5th Grade - Gateway 3

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing teacher 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.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials.

• A Curriculum Overview provides a chart of the components and description for the lessons, assessments, and Domain Review. The curriculum components are described briefly in the Overview section. 

• A Practical Approach to Using Assessments, Rubrics & Scoring Guidelines helps the teacher understand rubrics for the assessments.

• In the Teacher Guide, there is instruction on planning a lesson with a sample sequence for lessons and assessments. The materials provide pacing for the year.

• In the Teacher Guide, the instructional protocols used throughout the series are described and connected to the Mathematical Practices they support. 

• In the Detailed Lesson Plan, there is a section to help support Diverse Learners with a chart of Accommodations, Modifications, and Extensions, as well as Language Routines. 

• Common Misconceptions are listed in each Detailed Lesson Plan.

• In each Detailed Lesson Plan, teachers are given suggestions for vocabulary incorporation such as, “In your math classroom, make a Word Wall to hang and refer to vocabulary words throughout the lesson. As a whole-class exercise, create a visual representation and definition once students have had time to use their new words throughout a lesson.”

• Guidance is given to teachers for applying and reinforcing math practices in the Teacher Guide and in Detailed Lesson Plans. For example, MP8: “This practice is reinforced by having the students watch a complimentary video in which Jo Boaler has students modeling how to look for and identify patterns in real-life scenarios.” Guidance shared directly from Jo Boaler states, “Students need time and space to develop their capacity to ‘look for and express regularity in repeated reasoning.’ When you provide tasks that are specific to supporting MP8, explicitly tell students that it’s ok to slow down, and to think deeply.” Several “tips” to address the MP are also shared.

• “Detailed Lesson Plans provide a step-by-step guide with specific learning objectives for the math standard, lesson summary, prerequisite standards, vocabulary and vocabulary protocols, applying Standards for Mathematical Practice, Jo Boaler's SMP Tips, cluster connection, common misconceptions, instruction at a glance, and day-by-day teaching instructions with time allotments. Also included are suggestions for differentiation, and instructional moves as well as tips for the English Language Learner student.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Throughout each lesson’s Detailed Lesson Plan, there is narrative information to assist the teacher in presenting student material throughout all phases. Examples include:

• In 5.G.B.3 Squaring Off, Teacher Instruction: “Here are some examples you might make to the class. Zeus and Poseidon were arguing about whether or not a square is a rectangle: We saw that even though a square is a square, it’s also a special rectangle; Polygon hierarchies can help us understand the relationship among different polygons, especially quadrilaterals.”

• 5.NF.A.2 Acre Acquisition, Common Misconceptions: “Students may forget that when adding or subtracting fractions, the fractions must refer to the same whole. Often when students use fraction models to represent addition or subtraction, they use mixed models (rectangles and circles, for example), which makes it difficult to compare the fractional segments. Tell students to use one type of model for each problem.”

• 5.NF.B.7a Pirate Pay, Cluster Connection: “Direct Connection: In Pirate Pay, students extend their previous understanding that $$a+b$$ is the same as $$a×\frac{1}{b}$$ to fractions. Namely, that $$\frac{1}{a}÷b$$ is the same as $$\frac{1}{a}×\frac{1}{b}$$. Students will divide $$\frac{1}{3}$$ pound of gold by 4 galley crew members, $$\frac{1}{3}÷4=\frac{1}{3}×\frac{1}{4}$$.”

• 5.OA.A.2 Liftie Lesson, Part 3 Resolution: “1. Play Resolution video to the whole class, and have the students compare their solutions as they watch. 2. After the video, prompt students with the following questions: What did you do that was the same? What was different? What strategy do you think was more efficient to find the equation? Why? Students may respond aloud or in a journal.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 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.

Under the Resources Tab, there is a section dedicated to Adult-Level Resources. These contain adult-level explanations including examples of the more complex grade-level concepts so that teachers can improve their own knowledge of the subject. There are also professional articles provided on topics such as mathematical growth mindset, cultural diversity in math, and mathematical language routines.

The Teacher Guide contains a page at the beginning of each cluster section titled, “Cluster Refresher for the Teacher - Adult Level Explanation”. This provides a page of basic background information for the teacher including strategies to develop understanding. For example,

• “5.NBT.A works to extend the understanding of place value, the magnitude of digits in a number, and the relationship between adjacent places. By emphasizing the meaning behind our base-ten system, the mathematical relationship can be extended in both directions, relating places that are ten times or one tenth the size of any given place. Understanding place value and the basic relationships between places not only establishes a foundation for understanding the magnitude of numbers, but also reveals how numbers can be broken down, expanded, or compared. ... The key is the order of the digits and how they are in relation to the decimal point. Once digits are in order, the magnitude of each can be shifted by powers of ten in either direction. The shift along the place value spectrum can show how adjacent places are ten times or one-tenth the size depending on which direction we move.”

The Adult-Level Explanations booklet under the Resources tab includes a progression through each domain from Grade 5 through High School. The last section is Beyond Grade 8, which explains how concepts from the middle grades connect to high school standards. For example: Beyond Grade 8: Number and Quantity:

• “Understanding the relationship between numbers, the various ways to represent them, their place on the number line, and the familiarity of the properties that they possess describes the concept that is the number system. The middle grades allow for opportunities to discover and conceptualize different types of numbers and their place on the number line. Starting with counting numbers in early years eventually leads to integers, and then to all rational numbers. Rational numbers are all numbers that can be written as a fraction p/q, when p is an integer and q is a nonzero. Studying the placement of these numbers on the number line is of equal importance as understanding different representations of the same number. Once one has a firm understanding of all rational numbers, their relationships and different representations, irrational numbers are introduced. ... This notion and its properties extends to sets of polynomial and rational functions in later courses. The rules that govern these are as follows: The sum or product of two rational numbers is rational. The sum of a rational number and an irrational number is irrational. The product of a non-zero rational number and an irrational number is irrational. Understanding the relationships between rational and irrational numbers and how they relate to form real numbers leads to future work in understanding complex numbers and their role in the number system. These are not rules and procedures, but numerical relationships that provide the foundation to these rules.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series.

• Each course in this series includes a document called Planning the Year that provides the standards and pacing for each lesson. 

• There are standards correlations in the Scope and Sequence Chart that lists each Lesson, Domain Review, and Major Cluster Lessons throughout a year.

• Each lesson is designed to address a single standard.

Explanations of the role of the specific grade-level/course-level mathematics are present in the context of the series.

• The Teacher Guide contains a page at the beginning of each cluster section titled “Role of Mathematics” which clearly identifies the grade-level clusters and standards within a domain and describes the intent of the cluster. The Cluster Role Across Grade Levels describes the grade-level content in context of the domain progression from when the initial related skills were introduced to how the skills progress through high school. For example, “The 5.NBT.A cluster involves place value and using it to represent, round and compare decimal numbers to the thousandths. The development of these skills begins in Grade 2 where place value is introduced for three-digit numbers (2.NBT.A.1). In Grade 4, throughout Number and Operations - Fractions, students use place value to understand fractions with denominators of 10 and 100, relating them to their decimal equivalents (4.NF.C.6) as well as adding them (4.NF.C.5). Within the Numbers and Operations in Base Ten domain, students continue using their place value knowledge in the 5.NBT.B cluster as they learn operations with multi-digit whole numbers and decimals. Students expand their knowledge of powers to bases other than 10 as they write and evaluate numerical expressions involving whole number exponents in Grade 6 (6.EE.A.1) and work with radicals and integer exponents in Grade 8 (8.EE.A.3).” 

• The Detailed Lesson Plan for each lesson lists the Prerequisite Standards required for students to be successful in the lesson. For example, in 5.G.B.3 Squaring Off, the Prerequisite Standards listed are 3.G.A.1 and 4.G.A.2.

• The Detailed Lesson Plan for each lesson includes Cluster Connections that identify connections between clusters and coherence across grade levels. For example, in 5.G.B.4 It’s a Polygon World, Cross-Cluster Connection, “This activity connects 5.G to 6.G, 7.G, 8.G as students will continually use their knowledge of figures and their attributes to determine missing dimensions and to calculate area and volume, including figures in 3-dimensions. It also connects to High School Geometry, as students will know what assumptions can be made about figures, and they will be able to write better proofs.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Materials explain the instructional approaches of the program.

• The Curriculum Overview in the Teacher's Guide states that the curriculum is designed to “STOP THE DROP.” The materials state, “Core Curriculum by MidSchoolMath is developed to fix this problem through a fundamentally different approach... MidSchoolMath emphasizes structured, conceptual learning to prepare students for Algebra I... MidSchoolMath is specifically designed to address the ‘The Mid School Math Cliff’.”

• In the Teacher Guide, the overview on Scoring Guidelines states, “In coordination with Dr. Jo Boaler, MidSchoolMath has developed an approach to using rubrics and scoring with an emphasis on making them useful and practical for helping teachers support student learning. This is in contrast to the use of scoring guidelines for the primary purpose of giving grades.” 

• In the Letter to the Parent, the instructional approaches are summarized, “MidSchoolMath strives to help students see that math is relevant and holds value and meaning in the world. The curriculum is designed not only to enhance student engagement, but also to provide stronger visual representation of concepts with focus on logic structures and mathematical thinking for long-term comprehension. ... Peer Teaching: Students learning from other students is a powerful mechanism, wherein both the ‘teachers’ and the ‘learners’' receive learning benefits.”

Materials reference relevant research sources:

• “Hattie, J. (2017) Visible Learning

• Cooney, J.B., Laidlaw, J. (2019) A curriculum structure with potential for higher than average gains in middle school math

• Tomlinson (2003) Differentiated Instruction

• Dweck (2016) Growth Mindset

• Carrier & Pashler (1992) The influence of retrieval on retention: the testing effect

• Boaler, J. (2016) Mathematical MindSets

• Rohrer, D., & Pashler, H. (2007) Increasing retention without increasing study time

• Kibble, J (2017) Best practices in summative assessment

• Laidlaw, J. (2019) Ongoing research in simulators and contextualized math

• Lave, J. (1988) Cognition in practice: Mind, mathematics and culture in everyday life

• Schmidt and Houang (2005) Lack of focus in mathematics curriculum: symptom or cause.”

Materials include research-based strategies. Examples include:

• “Detailed Lesson Plans (Research Indicator: Teacher pedagogy and efficacy remains the highest overall factor impacting student achievement. Multiple instructional models show greater gains than ‘stand and deliver’.)

• The Math Simulator (Research Indicator: On randomized controlled trials, The Math SimulatorTM elicited high effect sizes for achievement gains across educational interventions. Contextual learning and Productive Failure are likely influences contributing to the large achievement gains.)

• Teacher Instruction (Research Indicator: Clarity of teacher instruction shows a large effect on student achievement.)

• Practice Printable (Research Indicator: Differentiation of instruction leads to higher effect sizes compared to full-time ‘whole-group’ instruction. Varied instructional approaches support a growth mindset, an indicator for student success.)”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

The Teacher Guide includes Planning the Year, Comprehensive Supply List which provides a supply list of both required and recommended supplies for the grade. For example: “Required: Markers, Chart, Paper, Colored Pencils, Dry-erase Markers, Graph Paper, Ruler, Protractor; Recommended: Individual white boards/laminated alternative, Calculator, Base-10 Blocks, Fraction Tiles, Multiplication Chart, Fake Money, Number magnets, Unit cubes."

Each Detailed Lesson Plan includes a Materials List for each component of the lesson. For example, in 5.OA.B.3 Snowfall:

• “Immersion: Materials -  Snowfall Immersion video; Chart paper/Interactive whiteboard

• Data & Computation: Materials - Copies of Snowfall Data Artifact, one per student 

• Resolution: Materials - Snowfall Resolution video

• Math Simulator: Materials - Snowfall Simulation Trainer; Student Devices; Paper and Pencil; Student Headphones

• Practice Printable: Materials - Copies of Snowfall Practice Printable, 1 per student

• Student Reflection: Materials - Copies of Student Reflection rubric, 1 per student; White Paper; Colored Pencils; Sticky Notes

• Clicker Quiz: Materials - Snowfall Clicker Quiz; Student Devices; Paper and Pencil”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the standards and Mathematical Practices addressed by formal assessments.

• In the Teacher Guide, Curriculum Components lists several assessments: Clicker Quiz, Test Trainer Pro, and the summative Milestone Assessment. Each cluster has a Pre-assessment and a Post-assessment (Milestone Assessment) which clearly identifies the standard(s) being assessed. The standard is part of the title, for example, “Milestone Post-Assessment 5.MD.A.”; individual tasks and items are not identified on the actual assessment. However, each problem is identified with the standard being assessed in the teacher answer key. 

• Standards are identified accurately and are from the appropriate grade level. 

• Assessment problems are presented in the same order as the lessons. They are sequential according to Domain and Cluster headings.

• The Milestone Assessments include a chart that aligns Mathematical Practices to each question on the assessment, including identifying if the assessment is online, print, or both.

• The end of each lesson includes a student self-assessment rubric that has students evaluate their understanding of the content standard and the mathematical practices that align with the lesson.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for including an assessment system that provides multiple opportunities throughout the grade to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. There is guidance provided to help interpret student performance and specific suggestions for following-up. 

The assessment system provides multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance.

• In the Teacher Guide, the Domain Curriculum Components lists two assessments: Test Trainer Pro (formative) and Milestone Assessment (summative). 

  • “Milestone Assessment is a summative evaluation following each cluster per grade. They are automatically graded, yielding the percentage of items answered correctly. The math items are crafted to include items of varying difficulty.” “Please note: Milestone Assessments should not be used to determine student growth. As summative assessments, they are not as sensitive nor as accurate as the adaptive tool, Test Trainer Pro, for providing individual student data for achievement gains over time.”

  • “Test Trainer Pro acts as a low-stakes, formative learning tool for students to practice testing under more relaxed and stress-free conditions. It is an adaptive tool and is designed to elicit the largest gains in student achievement possible in the shortest period of time.”

• The Teacher Guide contains a section titled “A Practical Approach to Using Assessments, Rubrics & Scoring Guidelines.” This section provides several assessment rubrics: 

  • The MidSchoolMath Rubric and Scoring Framework aligns a percentage “raw score” with a 4-point rubric and proficiency levels.

  • The Milestone Assessment Rubric aligns a percentage “raw score” with a 4-point rubric and has suggestions for follow-up.

  • The Student Self Assessment has students reflect and identify understanding for each lesson component. 

• An article by Jo Boaler, “Assessing Students in a Growth Mindset Paradigm with Jo Boaler” provides “recommendations for assessment and grading practices to encourage growth mindsets.”

• Each Curricular cluster contains a tab for Assessments which has a Milestone Assessment Overview & Rubric. There is a rubric from 0 to 3 provided for the open response section of the assessment. To earn all 3 points, students must demonstrate accuracy, show work, and may only have minor mistakes.

  • “Recommended Scoring for Milestone Assessments: A 3-point response includes the correct solution(s) to the question and demonstrates a thorough understanding of the mathematical concepts and/or procedures in the task. This response: Indicates that the student has completed the task correctly, using mathematically sound procedures; Contains sufficient work to demonstrate a thorough understanding of the mathematical concepts and/or procedures; May contain inconsequential errors that to not detract from the correct solution(s) and the demonstration of a thorough understanding.”

• The Overview states, “All items in Milestone Assessments are at grade level and evaluate student understanding of the content at the ‘cluster’ level. Milestone Assessments should only be administered to students after all lessons are completed within the cluster, following recommended sequence and pacing.”

• The answer key for each Milestone Assessment provides examples of correct responses for each problem. There is a sample response for the open-ended questions. 

• Several of the other lesson components could be used as formative assessments or for progress monitoring such as the Clicker Quiz.

The assessment system provides task-specific suggestions for following-up with students. There are suggestions for follow-up that are generic strategies, and there are some that direct students to review specific content.

• The Milestone Assessment Rubric includes Recommendations for Follow Up. These are found in the front matter of the Teacher Guide. They are generic to all assessments and align with the 4 points of the rubric: 

  • “Review and correct any mistakes that were made. Participate in reteaching session led by teacher.

  • Review and correct any mistakes that were made. Identify common mistakes and create a ‘Top-3 Tips’ sheet for classmates. 

  • Review and correct any mistakes that were made. Participate in the tutorial session. 

  • Review and correct any mistakes that were made. Plan and host a tutorial session for the Nearing Proficient group.”

• The Milestone Assessment also includes suggestions based on which problems are missed. The guidance directs students to review the worked example and Clicker Quiz in the lessons that align to the missed problems and then revise the problems they missed in the assessment. This provides specific feedback to review the content of the lesson. 

• The Student Self-Assessment provides a generic strategy for follow-up: “Recommended follow-up: When students self-identify as ‘Don’t get it!’ Or ‘Getting there!’ on an assignment, is it essential for teachers to attempt to provide support for these students as soon as possible. Additionally, it is helpful for teachers to use scoring on Practice Printables and Clicker Quizzes to gauge student comprehension. Use the general scoring guidelines to determine approximate proficiency. It is highly recommended that all assignments may be revised by students, even those which are scored.”

• The Student Self-Assessment provides suggestions based on where the students rate themselves. Students are directed to review specific parts of the lesson to reinforce the parts they do not feel successful with. There are also more generic strategies suggested that go across lessons and grade levels.

• The materials state that “Test Trainer Pro automates assessment and recommendations for follow-up under the score. As an assessment, Test Trainer Pro is the most specific, and most accurate measure available in MidSchoolMath to determine how students are performing in terms of grade and domain level performance.” A teacher can view the Test Trainer Pro question bank; however, there is no way to review the specific follow-up recommendations provided since they are adapted to each student. 

• Exit Ticket results are sometimes used to suggest grouping for instructional activities the following day.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. Assessments include opportunities for students to demonstrate the full intent of grade-level standards and the mathematical practices across the series.

• Assessments are specific to each standard, so there is opportunity for students to demonstrate to the full intent of grade-level standards. 

• Considering both formative and summative assessments, there are a variety of item types offered including Exit Tickets, Clicker Quizzes, Test Trainer Pro, Lesson Reflection, Self-Reflection, and Milestone Assessments.

• Most assessments are online and multiple choice in format, though there is a print option for milestone assessments that includes open response. 

• Students have the opportunity to demonstrate the full intent of the practices in assessments; practices are aligned in Milestone assessments and addressed in the student self-assessments for each lesson.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In each Detailed Lesson Plan, Supporting Diverse Learners, there is a chart titled Accommodations, Modifications, and Extensions for English Learners (EL) and Special Populations that provides accommodations for each component of the lesson. Many of these are generic, but some are specific to the content of the lesson. For example, the components of 5.NBT.A.4, Round and Round, include:

• The Math Simulator Data & Computation: “Provide students with a place value chart. Provide students with manipulatives such as base ten blocks and fake money. Provide a rounding rules chart: (two rounding rules included).”

• Simulation Trainer: “Pair students to allow for peer teaching and support.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Make tracing paper available. Consider allowing students to answer questions verbally to a scribe. Students may benefit from doing fewer problems or receiving extended time to complete this assignment. Provide students with a place value chart, manipulatives chart, a number line, and a rounding rules chart. Underline the place that each question is asking the students to round to, and draw an arrow above every digit to the right of the underlined number.” The Practice Printable also has interactive buttons that allow students to complete work online through draw and text tools as well as a work pad that includes an opportunity to chat with the teacher. 

• Clicker Quiz: “Provide students with a  place value chart, manipulatives chart, a number line, and a rounding rules chart.”

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Materials provide multiple opportunities for advanced students to investigate the grade-level content at a higher level of complexity. The Exit Ticket in each lesson provides a differentiation plan that includes extension. While some strategies are the same across lessons, there are a variety of tasks offered. Examples include:

• 5.MD.C.5 Polly Packs, “Distribute this figure to students. Challenge them to find as many ways as possible to find the total volume. This may include enclosing it in an even larger prism and subtracting out the missing volume.”

• 5.NBT.B.7 Bread & Butter, “Write and solve four word problems involving decimals, one for each operation (add, subtract, multiply and divide). Use diagrams or other strategies to illustrate your reasoning for each problem.”

• 5.OA.B.3 Snowfall, “Consider having students play act their own math problem with the rest of the group tasked to solve it (≈ 2-3 min each) if time allows. Have students create and graph rules that have more than one step. For example, the rule is multiply by 2, and then subtract 4. Start with 1.”

In each Detailed Lesson Plan under Supporting Diverse Learners, there is a chart titled Accommodations, Modifications and Extensions for English Learners (EL) and Special Populations that provides extensions for each component of the lesson. Many of these are generic, but some are specific to the content of the lesson. For example, the components of 5.NBT.A.4 Round and Round include:

• The Math Simulator Resolution: “Task the students with describing another real-life situation when rounding to the nearest whole would not work to estimate.”

• Practice Printable: “Upon completion of the first page (Procedure #1), consider following the Exit Ticket Differentiation Plan. Give the students multi-operational problems involving decimals and estimation. Ex: Estimate each number to the nearest whole and solve using order of operations. 67.8 + 9.34 + 12.3.”

• Clicker Quiz: “Task students with writing and solving their own ‘clicker quiz’ question.”

• Extensions are optional; there are no instances of advanced students doing more assignments than their classmates.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 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.

Materials consistently provide strategies and supports for students who read, write, and/or speak in a language other than English to meet or exceed grade-level standards through regular and active participation in grade-level mathematics. Examples include:

• In the Detailed Lesson Plan for every lesson, the same two strategies are suggested: “Access Closed Caption and Spanish Subtitles within the video.” and “Pair students to allow for peer teaching and support. Consider allowing EL students to write the narrative in their native language, then use a digital translator to help them transcribe it into English.” 

• Each Detailed Lesson Plan makes a connection with one of the eight identified Math Language Routines (MLR), listed and described in the Teacher Guide. The MLRs include: Stronger and Clearer Each Time, Collect and Display, Critique, Correct, and Clarify, Information Gap, Co-Craft Questions and Problems, Three Reads, Compare and Connect, Discussion Supports.

• All materials are available in Spanish. 

• The use of protocols such as Think-Pair-Share, Quick Write, and I Wonder I Notice provides opportunities for developing skills with speaking, reading, and writing.

• Vocabulary is provided at the beginning of each lesson and reinforced during practice and lesson reflection, “In the Practice Printable, remind students that key vocabulary words are highlighted.” In the Student Reflection, the rubric lists the key vocabulary words for the lesson. Students are required to use these vocabulary words to explain, in narrative form, the math experienced in the lesson. 

• There is teacher guidance under the Resources tab - Math Language Routines. “Principles for the Design of Mathematics Curricula: Promoting Language and Content Development”, from the Stanford University Graduate School of Education, provides background information, philosophy, four design principles, and eight math language routines with examples. 

• There are no strategies provided to differentiate the levels of student progress in language development.

The materials reviewed for Core Curriculum by MidSchoolMath Grade 5 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. 

Examples where manipulatives are accurate representations of mathematical objects include:

• The students have access to virtual manipulatives on the Work Pad which is available online in their Simulator Trainer, Practice Printable, Assessments, and Clicker Quiz. These include shapes, 2-color counters, base 10 blocks, algebra tiles, protractor, and ruler. In addition, there are different styles of digital graph paper and dot paper on the digital whiteboard.

• Throughout the materials, there are visual models with number lines, graphs, or bars, though these cannot be manipulated. 

• During the Immersion and Resolution videos, items from the real world are used to represent mathematical concepts. 

• The Teacher Guide has a section titled “Guidance on the Use of Virtual Manipulatives.” This section includes sub-sections titled: Overview, General Guidance, During Lessons, Manipulative Tools, and Examples of their Use & Connecting to Written Methods. The “Examples of their Use & Connecting to Written Methods” provides teachers with guidance about how to use and make connections with the manipulatives.

• In the Detailed Lesson Plan, Practice Printable, there is a “Manipulative Task!” where students use the virtual tools in the Work Pad and specifically connect manipulatives to written methods. For example, in 5.G.4 It’s a Polygon World, Manipulative Task, “Create a polygon! It's a Polygon World provides a good opportunity for students to use the WorkPad manipulatives. Have students create a polygon of their choice (with 5 or fewer vertices), name the polygon, and provide a definition.”