 2, “Students use a variety of procedures for using mental math to add and subtract multi-digit whole numbers.  They also use rounding to estimate sums and differences. Students develop fluency in adding and subtracting multi-digit whole numbers using the standard algorithms. The standard addition and subtraction algorithms are shortened versions of strategies involving partial sums and partial differences. Students begin by lining up the numbers by place value. Then they add (or subtract) the ones, then the tens, then the hundreds, and so on, regrouping as necessary."
• Each lesson contains a Visual Learning Bridge which provides instruction on  procedural skills. Students make and interpret line plots. In Lesson 11-2, the Visual Learning Bridge states, “The manager of a shoe store kept track of the lengths of the shoes sold in a day. Complete the line plot using the data from the shoe store."
• Fluency Practice Activities are found at the end of Topics 2-16 to support adding and subtracting fluently within 1,000,000. In Topic 4, the Fluency Activity provides a numbered chart and states, “Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match.” For example, “The sum is greater than 300 and less than 400” matches “283 + 38."
• The Performance Task for Topic 2 provides students the opportunity to demonstrate fluency when adding and subtracting multi-digit whole numbers.  Students are given a table displaying items ordered in April and May and asked to complete tasks such as, “Use an algorithm to find how many more items of fruit and yogurt were ordered to fill the vending machine in April than in May."
• Students can practice fluency skills when accessing the Game Center Online at PearsonRealize.com.

The instructional materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade-level. Examples include, but are not limited to: 

• In Topic 2 there are six Fluency Practice/Assessment worksheets for students to complete to practice 4.NBT.4. In Lesson 2-2, Question 11, students estimate sums and differences, “485,635 - 231,957= __."
• In Topic 6: Fluency Practice Activity: Students subtract 3-digit numbers and then find a matching clue on the page: “917-365 = _____.” (4.NBT.4) 
• In Lesson 12-5, Independent Practice, students work on adding and subtracting multi-digit numbers in world problems. Problem 5 states, "Carlos spends $14.38 on equipment.  How much change should Carlos receive if he gives the clerk $20.00?” (4.NBT.B.4, 4.MD.A.2)

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of grade-level mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Materials provide opportunities for students to solve a variety of problem types requiring the application of mathematics in context. Additionally, the materials support teachers by explaining how the students will apply concepts they have learned within each topic in the Math Background: Rigor section of the Topic Overview. 

Students are provided opportunities to work with routine problems presented in context that require application of mathematics.  Examples include, but are not limited to:

• In Lesson 6-1, the Visual Learning Bridge states, “Max said the Rangers scored 3 times as many runs as the Stars. Jody said the Rangers scored 8 more runs than the Stars. Could both Max and Jody be correct?”
• In Lesson 12-6, the Problem Solving Performance Task states, “Tomas deposits money in his savings account every month. If he continues to save $3.50 each month, how much money will he have at the end of 6 months? 12 months?”  

Students are provided opportunities to work with non-routine problems presented in context that require application of mathematics.  Examples include, but are not limited to:

• In Lesson 1-3, Solve & Share, students use a chart of ocean depths to solve,  “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 anyway you choose.”
• In Lesson 6-5, the Performance Task states, “Rainey’s group designed the flag shown for a class project. They use 234 square inches of green fabric. After making one flag, Rainey’s group has 35 square inches of yellow fabric left. How can Rainey’s group determine the total area of the flag?” 

Students are provided opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples include, but are not limited to:

• In Lesson 2-5, Question 19, students are provided a map containing information about the areas of certain counties.  Students determine, “How much greater is the area of Hernando County than Union County?”
• In Lesson 4-2, Question 11, students use place value blocks, area models, or arrays to solve, “During a basketball game, 75 cups of fruit punch were sold. Each cup holds 20 fluid ounces. How many total fluid ounces of fruit punch were sold?” 
• In Lesson 8-7, Question 3 states, “In the after-school club, Dena, Shawn, and Amanda knit scarves that are all the same size with yellow, white and blue yarn.  Dena’s scarf is $$\frac{3}{5}$$ yellow, Shawn’s scarf is $$\frac{2}{5}$$ yellow, and Amanda’s scarf is $$\frac{3}{4}$$  yellow. The rest of each scarf has an equal amount of white and blue. Describe how Amanda could make the argument that her scarf has the most yellow.”
• In Lesson 10-4, Question 9 states, “There are 55 minutes between the time dinner ends and the campfire begins. What is the elapsed time from the beginning of dinner to the beginning of the campfire?” 

The instructional materials for enVision Mathematics Common Core Grade 4 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.  

All three aspects of rigor are present independently throughout the program materials. Examples include, but are not limited to:

• Conceptual Understanding is needed to solve Lesson 2-2, Question 14. Students use conceptual understanding of number sense to estimate when asked, “The table shows the number of students at each school in the district. Is 2,981 reasonable for the total number of students at Wilson Elementary and Kwame Charter School? Explain."
• Fluency is practiced in Lesson 2-5, Questions 7-18. Students use the standard algorithm to subtract 3-digit numbers and use estimation to check reasonableness. Question 7 states, “289 - 145."
• Students apply mathematics to solve problems in context. In Lesson 10-3, Question 19 states, “Oscar wants to make 4 chicken pot pies. The recipe requires $$\frac{2}{3}$$ pound of potatoes for each pot pie. How many pounds of potatoes will Oscar need?”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include, but are not limited to:

• In Lesson 3-4, students use conceptual understanding of area models to build procedural skill using the distributive property to find products. In Question 5, students use area models and partial products to solve, “3 x 185."
• In Lesson 7-5, students develop conceptual understanding of multiples and use procedural skill to list multiples of a whole number.  Question 25 states, “Latifa and John played a game of multiples. Each player picks a number card and says a multiple of that number. Latifa picked a 9. Write all the multiples of 9 from the box." 
• In Lesson 12-5, students use their understanding of bills and coins to represent money amounts to solve problems in context. Question 4 states, “Sarah bought 3 wool scarves. The price of each scarf was $23.21. How much did 3 scarves cost?”  
• In Lesson 15-6, students use their understanding of rulers and protractors to solve problems in context. Question 2 states, “Lee brought $$1\frac{3}{5}$$ pounds of apples to the picnic. Hannah brought $$\frac{4}{5}$$ pound of oranges. Less said they bought $$2\frac{2}{5}$$ pounds of fruit in all. Lee needs to justify that $$1\frac{3}{5}+\frac{4}{5}=2\frac{2}{5}$$ . How can Lee use a tool to justify the sum? Draw pictures of the tool you used to explain."

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

All eight Standards for Mathematical Practice (MPs) are clearly identified throughout the materials. Math Practices are identified in the Topic Planner by lesson. In addition, Math Practices and Effective Teaching Practices (ETP)  are identified for each topic, within each lesson, and for specific problems.

• In Topic 4, the Topic Planner states, “MP.4, and MP.7 are addressed in Lesson 5.”
• In Topic 4, the Math Practices and ETP addresses MP4: “Model with mathematics. Students model with math when they use arrays and equations to represent multiplication. (e.g., p. 142, Convince Me!).”
• In Lesson 3-6, Mathematical Practice states, “MP.2 Reason Abstractly and Quantitatively: Students make sense of quantities and their relationships in problem situations by representing a multiplication problem symbolically and manipulating the symbols. Also MP.3, MP.7."

The MPs are used to enrich the mathematical content and are not treated separately. A Math Practices and Problem Solving Handbook is available online at PearsonRealize.com.  This resource provides a page on each math practice for students and teachers to use throughout the year. Math Practice Animations are also available for each practice to enhance student understanding. For example:

• MP1: In Lesson 6-4, Question 8, students make sense of problems and persevere in solving them by using modeling to solve a multi-step problem, and checking reasonableness of the answer. For example, “Cody and Max both solve the problems below correctly. Explain how each solve. Emma has $79 to spend at the toy store. She wants to buy a building set, a board game, and 2 action figures from her favorite movie. What else can she buy?” A picture providing the work of both Cody and Max is provided along with a table of toys sold and their cost.
• MP2: In Lesson 3-2, Visual Learning Bridge, students reason abstractly and quantitatively when estimating products of a 1-digit number multiplied by numbers of up to 4-digits.  The student task states, “Mr. Hector’s class sold calendars and notepads for 3 weeks as a class fundraiser. About how much did Mr. Hector’s class make selling calendars? Selling notepads?” Teachers are prompted to ask, “Is the estimated earnings for the notepads greater than or less than the actual amount? How do you know?” 
• MP3: In Lesson 11-4, Solve & Share Activity, students critique the reasoning of others. For example: “A class made a line plot showing the amount of snowfall for 10 days. Nathan analyzed the line plot and said, ‘The difference between the greatest amount of snowfall recorded and the least amount of snowfall recorded is 3 because the first measurement has one dot and the last measurement has 4 dots.’ How do you respond to Nathan’s reasoning?"
• MP7: In Lesson 3-2, Look Back, students use place value structure to multiply. The student task states, “A theatre contains 14 rows of seats with 23 seats in each row. How many seats are in the theater? Solve this problem using any strategy you choose. Theater seating is an example of objects that are arranged in rows and columns, or arrays. How do the number of rows and the number of seats in each row relate to the total number of seats?”
• MP8: In Lesson 13-1, Convince Me!, students use repeated reasoning to generalize about multiplying to get a greater number of units when converting from a larger unit to a smaller unit. For example: “Maggie has a tree swing. How many inches long is each rope from the bottom of the branch to the swing? How do you know the answer is reasonable when converting a larger unit to a smaller unit?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. 

Materials do not attend to the full meaning of MP4 because students are given a model or told what model to use rather than having to model their mathematical thinking. Examples of questions labeled MP4: Model with Math that do not attend to the full meaning of the standard include, but are not limited to:

• In Lesson 5-1, Question 28 states, “On Saturday afternoon, 350 people attended a play.  The seating was arranged in 7 equal rows. Draw a bar diagram and solve an equation to find p, how many people sat in each row”   
• In Lesson 6-1, the Look Back states, “Sarah is making a square pillow with edges that each measure 18 inches long. She needs a strip of fabric 4 times as long as one edge of the pillow to make a border around the pillow. How long does the strip of fabric need to be? How could a bar diagram help you write an equation for the problem?” In Lesson 10-5, Question 5 states, “Draw bar diagrams and write equations to find g, how many gallons of paint are in a batch and b, how many batches Perry needs to make."  

Materials do not attend to the full meaning of MP5 because students are given tools rather than being able to choose a tool to support their mathematical thinking. Examples of questions labeled MP5 that do not attend to the full meaning of the standard include, but are not limited to:

• In Lesson 8-2, the Visual Learning Bridge states, “Show $$\frac{3}{4}$$ on the number line. Divide each fourth into two equal parts to show eighths. Divide each fourth into three equal parts to show twelfths” (Teacher’s Edition, page 298). 
• In Lesson 8-1, Visual Learning Animation Plus states, “James ate part of the pizza shown in the picture at the right. He said $$\frac{5}{6}$$ of the pizza is left. Cardelll said $$\frac{10}{12}$$ of the pizza is left. Who is correct? Why is the first area model labeled $$\frac{5}{6}$$? Why is the second area model labeled $$\frac{10}{12}$$? Why are $$\frac{5}{6}$$ and $$\frac{10}{12}$$ equivalent?”
• In Lesson 9-6, Convince Me! states, “Use the number line below to find $$\frac{5}{8}+\frac{2}{8}$$.  Can you also use the number line to find $$\frac{5}{8}-\frac{2}{8}$$? Explain.” 

Materials attend to the full meaning of MP1, MP2, MP6, MP7, and MP8. Examples include, but are not limited to:

• MP1: In Lesson 3-8, Question 7, students make sense of problems and persevere in solving them. For example, “A truck like the one shown delivers a load of gasoline to a gas station 3 times a week. The storage tank at the gas station holds 9 loads of fuel. How much more gas does the storage tank hold than the truck? What do you know and what do you need to determine?” 
• MP2: In Lesson 1-3, Convince Me!, students use quantitative reasoning to analyze relationships between place value positions to compare numbers. For example, “Is a whole number with 4 digits always greater than or less than a whole number with 3 digits? Explain."
• MP6: In Lesson 3-5, Question 9, students attend to precision. For example, “There are usually 365 days in each year.  Every fourth year is called a leap year and has one extra day in February. How many days are there in 8 years if 2 of the years are leap years?”
• MP7: In Lesson 2-5, Convince Me!, students use the structure of the place-value system as a basis for understanding the standard algorithm for subtraction. For example, “How many times do you need to regroup to subtract 483 - 295? Explain.” 
• MP8: In Lesson 7-3, Solve & Share, students look for and express regularity in repeated reasoning. For example, “A closet company sells wooden storage cubicles. Jane bought 24 cubicles. She wants to arrange them in a rectangular array. What are all the different ways Jane can arrange them, using all of her cubicles? Explain how you know you found them all." 

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The Solve & Share activities, Visual Learning Bridge problems, Problem Sets, 3-Act Math activities, Problem Solving: Critique Reasoning problems, and Assessment items provide opportunities throughout the year for students to both construct viable arguments and analyze the arguments of others.

Student materials consistently prompt students to construct viable arguments. Examples include, but are not limited to:

• In Lesson 7-2, the Visual Learning Bridge states, “Jean wants to arrange her action figures in equal-size groups. What are all the ways Jean can arrange her action figures?” Students are presented with three different methods for arranging the action figures. The next question in the Convince Me! section states, “How do you know there are no other factors for 16 other than 1, 2, 4, 8, and 16? Explain." 
• In Lesson 8-7, the Solve & Share states, students determine how full 3 bottles are with water by comparing fractions. The problem states, “If Tia’s bottle was $$\frac{1}{3}$$ filled with water at the end of the hike, would you be able to decide who had the most water left? Construct an argument to support your answer." 
• In Lesson 16-4, Problem Solving, students answer several questions about symmetry. Question 22, labeled as “Construct Arguments,” states, “How can you tell when a line is NOT a line of symmetry?” 

Student materials consistently prompt students to analyze the arguments of others. Examples include, but are not limited to:

• In Lesson 1-2, Question 9 states, “Vin says in 4,346, one 4 is 10 times as great as the other 4. Is Vin correct? Explain.”
• In Lesson 3-6, Question 14 states, “Quinn used compensation to find the product of 4 x 307. First, she found 4 x 300 = 1,200. 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 Lesson 8-6, Convince Me! states, “The fractions on the right refer to the same whole. Kelly said, ‘These are easy to compare. I just think about $$\frac{1}{8}$$ and $$\frac{1}{6}$$.’ Circle the greater fraction. Explain what Kelly was thinking." 
• In Lesson 15-2, Convince Me! states, “Susan thinks the measure of angle B is greater than the measure of Angle A. Do you agree? Explain."

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Materials assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others in a variety of problems and tasks presented to students. Examples include, but are not limited to:

• In Lesson 1-5, Convince Me! states, “Construct Arguments, teachers are given an Essential Teaching Practice (ETP), Construct Arguments; providing a clear and complete explanation for conjecture involves using objects, drawings, diagrams, and actions to support the argument. When constructing a viable argument students use mathematical terms and symbols correctly. They should use definitions and previously solved problems when deciding when another students' explanation makes sense. Point out that Bella’s conjecture makes sense because it can be supported with a correct and clear explanation including diagrams, mathematical terms, and symbols.”
• In Lesson 2-3, Question 17 states, “Harmony solved this problem using the standard algorithm, but she made an error. What was her error, and how can she fix it? 437 + 175 = 5,112.” The teacher edition states, “Encourage students to estimate their sum before trying to determine Harmony’s error. Students should see because 400 + 200 = 600, Harmony’s sum is not reasonable."
• In Lesson 7-4, Convince Me! states, “Generalize: Can a number be both prime and composite? Explain.” The teacher edition states, “Students use the definitions of prime and composite numbers to generalize that all whole numbers greater than 1 are classified as either prime or composite."

The instructional materials reviewed for enVision Mathematics Common Core Grade 4 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Examples include, but are not limited to:

• Each topic contains a Vocabulary Review providing students the opportunity to show their understanding of vocabulary and use vocabulary in writing.
• The Teacher Edition provides teacher prompts to support oral language. In Topic 3, the Oral Language prompt states, “Before students complete the page, you might reinforce oral language through a class discussion involving one or more of the following activities. Have students say math sentences that use the words." 
• The Game Center at PearsonRealize.com contains an online vocabulary game. 
• A vocabulary column is provided in the Topic Planner that lists words addressed with each lesson in the topic. In Lesson 2-1, the Vocabulary List includes: Commutative Property of Addition, Associative Property of Addition, Identity Property of Addition, Count Up, Count Down, and Compensation. These words are also listed in the Lesson Overview. 
• Online materials contain an “Academic Vocabulary” and an “Academic Vocabulary Teacher’s Guide” section. The guide supports vocabulary instruction by providing information on how teachers can develop word meaning and build word power. The Academic Vocabulary section provides a variety of academic words with definitions and activities to help students learn the words. For instance, when clicking on the word, conjecture, the definition is provided: “a statement that is thought to be true but has not been proven." Next, the word is used in context: “Make a conjecture about which expression has a lesser sum. 205 + 627 or 354 + 428." Lastly, students are provided a task to help build word power: “Use the word in a sentence."
• A glossary exists in both the Student Edition and the Teacher’s Edition Program Overview. In the glossary, breaking apart is defined as: “Mental math method used to rewrite a number as the sum of the numbers to form an easier problem."
• Visual Learning Bridge activities provide explicit instruction in the use of mathematical language. The words are highlighted in yellow and a definition is provided. 
• A bilingual animated glossary is available online which uses motion and sound to build understanding of math vocabulary.

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. Examples include, but are not limited to:

• In Lesson 6-1, Visual Learning Bridge, students learn how to display data using a line plot. The materials define line plots as: “A line plot shows data along a number line. Each dot above a point on the line represents one number in the data set.” To further explain, a line plot is displayed and the materials state, “Here is how the data look on a line plot."
• In Lesson 12-1, the Visual Learning Bridge introduces factor pairs by highlighting in yellow and giving the definition. It states, “Pairs of whole numbers multiplied together to find a product are called factor pairs. Think about multiplication to decompose a number into its factors.” 
• In Lesson 15-1, Questions 16-18 directions state, “For 16-18, use the map of Nevada. Write the geometric term that best fits each description. Draw an example." 

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 the visual design is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

Student print and digital materials follow a consistent format. Tasks within a lesson are numbered to match the teacher guidance. The print and visuals on the materials are clear without any distracting visuals. Student practice problem pages include space for students to write their answers and demonstrate their thinking. In the student’s digital textbook, audio support is provided for Solve & Share and Convince Me! problems. Vocabulary is highlighted when used in the textbook, provided in bold print in independent practice, and an icon reminds students that vocabulary can be found in the glossary. 

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 partially meet expectations that materials contain a teacher’s edition with full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The Teacher Edition Program Overview includes resources to help teachers understand the mathematical content, the overarching philosophy of the program, a user’s guide, and a content guide. Additionally, each topic contains a Professional Development Video explaining the mathematical concepts of the lessons with examples that are clearly explained. 

A Math Background is provided for each topic and lesson identifying the connections between previous grade, grade level, and future-grade mathematics. However, these do not support teachers in understanding the underlying Mathematical Progressions.

The Assessment Source Book, Teacher Edition, and Mathematical Practices and Problem Solving Handbooks provide answers and sample answers to problems and exercises presented to students. However, there are no adult-level explanations to build understanding of the mathematics of these tasks.

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 provide a list of lessons in the teacher's edition, cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter, and unit.

The Teacher Edition Program Overview provides a visual showing the number of lessons per topic by domains. It also provides a Pacing Guide showing how many total days, by topic, the material will take. Support for lessons requiring additional time is provided: “Each Core lesson, including differentiation, takes 45-75 minutes. The Pacing Guide above allows for additional time to be spent on the following resources during topics and/or at the end of the year."  A resource list is provided.

The instructional materials for enVision Mathematics Common Core Grade 4 contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

A Home-School Connection letter to families and caregivers is provided for each topic. The letter provides an overview of what students will be learning and an activity that the family can complete together. These letters are available in both Spanish and English.

The instructional materials for enVision Mathematics Common Core Grade 4 contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials draw on research to explain and contextualize instructional routines and lesson activities. The Teacher Edition Program Overview contains specifics about the instructional approach. Additionally, the Teacher Edition Program Overview explains Instructional Routines. Examples include, but are not limited to:

• The Efficacy Research section states, “First, the development of enVision Mathematics started with a curriculum that research has shown to be highly effective."
• The Research Principles for Teaching with Understanding section states, “The second reason we can promise success is that the enVision Mathematics fully embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously-learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle."
• Problem Solving Lessons are explained: “Throughout enVision Mathematics, the eight math practices are infused in lessons. Each Problem Solving lesson gives special focus to one of the eight math practices. Features of these lessons include the following: Solve & Share, Visual Learning Bridge, Convince Me!, Guided Practice, Independent Practice, Performance Task, and Additional Practice."

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 partially meet expectations that assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

End of Topic Assessments and Topic Performance Tasks contain a Scoring Guide assigning point values to each question. However, there is no rubric or sample answers to assist the teacher in scoring student written responses.

Assessments can be taken online where they are automatically scored, and students are assigned appropriate practice, enrichment, or remediation based on their results.  However, teachers must interpret the results on their own and determine materials for follow-up when students take print assessments.

The instructional materials for enVision Mathematics Common Core Grades 3, 4, and 5 do not include opportunities for students to monitor their own progress. Materials do not provide support or components for students to monitor their progress.

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. 

he instructional materials for enVision Mathematics Common Core Grade 4 provide opportunities for teachers to use a variety of grouping strategies.

Materials include teacher-led instruction that present limited options for whole-group, small-group, partner, and/or individual work. When suggestions are made for students to work in small groups, there are no specific roles suggested for group members, but teachers are given suggestions and questions to ask to move learning forward. The Visual Learning Bridge Animation Plus focuses on independent work, while the Pick a Project and 3-Act Math activities have opportunities to work together in small groups or partners.

The instructional materials for enVision Mathematics Common Core Grade 4 encourage teachers to draw upon home language and culture to facilitate learning.

The Teacher Edition Program Overview includes Supporting English Language Learners, which contains ELL Instruction and Visual Learning. English Language Learners' support for each lesson is provided for the teacher throughout lessons to provide scaffolding for reading, as well as differentiated support based on student language proficiency levels (emerging, expanding, or bridging). The Home-School Connection letters for each topic are available in both English and Spanish. There is also an English Language Learners Toolkit available that consists of Professional Development Articles and Graphic Organizers.

The instructional materials for enVision Mathematics Common Core Grade 4 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Teachers and students have access to tools and virtual manipulatives within a given activity or task, when appropriate. Pearson Realize provides additional components online such as games, practice, instructional videos, and links to other websites. In the print Teacher Edition, there are statements in each lesson noting when resources are available online.

Examples include, but are not limited to:

• Animated videos explaining each of the eight Math Practices are provided. At this time only Spanish versions of these videos are provided at Pearsonrealize.com.
• An Animated Glossary embedded in the program helps students internalize the meaning of key concepts, and sometimes visual models are provided.
• The Interactive Additional Practice book provides opportunities for students to engage in the Mathematical Practices.
• Problem-Based Learning activities provide repeated opportunities for students to use precise language to explain their solutions (MP6).
• Visual Learning Animation Plus videos provided at the beginning of each lesson in the Visual Learning Bridge is an interactive way for students to understand conceptually.

The digital instructional materials for enVision Mathematics Common Core Grade 4 are web-­based and compatible with multiple internet browsers. In addition, materials are “platform neutral” and allow the use of tablets and mobile devices.

The digital materials are platform neutral and compatible with multiple operating systems, such as Windows and Apple, and are not proprietary to any single platform. Materials are also compatible with multiple internet browsers such as Internet Explorer, Firefox, Google Chrome, and Safari. Finally, materials are compatible with various devices including iPads, laptops, Chromebooks, and other devices that connect to the internet with an applicable browser.

The instructional materials for enVision Mathematics Common Core Grade 4 include opportunities to assess students' mathematical understandings and knowledge of procedural skills using technology. Examples include, but are not limited to:

• PearsonRealize.com offers online assessments and data which are found in ExamView. Teachers can assign and score material, and analyze assessment data through dashboards.
• PearsonRealize.com offers online fluency games and other program games requiring procedural skills to solve problems.
• Virtual Nerd offers tutorials on procedural skills, but there are no assessments or opportunities to practice procedural skills within the tutorials.
• Skill and Remediation activities in the Topic Readiness online assessment tab include tutorials and opportunities for students to practice procedural skills using technology. There is also a Remediation button to see online activities.

The instructional materials for enVision Mathematics Common Core Grade 4 can easily be customized for individual learners and  include digital opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. 

Teachers can select and assign individual practice items for digital student remediation based on the Topic Readiness assessment. Teachers can also create and assign classes online for students through the Accessible Student Edition. Closed Captioning is included in STEM and 3-Act Math videos. Examples include, but are not limited to:

• Math problems in the digital Student Edition have a read aloud option. Students press the speaker button to have it read aloud. 
• Some lessons and resources are provided in English and Spanish for students such as the Math Practice Animations, Interactive Additional Practice, Game Center, and Animated Glossary.  
• Students have access to digital Math Tools to solve problems in the digital Student Edition such as counter stamps, place value block stamps, erasers, shapes, number lines, grids, fraction strips, and decimal strips.   

The instructional materials for enVision Mathematics Common Core Grade 4 can be easily customized for local use and provide a range of lessons to draw from on a topic. 

There are digital materials correlated to the topic lesson of the print materials. Also, teachers can create and upload files, attach links, and attach documents from Google Drive that can be assigned to students. Additionally, teachers can create assessments from a bank of test items or teacher-written items and assign them to students.

The instructional materials for enVision Mathematics Common Core Grade 4 include or reference technology that provides opportunities for teachers and/or students to collaborate with each other.

At PearsonRealize.com, teachers can assign a discussion from a list of prompts under the  “Discuss” tab. Teachers can also go to "Classes" and attach files for students.