4th Grade - Gateway 3

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for providing teachers guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

Within the Course Guide, several sections (Design Principles, A Typical Lesson, How to Use the Materials, and Key Structures in This Course) provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include but are not limited to:

• Resources, Course Guide, Design Principles, Learning Mathematics by Doing Mathematics, “A problem-based instructional framework supports teachers in structuring lessons so students are the ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them. The teacher has many roles in this framework: listener, facilitator, questioner, synthesizer, and more.”

• Resources, Course Guide, A Typical Lesson, “A typical lesson has four phases: 1. a warm-up; 2. one or more instructional activities; 3. the lesson synthesis; 4. a cool-down.” “A warm-up either: helps students get ready for the day’s lesson, or gives students an opportunity to strengthen their number sense or procedural fluency.” An instructional activity can serve one or many purposes: provide experience with new content or an opportunity to apply mathematics; introduce a new concept and associated language or a new representation; identify and resolve common mistakes; etc. The lesson synthesis “assists the teacher with ways to help students incorporate new insights gained during the activities into their big-picture understanding.” Cool-downs serve “as a brief formative assessment to determine whether students understood the lesson.”

• Resources, Course Guide, How to Use the Materials, “The story of each grade is told in eight or nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson in the unit also has a narrative. Lesson narratives explain: the mathematical content of the lesson and its place in the learning sequence; the meaning of any new terms introduced in the lesson; how the mathematical practices come into play, as appropriate. Activities within lessons also have narratives, which explain: the mathematical purpose of the activity and its place in the learning sequence, what students are doing during the activity, what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis, connections to the mathematical practices, when appropriate.”

• Resources, Course Guide, Scope and Sequence lists each of the nine units, a Pacing Guide to plan instruction, and Dependency Diagrams. These Dependency Diagrams show the interconnectedness between lessons and units within Grade 4 and across all grades.

• Resources, Course Guide, Course Glossary provides a visual glossary for teachers that includes both definitions and illustrations. Some images use examples and nonexamples, and all have citations referencing what unit and lesson the definition is from.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Section B, Lesson 8, Warm Up, “The purpose of an Estimation Exploration is to practice estimating a reasonable answer based on experience and known information. Students can identify fractions represented by the shaded portions in tape diagrams in which unit or non-unit fractions are marked. To estimate the shaded parts in an unmarked tape, students may rely on the size of benchmark fractions— \frac{1}{2}, \frac{1}{3}, or \frac{1}{4}—and partition those parts mentally until it approximates the size of the shaded ports. They may also estimate how many copies of the shaded part could fit in the entire diagram.”

• Unit 3, Extending Operations to Fractions, Section B, Lesson 7, Activity 1, provides teachers guidance in building students' understanding of fractions. "Previously, students considered non-unit fractions in terms of equal groups of unit fractions or as a product of a unit fraction and a whole number. This activity prompts students to think about non-unit fractions as being sums of other fractions. The given context—about measuring fractional amounts using measuring cups of certain sizes—allows students to continue thinking in terms of equal groups, but also invites them to consider a fractional quantity as a sum of two or more fractions with the same denominator."

• Unit 5, Multiplicative Comparison and Measurement, Overview, Throughout this Unit, assists teachers in presenting materials. “The Number Talks in this unit allow students to use what they previously learned about numbers and operations to support their current learning. Students use the relationship between multiplication and division to solve missing factor equations, which is helpful for multiplicative comparison work. They mentally find the value of expressions involving multiplication by 100 and 1,000, which is helpful when converting metric units of measurement. Some Number Talks offer ongoing practice toward end-of-year fluency goals, such as in multi-digit addition. Students also practice multiplying fractions by whole numbers, which supports the problem solving work in the unit.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Unit Overviews and sections within lessons include adult-level explanations and examples of the more complex grade-level concepts. Within the Course Guide, How to Use the Materials states, “Activities within lessons also have narratives, which explain: the mathematical purpose of the activity and its place in the learning sequence, what students are doing during the activity, what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis, connections to the mathematical practices, when appropriate.” Examples include:

• Unit 2, Fraction Equivalence and Comparison, Overview, “In this unit, students extend their prior understanding of equivalent fractions and comparison of fractions. In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction \frac{1}{b} results from a whole partitioned into b equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line. Here, students follow a similar progression of representations. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Students generalize that a fraction \frac{s}{b} is equivalent to fraction \frac{(n\times a)}{(n\times b)} because each unit fraction is being broken into n times as many equal parts, making the size of the part n times as small \frac{1}{(n\times b)} and the number of parts in the whole n times as many (n\times a). For example, we can see \frac{3}{5} is equivalent to \frac{6}{10}  because when each fifth is partitioned into 2 parts, there are 2 x 3 or 6 shaded parts, twice as many as before, and the size of each part is half as small \frac{1}{(2\times5)} or \frac{1}{10}.”

• Unit 4, From Hundredths to Hundred-Thousands, Section A, Lesson 1, Preparation, Lesson Narrative, “In this lesson, students rely on their knowledge of fractions to express tenths and hundredths as decimals. They begin to see connections between fraction notation, the names of fractions in words, and decimal notation. They also start to notice the structure of the decimal notation and how it relates to place value. Students use increasingly precise language to read decimals through this section (MP6). Students will develop this new understanding over several lessons so they are not expected to name the value of each place of a decimal at this time.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Section B, Lesson 9, Preparation, Lesson Narrative, “Students engage in quantitative and abstract reasoning (MP2) as they relate the partial products in a diagram and in an algorithm. Because this lesson offers an initial exposure to the new notation, students are not required to use an algorithm that uses partial products to multiply. They can rely on other methods they have learned so far.”

Also within the Course Guide, About These Materials, Further Reading states, “The curriculum team at Open Up Resources has curated some articles that contain adult-level explanations and examples of where concepts lead beyond the indicated grade level. These are recommendations that can be used as resources for study to renew and fortify the knowledge of elementary mathematics teachers and other educators.” Examples include:

• Resources, Course Guide, About These Materials, Further Reading, 3-5, “Fraction Division Parts 1–4. In this four-part blog post, McCallum and Umland discuss fraction division. They consider connections between whole-number division and fraction division and how the two interpretations of division play out with fractions with an emphasis on diagrams, including a justification for the rule to invert and multiply. In Part 4, they discuss the limitations of diagrams for solving fraction division problems.”

• Resources, Course Guide, About These Materials, Further Reading, Entire Series, “The Number Line: Unifying the Evolving Definition of Number in K-12 Mathematics. In this article, the authors (Lahme, McLeman, Nakamaye, and Umland) focus their attention on the selection of definitions, notation, and graphical conventions surrounding the development of the real numbers from kindergarten to grade 12, and address the work that students might do in later years.”

The materials reviewed for Open Up Resources K-5 Mathematics Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

 Correlation information can be found within different sections of the Course Guide and within the Standards section of each lesson. Examples include:

• Resources, Course Guide, About These Materials, CCSS Progressions Documents, “The Progressions for the Common Core State Standards describe the progression of a topic across grade levels, note key connections among standards, and discuss challenging mathematical concepts. This table provides a mapping of the particular progressions documents that align with each unit in the K–5 materials for further reading.”

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in the Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.”

• Resources, Course Guide, Scope and Sequence, Dependency Diagrams, All Grades Unit Dependency Diagram identifies connections between the units in grades K-5. Additionally, a “Section Dependency Diagram” identifies specific connections within the grade level.

• Resources, Course Guide, Lesson and Standards, provides two tables: a Standards by Lesson table, and a Lessons by Standard table. Teachers can utilize these tables to identify standard/lesson alignment.

• Unit 4, From Hundredths to Hundred-Thousands, Section B, Lesson 6, Standards, “Addressing: 4.NBT.A.1 Recognize that in a multi-digit whole number, a digit in the one place represents ten times what it represents in the place to its right. For example, recognize that 700\div70=10 by applying concepts of place value and division. Building Towards: 4.NBT.A.1.”

Explanations of the role of specific grade-level mathematics can be found within different sections of the Resources, Course Guide, Unit Overviews, Section Overviews, and Lesson Narratives. Examples include:

• Resources, Course Guide, Scope and Sequence, each Unit provides Unit Learning Goals, for example, “Students apply understanding of multiplication and area to work with factors and multiples.” Additionally, each Unit Section provides Section Learning Goals, “determine if a number is prime or composite.”

• Unit 2, Fraction Equivalence and Comparison, Overview, “In this unit, students extend their prior understanding of equivalent fractions and comparison of fractions. In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction \frac{1}{b} results from a whole partitioned into b equal parts.”

• Unit 5, Multiplicative Comparison and Measurement, Section A, Overview, "In this section, students expand on these concepts to convert measurements within the same system (metric or customary) from larger units to smaller units. These conversions require an understanding of the multiplicative relationship between units. Students begin by exploring lengths in metric units. To develop a sense of the multiplicative relationship between centimeters and meters, students build a length of 1 meter from centimeter grid paper. They recognize that 1 meter is 100 times as long as 1 centimeter and use this reasoning to convert meters to centimeters. Later, they make sense of 1 kilometer by relating it to multiples of shorter measurements, such as the length of a basketball court or a soccer field. Later, students learn the relationships between grams and kilograms, milliliters and liters, ounces and pounds, and hours, minutes, and seconds. As they solve problems and use multiplication to perform conversion, they develop a sense of the relative size of the units." (4.MD.1)

• Unit 7, Angles and Angle Measurement, Section B, Lesson 9, "Students then make sense of one-degree angles in terms of a fraction of a turn and are introduced to the protractor as a tool of measurement. They make sense of the numbers on the tool and how angles are shown. They learn to read the measurement of angles whose vertices have been pre-aligned to the center point of a protractor. Students will continue to add new vocabulary to their personal word walls. In the next lesson, students will further develop their ability to use a protractor by measuring a variety of angles with less support." (4.MD.6)

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The materials explain and provide examples of instructional approaches of the program and include and reference research-based strategies. Both the instructional approaches and the research-based strategies are included in the Course Guide under the Resources tab for each unit. Design Principles describe that, “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice.” Examples include:

• Resources, Course Guide, Design Principles, “In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.” Principles that guide mathematics teaching and learning include: All Students are Capable Learners of Mathematics, Learning Mathematics by Doing Mathematics, Coherent Progression, Balancing Rigor, Community Building, Instructional Routines, Using the 5 Practices for Orchestrating Productive Discussions, Task Complexity, Purposeful Representations, Teacher Learning Through Curriculum Materials, and Model with Mathematics K-5.

• Resources, Course Guide, Design Principles, Community Building, “Students learn math by doing math both individually and collectively. Community is central to learning and identity development (Vygotsky, 1978) within this collective learning. To support students in developing a productive disposition about mathematics and to help them engage in the mathematical practices, it is important for teachers to start off the school year establishing norms and building a mathematical community. In a mathematical community, all students have the opportunity to express their mathematical ideas and discuss them with others, which encourages collective learning. ‘In culturally responsive pedagogy, the classroom is a critical container for empowering marginalized students. It serves as a space that reflects the values of trust, partnership, and academic mindset that are at its core’ (Hammond, 2015).”

• Resources, Course Guide, Design Principles, Instructional Routines, “Instructional routines provide opportunities for all students to engage and contribute to mathematical conversations. Instructional routines are invitational, promote discourse, and are predictable in nature. They are ‘enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.’ (Kazemi, Franke, & Lampert, 2009)”

• Resources, Course Guide, Key Structures in This Course, Student Journal Prompts, Paragraph 3, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson & Robyns, 2002; Liedke & Sales, 2001; NCTM, 2000).”

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for including a comprehensive list of supplies needed to support the instructional activities.

In the Course Guide, Materials, there is a list of materials needed for each unit and each lesson. Lessons that do not have materials are indicated by none; lessons that need materials have a list of all the materials needed. Examples include:

• Resources, Course Guide, Key Structures in This Course, Representations in the Curriculum, provides images and explanations of representations for the grade level. “Fraction Strips (3-4): Fraction strips are rectangular pieces of paper or cardboard used to represent different parts of the same whole. They help students concretely visualize and explore fraction relationships. As students partition the same whole into different-size parts, they develop a sense for the relative size of fractions and for equivalence. Experience with fraction strips facilitates students’ understanding of fractions on the number line.” 

• Resources, Course Guide, Materials, includes a comprehensive list of materials needed for each unit and lesson. The list includes both materials to gather and hyperlinks to documents to copy. “Unit 4, Lesson 10 - Gather: Number cards 0-10; Copy: Greatest of Them All Stage 3 Recording Sheet, Mystery Number Stage 4 Gameboard.”

• Unit 2, Fraction Equivalence and Comparison, Section B, Lesson 8, Materials Needed, “Activities: Tape (painter’s or masking) (Activity 1); Centers: Dry erase markers (Get Your Numbers in Order, Stage 4), Sheet protectors (Get Your Numbers in Order, Stage 4).”

The materials reviewed for Open Up Resources Math Grade 4 meet expectations for having assessment information in the materials to indicate which standards are assessed. 

The materials consistently and accurately identify grade-level content standards for formal assessments for the Section Checkpoints and End-of-Unit Assessments within each assessment answer key. Examples from formal assessments include:

• Resources, Course Guide, Summative Assessments, End-of-Unit Assessments, “All summative assessment problems include a complete solution and standard alignment. Multiple choice and multiple response problems often include a reason for each potential error a student might make.”

• Unit 8, Properties of Two-DImensional Shapes, End-Of-Unit Assessment, Problem 2, “4.G.A.2, 4.G.A.3: Which statement is true? A. A right triangle never has a line of symmetry.; B. A right triangle sometimes has a line of symmetry.; C. A right triangle always has a line of symmetry.; D. If a triangle has a line of symmetry then it is a right triangle.”

• Unit 6, Multiplying and Dividing Multi-Digit Number, Section B, Lesson 9, Cool-down, “Assessing 4.NBT.B.5: Find the value of 5\times1023. Show your reasoning. Student Responses 5,115. Sample responses:5\times3=15, 5\times20=100, and 5\times1000=5000. The sum of 15, 100, and 5,000 is 5,115.”

• Unit 4, From Hundredths to Hundred-Thousands, Assessments, Section D Checkpoint, Problem 2, "4.NBT.B.4: Find the value of 100,058-86,249. Show your reasoning.”

Guidance for assessing progress of the Mathematical Practices can be found within the Resources, Course Guide, How to Use These Materials, Noticing and Assessing Student Progress in Mathematical Practices, How to Use the Mathematical Practices Chart, “Because using the mathematical practices is part of a process for engaging with mathematical content, we suggest assessing the Mathematical Practices formatively. For example, if you notice that most students do not use appropriate tools strategically (MP5), plan in future lessons to select and highlight work from students who have chosen different tools.” In addition, “...a list of learning targets for each Mathematical Practice is provided to support teachers and students in recognizing when engagement with a particular Mathematical Practice is happening…the ‘I can’ statements are examples of types of actions students could do if they are engaging with a particular Mathematical Practice.” Examples include:

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress In Mathematical Practices, Standards for Mathematical Practices Chart, Grade 4, MP5 is found in Unit 4, Lesson 16. 

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress In Mathematical Practices, Standards for Mathematical Practices Chart, Grade 4, MP7 is found in Unit 9, Lessons 1, 4, 5, 6. 

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress In Mathematical Practices, Standards for Mathematical Practice Student Facing Learning Targets, “MP6: I Can Attend to Precision. I can use units or labels appropriately. I can communicate my reasoning using mathematical vocabulary and symbols. I can explain carefully so that others understand my thinking. I can decide if an answer makes sense for a problem.”

The materials reviewed for Open Up Resources K-5 Math Kindergarten meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

Formative assessments include instructional activities, Practice Problems and Section Checkpoints in each section of each unit. Summative assessments include End-of-Unit Assessments and End-of-Course Assessments. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types including multiple choice, multiple response, short answer, restricted constructed response, and extended response. Examples include:

• Unit 3, Extending Operations to Fractions, Assessments, End-of-Unit Assessment, Problem 6, 4.NF.3, 4.NF.4, “Students find sums, differences, and products of fractions without context. The numbers (for sums and differences) are presented both as fractions and as mixed numbers. No reasoning is requested and this item and the next assess grade level skills calculating with fractions.” Problem 6, “Find the value of each expression. a. \frac{5}{6}+\frac{2}{6}+\frac{3}{6}; b. 3-\frac{7}{8}; c. 4\frac{3}{5}+3\frac{4}{5}; d. 8-\frac{8}{10}; e. 5\times\frac{3}{8}”.

• Unit 5, Multiplicative Comparison and Measurement, Assessments, End-of-Unit Assessment, Problem 3, 4.MD.1, “Students choose a correct conversion statement between inches and feet. Each distractor uses the correct conversion factor but the wrong operation, namely subtraction for 60, addition for 84, and multiplication for 864. None of the distractors are reasonable so students who miss this item need more work on visualizing length units.” Problem 3, “The length of the table in inches is 72. What is the length of the table in feet? A. 6; B. 60; C. 82; D. 864.”

• Unit 7, Angles and Angle Measurement, Section B, Lesson 7, Cool-Down, 4.G.A.3, 4.MD.A.3, “Here is a rectangle with two lines of symmetry. Find its perimeter. Write an expression to show how you find it.”

• Unit 8, Properties of Two-Dimensional Shapes, Assessments, Section B Checkpoint, Problem 2, supports the full intent of MP8 (Look for and express regularity in repeated reasoning) as students use their understanding of equilateral triangles, symmetry, and perimeter to solve a problem. An equilateral triangle is shown. “This figure has three lines of symmetry and a perimeter of 18 cm. What is the length of each of the sides? Explain or show your reasoning.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each lesson. According to the Resources, Course Guide, Universal Design for Learning and Access for Students with Disabilities, “Supplemental instructional strategies that can be used to increase access, reduce barriers and maximize learning are included in each lesson, listed in the activity narratives under ‘Access for Students with Disabilities.’ Each support is aligned to the Universal Design for Learning Guidelines (udlguidelines.cast.org), and based on one of the three principles of UDL, to provide alternative means of engagement, representation, or action and expression. These supports provide teachers with additional ways to adjust the learning environment so that students can access activities, engage in content, and communicate their understanding.” Examples of supports for special populations include: 

• Unit 2, Fraction Equivalence and Comparison, Section C, Lesson 12, Activity 1, Access for Students with Disabilities, “Engagement: Sustaining Effort and Persistence, Chunk this task into more manageable parts. Invite students to look at column A first, then column B, then row 1. Provide access to pre-made fraction strips for thirds and fifths to help them get started. Check in with students to provide feedback and encouragement after each chunk, particularly in terms of looking for and making use of structure. Provides accessibility for: Conceptual Processing, Organization, Social-Emotional Functioning.”

• Unit 5, Multiplicative Comparison and Measurement, Section C, Lesson 17, Access for Students with Disabilities, “Representation: Language and Symbols. Invite students to represent each problem as an equation to help them identify strategies for solving and to give them practice interpreting mathematical language. Provides accessibility for: Conceptual Processing, Language.”

• Unit 7, Angles and Angle Measurement, Section A, Lesson 5, Access for Students with Disabilities, “Engagement: Recruiting Interest Synthesis: Optimize meaning and value. Ask, “How might thinking about angles be useful in our lives?” Consider making a connection to sports. For example, it might be easier to score in soccer if the ball is in front of the goal rather than off to the side, because of the angles involved. Show pictures if applicable and possible. (Consider drawing or labeling a picture in which the soccer ball is the vertex and the posts are points along the rays.) Provides accessibility for: Conceptual Processing, Attention, Social- Emotional Functioning.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found where problems are labeled as “Exploration” at the end of practice problem sets within sections, where appropriate. According to the Resources, Course Guide, How To Use The Materials, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity directly related to the material of the unit that students can do either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.” Examples include:

• Unit 3, Extending Operations to Fractions, Section A, Practice Problems, Problem 12 (Exploration), “Diego walked the same number of miles to school each day. He says that he walked  48/5 miles in total, but does not say how many days that distance includes. What are some possible number of days Diego counted and the distance he walked each of those days?”

• Unit 4, From Hundredths to Hundred-Thousands, Section B, Practice Problems, Problem 7 (Exploration), “For each question, use only the digits 1, 0, 5, 9, and 3. You may not use a digit more than once and you do not need to use all the digits. a. Can you make three numbers greater than 3,000 but less than 3,500?; b. Can you make three numbers greater than 9,000 but less than 10,000?; c. Which numbers can you make that are greater than 39,500 but less than 40,000?”

• Unit 8, Properties of Two-Dimensional Shapes, Section A, Problem 11 (Exploration), “Draw each shape and all the lines of symmetry you can find in it. a. rectangle; b. rhombus; c. square.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided to teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Resources, Course Guide, Mathematical Language Development and Access for English Learners, “In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” Examples include:

• Unit 2, Fraction Equivalence and Comparison, Section B, Lesson 8, Activity 1, “Access for English Learners - Listening, Speaking: MLR8 Discussion Supports. During partner work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: ‘I heard you say …’ Original speakers can agree or clarify for their partner.”

• Unit 3, Extending Operations to Fractions, Section A, Lesson 4, Activity1, “Access for English Learners - Reading, Representing: Reading: MLR6 Three Reads. ‘We are going to read this 3 times.’ After the 1st Read: ‘Tell your partner what this situation is about.’ After the 2nd Read: ‘List the quantities. What can be counted or measured?’ (number of jars, number of friends, number of cups of jam). After the 3rd Read: ‘What strategies can we use to solve this problem?’”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 3, Activity 1, "Access for English Learners, Representing - Conversing: MLR7 Compare and Connect; Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, ‘What did the strategies have in common?’, ‘How were they different?’, and ‘Why did the different approaches lead to the same outcome?’”

The materials reviewed for Open Up Resources K-5 Math Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Suggestions and/or links to manipulatives are consistently included within materials to support the understanding of grade-level math concepts. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Section A, Lesson 1, Activity 1, “The purpose of this activity is for students to use fraction strips to represent halves, fourths, and eighths. The denominators in this activity are familiar from grade 3. The goal is to remind students of the relationships between fractional parts in which one denominator is a multiple of another. Students should notice that each time the unit fractions on a strip are folded in half, there are twice as many equal-size parts on the strip and that each part is half as large. Groups of 2. Give each group 4 paper strips and a straightedge. Hold up one strip for all to see. ‘Each strip represents 1.’ Label that strip with ‘1’ and tell students to do the same on one of their strips. ‘Take a new strip. How would you fold it to show halves?’ 30 seconds: partner think time. ‘Think about how to show fourths on the next strip and eighths on the last strip.’”

• Unit 4, From Hundredths to Hundred-Thousands, Section A, Lesson 2, Activity 1, “In this activity, students reinforce their understanding of equivalent fractions and decimals by sorting a set of cards by their value. The cards show fractions, decimals, and diagrams. A sorting task gives students opportunities to analyze different representations closely and make connections (MP2, MP7). ‘Work with your group to sort the set of cards by their value. One diagram has no matching cards. Write the fraction and decimal it represents.’ 6–7 minutes: group work on the first two problems. Monitor for the ways students sort the cards and the features of the representations to which they attend. ‘Work on the last problem independently.’ 2–3 minutes: independent work on the last problem. Your teacher will give you a set of cards. Each large square on the cards represents 1. Sort the cards into groups so that the representations in each group have the same value. Record your sorting decisions. Be prepared to explain your reasoning. One of the diagrams has no matching fraction or decimal. What fraction and decimal does it represent? Are 0.20 and 0.2 equivalent? Use fractions and a diagram to explain your reasoning.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Section B, Lesson 6, Activity 1, “This activity prompts students to make sense of base-ten diagrams for representing multiplication. The representation supports students in grouping tens and ones and encourages them to use place value understanding and to apply the distributive property (MP7). This activity is an opportunity for students to build conceptual understanding of partial products in a more concrete way. In the next activity, students will notice that working with these drawings can be cumbersome and transition to using rectangular diagrams, which are more abstract.”