## enVisionMATH California Common Core

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###### Usability
Our Review Process

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### Overall Summary

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for alignment to the CCSSM. The instructional materials partially meet expectations for focus and coherence in Gateway 1 as they do not meet expectations for focus and partially meet expectations for coherence. In Gateway 2, the instructional materials partially meet the expectations for rigor and balance, and they partially meet the expectations for practice-content connections. Since the instructional materials do not meet expectations for both Gateways 1 and 2, evidence was not collected regarding usability in Gateway 3.

###### Alignment
Partially Meets Expectations
Not Rated

### Focus & Coherence

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for focus on major work and coherence in Gateway 1. The instructional materials do not meet expectations for focus as they assess topics before the grade level in which the topic should be introduced, but they do devote the large majority of class time to the major work of the grade. The instructional materials partially meet the expectations for coherence by including an amount of content designated for one grade level that is viable for one school year and fostering coherence through connections at a single grade.

##### Gateway 1
Partially Meets Expectations

#### Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 assess topics before the grade level in which the topic should be introduced. There are assessment items that assess above grade level statistics and probability standards.

##### Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 do not meet expectations for assessing grade-level content. Most of the assessments include material appropriate for Grade 4; however, there are two assessment items that assess above grade-level statistics and probability standards.

In the teacher edition, a Topic Test is available for each of the sixteen topics. In Topic 3, the instructional materials assess content that aligns to 7.SP.8. For example:

• In the Topic 3 Topic Test, question 9 states, “Betsy is making a flag. She can choose three colors from red, white, blue, and yellow. How many choices does Betsy have?”
• In the Topic 3 Topic Test, question 16 states, “Tammy wants to get change for 30 cents. The only coins she can get are quarters, nickels, and dimes. How many different ways can she get 30 cents using only these coins?”

Examples of the instructional materials assessing grade-level content include:

• In the Topic 4 Topic Test, question 1 states, “Joe got 34,867 points playing a video game, and Carlos got 29,978 points. How many more points did Joe get than Carlos?” Students subtract multi-digit whole numbers using the standard algorithm. (4.NBT.4)
• In the Topic 8 Topic Test, question 7 states, “A school bought 28 new microscopes for its students to use. The price for each microscope was 87. How much did the microscopes cost in all?” Students multiply two-digit numbers by two-digit numbers. (4.NBT.5) • In the Topic 14 Topic Test, question 12 states, “Andrea ran 400 meters in gym class. How many centimeters did she run?” Students convert measurements within a single system of measurement. (4.MD.1) #### Criterion 1.2: Coherence Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade. Students and teachers using the materials as designed devote the large majority of class time to the major work of the grade. The instructional materials devote approximately 68 percent of class time to the major work of Grade 4. ##### Indicator {{'1b' | indicatorName}} Instructional material spends the majority of class time on the major cluster of each grade. The instructional materials reviewed for enVisionMATH California Common Core Grade 4 meet expectations for spending a majority of instructional time on major work of the grade. • The approximate number of topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 12 out of 16, which is approximately 75 percent. • The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 114 out of 168, which is approximately 68 percent. • The number of weeks devoted to major work (including assessments and supporting work connected to the major work) is approximately 23 out of 34, which is approximately 68 percent. A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work, and the assessments embedded within each topic. As a result, approximately 68 percent of the instructional materials focus on major work of the grade. #### Criterion 1.3: Coherence Coherence: Each grade's instructional materials are coherent and consistent with the Standards. The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for coherence. The instructional materials include an amount of content designated for one grade level that is viable for one school year and foster coherence through connections at a single grade. The instructional materials also miss some connections between major and supporting work and do not clearly identify content from prior and future grade levels. ##### Indicator {{'1c' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Supporting standards/clusters are not always used to support major work of the grade and often appear in lessons with few connections to the major work of the grade. Throughout the series, supporting standards/clusters are typically taught in isolation and rarely connected to the major standards/clusters of the grade. Students can often complete problems aligned to supporting work without engaging in the major work of the grade. The following examples illustrate missed connections in the materials: • In Topic 11 Lesson 11-1, students find factors of whole numbers. The supporting standard of 4.OA.4 has a natural connection with the major work standard 4.NF.1. Though these standards are not contained in the same lesson, the standards are within the same topic but do not simultaneously enhance coherence in the major work of the grade. In lesson 4, students do not use the understanding that a whole number is a multiple of its factors to help find equivalent fractions. • In Topic 15 Lesson 15-2, students analyze and create line plots to solve problems. The supporting standard of 4.MD.4 is aligned to this lesson. 4.MD.4 has a natural connection to the major work standard 4.NF.B, adding and subtracting fractions. The lesson contains line plots that have whole units as well as decimal units, which is respectively below and above grade level. There are three questions in the lesson related to finding the difference between numbers; however, two of the three questions refer to line plots with whole or decimal units. Examples that illustrate connections in the materials include: • In Topic 14 Lesson 14-9, students express measurements in a larger unit in terms of a smaller unit. Independent Practice question 12 states, “8 km = __ m, 8 x 1,000 = __ m” The supporting standard of 4.MD.1 is used to enhance the focus on the major work standard 4.OA.1. • In Topic 15 Lesson 15-3, students solve real-world area and perimeter problems. Problem Solving question 11 states, “Julie planted a rectangular garden that is 20 feet long. She placed 56 feet of fencing around her garden. Draw and label a sketch of her garden. What is the width of her garden? What is the area?” The supporting standard of 4.MD.3 is used to enhance the focus on the major work standard 4.NBT.5. ##### Indicator {{'1d' | indicatorName}} The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades. Instructional materials for enVisionMATH California 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 174 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. The instructional materials consist of 120 lessons that are listed in the Table of Contents. Lessons are structured to contain a Daily Review, Develop Concept-Interactive, Develop Concept-Visual, Close/Assess and Remediate, and Center Activities. The instructional materials consist of 54 reteaching lessons and assessments that are listed in the Table of Contents. These include Reteaching, Topic Tests, Performance Assessments, Placement Test, Benchmark Tests, and End-of-Year Test. The publisher does not provide information about the suggested time to spend on each lesson or the components within a lesson. The Implementation Guide has a chart that suggests times for a multi-age classroom. The lessons within the multi-age classroom are structured differently than a single-age classroom. The multi-age lessons are structured to contain Problem Based Interactive Learning, Guided Practice, Center Activities, Independent Practice, Small Group Strategic Intervention, and Digital Assignments/Games. The suggested time for the multi-age lesson is 50-75 minutes per lesson. ##### Indicator {{'1e' | indicatorName}} Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades. The instructional materials for enVisionMATH California Common Core Grade 4 partially meet expectations for the materials being consistent with the progressions in the standards. The instructional materials do not clearly identify content from prior and future grade levels and do not use it to support the progressions of the grade-level standards. Prior and future grade-level work is not clearly identified within each lesson. For example: • In Topic 1 Lesson 1-1, the Teacher Edition lists the standard 4.OA.1 as the focus of the lesson. Students write addition sentences as well as multiplication sentences for a given set of pictures arranged in rows or groups. Independent Practice question 6 states, “Write an addition sentence and a multiplication sentence for each picture.” This is prior grade-level content aligned to 3.OA.1. • In Topic 3 Lesson 3-6, the Teacher Edition lists the standard 4.NBT.1 as the focus of the lesson. Students find and record all possible outcomes for a situation. This is future grade-level content aligned to 7.SP.8. • In Topic 13 Lesson 13-5, the Teacher Edition lists the standards 4.NF.5 and 4.NF.6 as the focus of the lesson. Students name the point on a number line from a given fraction. The Guided Practice question 1 states, “Use the number line below to name the fraction. 1/8” This is prior grade-level content aligned to 3.NF.2. • In Topic 13 Lesson 13-5, the Teacher Edition lists the standards 4.NF.5 and 4.NF.6 as the focus of the lesson. Students name the point on a number line from a given decimal. The decimal is not a multiple of 10. The Guided Practice question 3 states, “Name the point on the number line for each decimal. 1.33” This is future grade-level content aligned to 5.NBT.3. Some of the lessons include a section in the Teacher Edition called, Link to Prior Knowledge. The Link to Prior Knowledge poses a question or strategy that has previously been learned for students to connect to the current lesson. The Link to Prior Knowledge does not explicitly identify standards from prior grades. For example: • In Topic 11 Lesson 11-5, the Link to Prior Knowledge states, “Have students fold one of the paper strips into halves, another into fourths, and the third into eighths. They should mark the ends with 0 and 1, and mark each fold with the appropriate fractions: e.g., 1/4, 2/4, 3/4. Have each student lay their fraction strips next to a number line strip and transfer their fractions on to the number line. Do any of the marks from the fraction strips line up? Sample answers: Yes, some of the folds from the paper strips name the same point on the number line.” The publisher does not connect this prior knowledge to a specific prior grade level. The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The majority of lessons within the 16 topics focus on and provide students with extensive opportunities to practice grade-level problems. Within each lesson, students practice grade-level problems within Daily Common Core Review, Practice, Reteaching, Enrichment, and Quick Check activities. For example: • In Topic 5 Lessons 1-4, the Teacher Edition lists the standard 4.NBT.5, Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models, as the focus of the lesson. Students multiply whole numbers using strategies based on place value and properties of operations. In lesson 5-1, Independent Practice question 9 states, “Draw and array and find each product. 9 x 10” • In Topic 9 Lesson 9-3, the Teacher Edition lists the standard 4.NBT.6, find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation using equations, rectangular arrays, and/or area models, as the focus of the lesson. Students find whole-number quotients using strategies based on place value. Guided Practice question 1 states, “Use multiplication facts to help estimate each quotient. 3,340/8” • In Topic 13 Lesson 13-1, the Teacher Edition lists the standard 4.NF.4a, apply and extend previous understandings of multiplication to multiply a fraction by a whole number, as the focus of the lesson. Students use models when multiplying fractions by a whole number. Independent Practice question 7 states, “Write the fraction as a multiple of a unit fraction. Use fraction strips to help. 3/4 = ___ x 1/4” The instructional materials contain a Common Core State Standards Skills Trace for each topic that can be found in the Printable Resources section of the Program Resources Document. This document contains the grade-level standards for each topic and the standards from previous and future grade levels that are related to the standards focused on in the specified topic. The document states the specific topic numbers from previous and future grades to which the grade-level standards are related. • In Topic 15, the Skills Trace lists the standard 4.MD.2 as the focus of the topic. This standard is linked to a “Looking Back” list where it lists the standard 3.MD.2 as the focus in Topic 15 within the Grade 3 instructional materials. The standard 4.MD.2 is also linked to a “Looking Ahead” list where it lists the standard 5.MD.1 as the focus in Topic 13 within the Grade 5 instructional materials. ##### Indicator {{'1f' | indicatorName}} Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to 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. The instructional materials for enVisionMATH California Common Core Grade 4 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards. Each topic is structured by a specific domain and the learning objectives within the lessons are clearly shaped by CCSSM cluster headings. For example: • In Topic 1 Lesson 1-7, the lesson objective states, “Students will multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison.” This is shaped by the cluster 4.OA.A, Use the four operations with whole numbers to solve problems. • In Topic 5 Lesson 5-2, the lesson objective states, “Students will use basic multiplication facts and numbers patterns to multiply by multiples of 10 and 100.” This is shaped by the cluster 4.NBT.B, Use place-value understanding and properties of operations to perform multi-digit arithmetic. • In Topic 12 Lesson 12-2, the lesson objective states, “Students use computational procedures to add fractions with like denominators and solve problems.” This is shaped by the cluster 4.NF.B, Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Instructional materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade in cases where the connections are natural and important. • In Topic 4 Lesson 4-6, cluster 4.NBT.B connects to cluster 4.OA.A when students use models/diagrams and equations involving multi-digit numbers to solve word problems. Guided Practice question 1 states, “Solve. Draw a picture to help you. Sandy earned36 from babysitting and $15 for doing her chores. Write an equation and find the total amount, t, that Sandy earned.” • In Topic 7 Lesson 7-4, cluster 4.NBT.A connects to cluster 4.NBT.B when students use estimation and rounding to solve multiplication problems involving two two-digit numbers. • In Topic 10 Lesson 10-7, cluster 4.OA.A connects to cluster 4.NBT.B when students solve multi-step word problems posed with multi-digit whole numbers and having whole-number solutions. Independent Practice question 10 states, “Use the data at the right for 8 through 11. How much more would it cost to buy 24 T-shirts at Just Jerseys than at Shirt Shack?” ###### Overview of Gateway 2 ### Rigor & Mathematical Practices The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for rigor and mathematical practices. The instructional materials partially meet expectations for rigor by meeting expectations on giving attention to the development of procedural skill and fluency and balancing the three aspects of rigor. The instructional materials also partially meet the expectations for practice-content connections by meeting expectations on explicitly attending to the specialized language of mathematics and prompting students to construct viable arguments and analyze the arguments of others. ##### Gateway 2 Partially Meets Expectations #### Criterion 2.1: Rigor Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application. The instructional materials for enVisionMATH California Common Core Grade 4 partially meet expectations for rigor and balance. The instructional materials meet expectations for giving attention to the development of procedural skill and fluency and balancing the three aspects of rigor. However, the instructional materials partially meet expectations for giving attention to conceptual understanding and applications. ##### Indicator {{'2a' | indicatorName}} Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The instructional materials for enVisionMATH California Common Core Grade 4 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The instructional materials present a Problem-Based Interactive Learning activity (PBIL) and a Visual Learning Bridge (VLB) within each lesson to develop conceptual understanding. However, the PBIL and VLB are teacher-directed and do not offer students the opportunity to practice conceptual understanding independently through the use of pictures, manipulatives, and models. Overall, the instructional materials do not consistently provide students opportunities to independently demonstrate conceptual understanding throughout the grade level. • In Topic 4 Lesson 4-3, the Overview of the PBIL states, “Students will use place-value blocks to add whole numbers.” In the teacher-directed PBIL activity, students use place-value blocks to add multi- digit numbers. The Develop the Concept: Visual section of the lesson describes three separate steps to add multi-digit numbers without place-value blocks. Step 3 states, “Add the hundreds, regroup, and then add the thousands.” The directions for the Independent Practice state, “In 9 through 24, find each sum.” Students do not demonstrate conceptual understanding of adding multi-digit numbers independently as sums are shown as sample answers in the 16 problems in the Independent Practice. • In Topic 8 Lesson 82, the Overview of PBIL states, “Students will use arrays and an expanded algorithm to multiply two-digit numbers.” In the teacher-directed PBIL activity, students draw arrays to demonstrate partial products when multiplying two-digit numbers. The directions for the Independent Practice state, “In 5 through 8, find all the partial products. Then add to find the product.” Students do not demonstrate conceptual understanding of partial products independently as products are shown as sample answers in the four problems in the Independent Practice. • In Topic 11 Lesson 11-4, the Focus question of PBIL states, “How can you name two fractions that name the same part of the whole?” Fraction strips are used during this teacher-directed activity to find equivalent fractions. In the Develop the Concept: Visual section of the lesson, fraction strips are used to check if fractions are equivalent. However, in the Guided and Independent practice, students multiply or divide to find equivalent fractions. The directions for the Independent Practice state, “For 9 through 16, multiply or divide to find equivalent fractions.” • In Topic 13 Lesson 13-1, the Overview of PBIL states, “Students fold paper strips to investigate the representation of a fraction as a multiple of a unit fraction.” In the teacher-directed PBIL activity, students fold into fourths, color 3 out of 4 of the sections, and label. The process is repeated with a second strip, but each section is cut, thus showing students an equivalent fraction. The directions for the Independent Practice state, “For 7 through 22, write the fraction as a multiple of a unit fraction. Use fraction strips to help.” Students do not demonstrate conceptual understanding of fractions as multiples of unit fractions independently as expressions are shown as sample answers in the 16 problems in the Independent Practice. ##### Indicator {{'2b' | indicatorName}} Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The instructional materials for enVisionMATH California Common Core Grade 4 meet expectations for attending to those standards that set an expectation of procedural skill and fluency. The instructional materials provide regular opportunities for students to attend to the standard 4.NBT.4, Fluently add and subtract multi-digit whole numbers using the standard algorithm. The instructional materials develop procedural skill and fluency throughout the grade level. • In Topic 4 Lesson 4-1, the instructional materials demonstrate several mental math strategies such as breaking apart, counting on, and compensation to add and subtract multi-digit numbers. The Guided Practice includes opportunities for students to practice solving addition and subtraction problems by choosing a mental math method. (4.NBT.4) • In Topic 4 Lesson 4-2, the Develop the Concept: Visual section of the lesson models the use of rounding to estimate sums and differences of multi-digit whole numbers. The materials develop procedural skill when showing an addition problem as well as the rounded estimate next to each addend of the problem. (4.NBT.4) • In Topic 4 Lesson 4-4, the Develop the Concept: Visual section of the lesson models the standard algorithm in three separate steps to find the difference of multi-digit numbers. Step 2 states, “Subtract the tens. Subtract the hundreds. Regroup if necessary.” (4.NBT.4) • In Topic 12 Lesson 12-7, the Guided Practice section of the lesson develops procedural skill by modeling the use of fraction strips when solving problems involving the addition and subtraction of mixed numbers with like denominators. Question 2 states, “Use fraction strips to find each sum or difference. Simplify, if possible. 4 1/4 - 3 3/4 = ____” (4.NF.3c) • In Topic 16 Lesson 16-5, the Guided Practice section of the lesson develops procedural skill when students draw an example of a right isosceles triangle. Question 6 states, “Is it possible to draw a right isosceles triangle? If so, draw an example.” (4.G.2) The instructional materials provide opportunities to demonstrate procedural skill and fluency independently throughout the grade level. • In Topic 4 Lesson 4-5, the Independent Practice section of the lesson provides multi-digit subtraction practice problems for students to demonstrate knowledge of procedural skill. Question 17 states, “450 - 313” (4.NBT.4) • In Topic 5 Lesson 5-3, the Common Core Review includes a subtraction word problem. Question 5 states, “The distance between Boston and Cincinnati is 840 miles. The distance between Boston and Philadelphia is 296 miles. How many miles closer is Philadelphia to Boston than Cincinnati to Boston?” (4.NBT.4) • In Topic 6 Lesson 6-2, the Common Core Review includes an addition word problem. Question 5 states, “There are 395 adults, 137 children, and 78 dogs living in an apartment building. How many people live in the building?” Students add three multi-digit numbers to find the solution. (4.NBT.4) • In Topic 12 Lesson 12-8, the Independent Practice section of the lesson provides practice solving addition of mixed numbers with like denominators. Question 7 states, “2 5/6 + 5 4/6 = ____” (4.NF.3c) • In Topic 15 Lesson 15-1, the Problem Solving section of the lesson includes questions that involve creating line plots. Question 9 states, “For 9-11, measure the lengths to the nearest quarter inch of 12 classroom objects that are between 1 and 6 inches long. Record your measurements. Draw a line plot to show the data.” (4.MD.4) ##### Indicator {{'2c' | indicatorName}} Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade The instructional materials for enVisionMATH California Common Core Grade 4 partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. Each topic includes at least one Problem Solving lesson that can be found at the end of the topic. These lessons offer students opportunities to integrate and apply concepts and skills learned from earlier lessons. Within each individual lesson, there is a section titled, Problem Solving, where students practice the application of the mathematical concept of the lesson. However, the applications of mathematics in Problem Solving are routine problems. The instructional materials have few opportunities for students to engage in non-routine application throughout the grade level. Examples of routine applications, where a solution path is readily available, are: • In Topic 2 Lesson 2-6, students use the four operations to solve a multi-step word problem with whole numbers. Independent Practice problem 4 states, “Find the number of each kind of object in Anya’s collection. 6 minerals, 3 fewer gemstones than rocks. 15 objects in all.” • In Topic 7 Lesson 7-5, students use the four operations to solve multi-step word problems involving whole numbers. Independent Practice problem 5 states, “Abby buys 15 sunflower plants and 12 petunia plants to plant in her garden. She plans to plant 3 flowers in each row. How many rows of flowers will Abby plant?” • In Topic 8 Lesson 8-5, students use multiplication and addition to solve a multi-step word problem with whole numbers. Independent Practice problem 5 states, “Dave plans to retile his porch floor. He wants to buy 25 black tiles and 23 white tiles. Each tile costs$2. How much money, m, will it cost Dave to retile his porch floor? Write an equation for each problem and then solve.”
• In Topic 12 Lesson 12-3, students solve subtraction word problems involving fractions with like denominators. Problem Solving problem 31 states, “To avoid their predators, ghost crabs usually stay in burrows most of the day and feed mostly at night. Suppose a ghost crab eats 1/8 of its dinner before 10:00 pm. By midnight, it has eaten 5/8 of its food. How much of its food did it eat between 10:00 pm and midnight?”
• In Topic 13 Lesson 13-3, students use multiplication of fractions to solve real-world problems dealing with making fruit punch, batches of pudding, and bread. Problem Solving problem 22 states, “Sun is making 7 fruit tarts. Each tart needs 3/4 cup of strawberries and 1/4 cup of blueberries. What is the total amount of fruit that Sun needs for her tarts?”
##### Indicator {{'2d' | indicatorName}}
Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

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

Lessons include components that serve to develop the three aspects of rigor. These include a Daily Common Core Review, Problem-Based Interactive Learning, Develop the Concept: Visual, Guided and Independent Practice, and Problem Solving. All three aspects of rigor are present independently throughout each topic in the materials. For example, in Topic 6:

• In Lesson 6-1, students develop conceptual understanding of partial products when using models and place value blocks to multiply 12 x 3.
• In Lesson 6-3, students practice the procedural skill of the standard algorithm of multiplication when multiplying two-digit numbers by one-digit numbers.
• In Lesson 6-6, students apply knowledge of multiplication properties to solve word problems.

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

• In Topic 13 Lesson 13-6, students develop conceptual understanding of fractions and equivalent decimals using decimal grids while applying that knowledge to solve word problems. Problem Solving problem 33 states, “A band has 20 instruments. Tyler says that 2/5 of the instruments are string instruments and 0.5 of the instruments are wind instruments. Does the band have the same number of wind instruments and string instruments? Explain.”
• In Topic 10 Lesson 10-4, students develop conceptual understanding of dividing a multi-digit number by a one-digit divisor while practicing the procedural skill of the standard algorithm of division when using a template or drawing a picture to solve the problem. Independent Practice problem 6 states, “In 5 through 13, divide. You may draw a picture to help you. 832/2”
• In Topic 8 Lesson 8-4, students practice procedural skill of multiplying two-digit numbers by a two-digit number while solving a word problem. Problem Solving problem 27 states, “In 2005, an ultra-light airplane tracked Monarch butterflies migrating to Mexico. Over 13 days, how many miles did the butterflies travel? Average distance each day: 45 miles.”

#### Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for practice-content connections. The instructional materials explicitly attend to the specialized language of mathematics and prompt students to construct viable arguments and analyze the arguments of others. The instructional materials partially meet expectations for identifying and using the mathematical practices to enrich mathematics content within and throughout the grade and assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others.

##### Indicator {{'2e' | indicatorName}}
The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

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

Mathematical Practice standards are identified in three places within the Teacher Edition: Problem Based Interactive Learning activity, Guided Practice exercises, and Problem-Solving exercises.

Throughout the teacher and student editions, there is a symbol that indicates that one or more MP is being used. Key phrases such as “Look for Patterns,” “Use Tools,” and “Reason” identify which practice is being highlighted. At the beginning of each lesson, all eight mathematical practices are listed. A check mark is placed beside each practice that is to be addressed within the lesson. For example:

• In Topic 9 Lesson 9-1, MP7 is identified with the icon and the key words “Use Structure.” Question 26 states, “At the North American Solar Challenge, teams use up to 1,000 solar cells to design and build solar cars for a race. If there are 810 solar cells in 9 rows, how many solar cells are in each row?” Teachers are told to “Guide students to look at the first 2 digits in the dividend and the 1-digit divisor. What basic fact can you use to help solve the problem? How will the answer to the problem be different than the basic fact?”
• In Topic 9 Lesson 9-5, MP4 is identified with the icon and the key word “Model.” Question 16 states “Tina is making flag pins like the one shown below. How many of each color bead are needed to make 8 pins?” Teachers are given the information that, “If students need additional help with this problem, they can draw a picture or use counters to model the number of beads of each color needed to make 8 pins.”

An example where the MPs are incorrectly labeled:

• In Topic 9 Lesson 9-4, MP5 is identified with the icon and the key words “Use Tools.” Question 37 states, “At the school concert, there were 560 people seated in 8 rows. If there were no empty seats, how many people were in each row?” Answer choices were: A. 553 people, B. 480 people, C. 70 people, D. 60 people

Overall, all eight math practices are included within the curriculum and are not treated as separate standards. However, the standards are not always used to enrich the content. They are aligned to some of the problems as an explanation to what math practice students might need to use to solve the problem.

##### Indicator {{'2f' | indicatorName}}
Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 do not meet expectations for carefully attending to the full meaning of each practice standard.

The materials do not attend to the full meaning of each of the eight MPs. The MPs are defined in both the topic and lesson narratives, as appropriate, but are not fully attended to when students interact with the aligned problems in the materials.

The materials do not attend to the full meaning of three or more MPs. Examples that demonstrate this include:

MP1 Make sense of problems and persevere in solving them.

• In Topic 16 Lesson 16-1, MP1 is identified for question 19 in the Problem Solving section. Question 19 states, “I have 6 square faces and 8 vertices. What am I?” The answer is a cube, which is the Grade 1 Standard 1.G.2.
• In Topic 3 Lesson 3-3, MP1 is identified for question 22 in the Problem Solving section. Question 22 states, “Which building is taller, the Willis Tower or the Empire State Building? How do you know?” The teacher support suggests locating the heights of the buildings and comparing the numbers. This problem needs little perseverance to solve.

MP4 Model with mathematics.

• In Topic 12 Lesson 12-4, MP4 is identified for question 33 in the Problem Solving section. Question 33 states, “Chris mowed 1/4 of the yard in the morning and 2/4 before football practice. How much of the yard does Chris have left to mow that night? Explain how you found your answer.” The teacher note says: “Students may draw a picture to represent the information given in this exercise."
• In Topic 14 Lesson 14-9, MP4 is identified for question 32 in the Problem Solving section. Question 32 states, “Use the diagram below to write a subtraction sentence.” A bar diagram is given; the student looks at what is shown pictorially and writes the solution as a mathematical equation.

MP5 Use appropriate tools strategically.

• In Topic 12 Lesson 12-7, MP5 is identified in the PBIL section and states, “Students will need to choose the appropriate fraction strips to use for finding the solution to the mathematical problem.” Students are given a select group of paper fraction bars to choose from to perform this learning.
• In Topic 13 Lesson 13-9, MP5 is identified for question 5 in the Guided Practice section. Question 5 states, “Gina’s allowance is $2.50. How much is this in dollars and dimes?” Students are given the following hint, “Tip: Remember, the number of dimes is the same as the number of tenths.” ##### Indicator {{'2g' | indicatorName}} Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by: ##### Indicator {{'2g.i' | indicatorName}} Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. The instructional materials reviewed for enVisionMATH California Common Core Grade 4 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Students justify their work and explain their thinking; however, evaluating and critiquing the work of others are found less often in the materials. Students critique the reasoning of others in problems that ask them if they agree or disagree with a statement or solution. Student materials prompt students to both construct viable arguments and analyze the arguments of others. Examples that demonstrate this include: • In Topic 10, Lesson 10-1, Independent Practice Question 19 states, “Critique Reasoning. Amanda thinks that she can separate her books into 7 equal piles. Amanda has a total of 42 books. Is Amanda’s reasoning correct?” • In Topic 12, Lesson 12-6, Problem Solving Question 15 states, “Critique Reasoning. Kathy wrote the mixed number for 35/5 as 7/5. Is she correct? Why or why not?” • In Topic 14, Lesson 14-1, Guided Practice Question 6 states, “Construct Arguments. Greg wants to measure how tall his 2-year-old sister is. What two units could he use? Explain your answer”. Examples where there are missed opportunities to construct viable arguments and analyze the arguments of others include: • In Topic 5, Lesson 5-2, Guided Practice Question 8 states, “Peter said the product of 4 x 500 is 2,000. Bob said it is 200. Who is correct?” Students critique the reasoning of others; however, students are not asked to justify or explain their answer. • In Topic 11, Lesson 11-2, Problem Solving Question 35 states, “Greta says that the product of two prime numbers must also be a prime number. Joan disagreed. Who is correct?” Students critique the reasoning of others; however, students are not asked to justify their conclusion. ##### Indicator {{'2g.ii' | indicatorName}} Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. The instructional materials reviewed for enVisionMATH California Common Core Grade 4 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The Teacher Edition contains a Mathematical Practice Handbook which defines each math practice and includes question stems for each MP to help the teacher engage students. MP3 offers the following questions stems: “How can I use math to explain why my work is right?”, “How can I use math to explain why other people’s work is right or wrong?”, and “What questions can I ask to understand other people’s thinking?” The materials label multiple questions as MP3 or parts of MP3; however, those labeled have little information assisting teachers to engage students in constructing viable arguments or to critique the reasoning of others. The information that the materials provide is not specific and are often hints or reminders to give students while they are solving a problem. There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example: • In Topic 2, Lesson 2-1, Problem Solving Question 18 states, “Writing to Explain. Balloons are sold in bags of 30. There are 5 giant balloons in each bag. How many giant balloons will you get if you buy 120 balloons? Explain.” No teacher guidance is given for this question. • In Topic 4, Lesson 4-3, Guided Practice Question 7 states, “Construct Arguments. When adding 36,424 and 24,842 above, why is there no regrouping in the final step?” No teacher guidance is given for this question. Examples where teachers are supported, although generally, to assist students in constructing viable arguments and analyzing the arguments of others include: • In Topic 6, Lesson 6-5, Problem Solving Question 32 “Construct Arguments. Mr. Tran would like to buy a new sofa that costs$934. He can pay the total all at once, or he can make a \$125 payment each month for 8 months. Which plan costs less? Explain.” The teacher notes for that question say “Guide students to understand that they need to compare exact amounts to solve this problem.”
• In Topic 7, Lesson 7-3, PBIL identifies MP3 as the mathematical practice being used in the activity. The teacher note states, “When students work with a partner they explain their thinking to others. This helps them both to construct arguments as well as critique the reasoning of others.”
##### Indicator {{'2g.iii' | indicatorName}}
Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for enVisionMATH California Common Core Grade 4 meet expectations for attending to the specialized language of mathematics.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols.

• Within the Yearlong Curriculum Guide, a list is provided for the Key Math Terms that are used each month of the school year.
• The teacher and student editions include a Review What You Know section at the beginning of every topic. This section reviews vocabulary used in prior topics along with introducing the vocabulary in the current topic. Students complete this activity by inserting the correct vocabulary word into a sentence to correctly identify its definition.
• Within Review What You Know, the new vocabulary listed for Topic 3 includes: digits, place value, expanded form, standard form, word form, and compare.

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them.

• In the Student Edition, vocabulary terms can be found highlighted in yellow within the Visual Learning Bridge across the top of the pages. A definition in context is provided for each term and is used in context during instruction, practice, and assessment.
• In the Implementation Guide, Teacher Edition, as well as the Student Edition, a complete Glossary is included and can be referred to at any time.
• No examples of incorrect use of vocabulary, symbols, or numbers were found within the materials.

Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of Mathematics.

### Usability

This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two
Not Rated

#### Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
##### Indicator {{'3a' | indicatorName}}
The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.
##### Indicator {{'3b' | indicatorName}}
Design of assignments is not haphazard: exercises are given in intentional sequences.
##### Indicator {{'3c' | indicatorName}}
There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.
##### Indicator {{'3d' | indicatorName}}
Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
##### Indicator {{'3e' | indicatorName}}
The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

#### Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
##### Indicator {{'3g' | indicatorName}}
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. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.
##### Indicator {{'3h' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains 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.
##### Indicator {{'3i' | indicatorName}}
Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.
##### Indicator {{'3j' | indicatorName}}
Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).
##### Indicator {{'3k' | indicatorName}}
Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.
##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

#### Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.
##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.
##### Indicator {{'3n' | indicatorName}}
Materials provide strategies for teachers to identify and address common student errors and misconceptions.
##### Indicator {{'3o' | indicatorName}}
Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.
##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing formative and summative assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.
##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
##### Indicator {{'3u' | indicatorName}}
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 (e.g., modifying vocabulary words within word problems).
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.
##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.
##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

#### Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.
##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).
##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

## Report Overview

### Summary of Alignment & Usability for enVisionMATH California Common Core | Math

#### Math K-2

The instructional materials reviewed for enVisionMATH California Common Core Kindergarten through Grade 2 have varied results for alignment to the CCSSM. The instructional materials reviewed for Kindergarten partially meet expectations for alignment as they meet expectations for focus, partially meet expectations for coherence and rigor and balance, and do not meet expectations for practice-content connections. The instructional materials reviewed for Grade 1 do not meet expectations for alignment as they do not meet expectations for focus and practice-content connections and partially meet expectations for coherence and rigor and balance. The instructional materials reviewed for Grade 2 do not meet expectations for alignment as they meet expectations for focus, partially meet expectations for coherence and rigor and balance, and do not meet expectations for practice-content connections. None of the materials for Kindergarten through Grade 2 were reviewed for usability in Gateway 3.

##### Kindergarten
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated

#### Math 3-5

The instructional materials reviewed for enVisionMATH California Common Core Grades 3 through 6 have varied results for alignment to the CCSSM. The instructional materials reviewed for Grade 3 do not meet expectations for alignment as they do not meet expectations for focus and partially meet expectations for coherence. The instructional materials reviewed for Grades 4 and 5 partially meet expectations for alignment as they do not meet expectations for focus and partially meet expectations for coherence, rigor and balance, and practice-content connections. The instructional materials reviewed for Grade 6 partially meet expectations for alignment as they meet expectations for focus and coherence and partially meet expectations for rigor and balance and practice-content connections. None of the materials for Grades 3 through 6 were reviewed for usability in Gateway 3.

###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated

#### Math 6-8

The instructional materials reviewed for enVisionMATH California Common Core Grades 3 through 6 have varied results for alignment to the CCSSM. The instructional materials reviewed for Grade 3 do not meet expectations for alignment as they do not meet expectations for focus and partially meet expectations for coherence. The instructional materials reviewed for Grades 4 and 5 partially meet expectations for alignment as they do not meet expectations for focus and partially meet expectations for coherence, rigor and balance, and practice-content connections. The instructional materials reviewed for Grade 6 partially meet expectations for alignment as they meet expectations for focus and coherence and partially meet expectations for rigor and balance and practice-content connections. None of the materials for Grades 3 through 6 were reviewed for usability in Gateway 3.

###### Alignment
Partially Meets Expectations
Not Rated

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### Overall Summary

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###### Usability
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