6th Grade - Gateway 3

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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.

• The Program Overview provides the teacher with information on program components and description about i-Ready classroom Mathematics implementation.

• Each unit has a Math Background document that provides the teacher with information to unpack the learning progressions and make connections between key concepts.

• Each Unit has an Unit Opener that provides the teacher with Unit Big Ideas and describes the themes of the unit.

• Each Unit has a Unit Flow and Progression video that describes how concepts are developed in the unit.

• Each Unit has a Professional Development document that provides guidance on instructional strategies, such as Supporting Math and Academic Vocabulary Development, Establishing Classroom Environments That Support Mathematical Discourse for ALL Learners, Knowing and Valuing Every Learner: Culturally Responsive Mathematics Teaching.

• Each Unit has a Unit Overview that provides the teachers with pacing, objectives, standards, vocabulary and lesson-level differentiation for each of the lessons in the unit.

• The Teacher’s Guide provides in-class instruction and practice included in the teacher’s edition.

• The Teacher’s Guide for Assessments and Reports supports whole group/partner discussion, ask/listen fors, common misconceptions, error alerts, etc.

• DIfferentiation strategies are included before and during the unit/lesson for the teacher. There are recommended resources to support students’ learning needs that are highlighted in the Prerequisites report.

• Unit and Lesson Support includes information about prerequisite lessons to focus on, and identifies the important concepts within those lessons.

• On the Spot Teaching Tips suggest additional scaffolding to support students with unfinished prerequisite learning as they engage with on-level work. 

• Digital Math Tools contain support videos that explain how to use their digital tools.

• Ready Classroom Central is an online teacher portal with resources for professional support such as training videos, planning tools, implementation tips, whitepapers, and discourse support.

• Language Expectations identify examples of what English learners at each level of language proficiency can do in connection with a one grade-level standard. 

• The Unit Prepare For provides teachers with guidance to support students when completing the graphic organizer in the beginning of the unit, Prepare for Unit. There is additional guidance to Build Academic Vocabulary through the use of identified cognates and specified academic terms. 

• The Unit Review includes problem notes for teachers identifying the Depth of Knowledge level of each problem and the standard, along with suggested strategies, and possible misconceptions based on the selected answer. 

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Throughout each lesson planning information, there is narrative information to assist the teacher in presenting student materials throughout all phases of the unit and lessons. Examples include:

• Unit 2, Lesson 8, Session 1, Connect It, Problem 3, “How is using the standard algorithm similar to using partial quotients to divide a three-digit number by a one-digit number? How is it different?” The teacher’s edition provides guidance for the teacher, “Look for understanding that the standard algorithm does not use zeros as placeholders in the quotient; instead each part of the quotient is written in its corresponding place value as the division occurs. Error Alert: If students struggle to identify the place value of each digit in the quotient of the standard algorithm and place the digits of the quotient in the wrong position, then have them organize their work on grid paper to maintain alignment. Provide place-value charts so students can match the digits in their quotients to the corresponding place values.

• Unit 3, Lesson 13, Session 2, Apply It, students answer questions about equivalent ratios. The Teacher’s Edition provides guidance for the teacher, “For all problems, encourage students to use a model to support their thinking. Allow some leeway in precision; drawing number lines with equal spacing between tick marks can be difficult, and precise measures are not necessary to determine a solution to the problem.”

• Unit 4, Prepare For, teachers are provided with guidance in using the graphic organizer included for students in Prepare for Unit Rates and Percent. This includes guidance in analyzing the term “comparing ratios” and how to build academic vocabulary throughout each lesson in the unit. “Next, have students meet with a partner to share ideas and add new information to the organizer. Circulate and validate responses and clarify any misconceptions.”

• Unit 5, Lesson 22, Session 2, Develop, Discuss It, teachers are instructed to support partner discussion by “After students work on Try It, encourage them to respond to Discuss It with a partner. If students need support in getting started, prompt them to ask each other questions such as: In your equation, how did you represent the number of days? The cost for a day? How can you organize the information to find a pattern? How can you test whether your equation makes sense?” Common Misconception: “Listen for students who are not precise when defining variables. For example, students may say d = dollars, instead of d = dollars to feed 8 meerkats for n days. As students share their strategies, have partners discuss and reinforce understandings of variables and the need for precise descriptions of the meaning of the variable.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

In the Teacher’s Guide, a Lesson Progression table is provided that links each lesson within the current unit to a prior and future grade level lesson. Within the Math Background section, detailed explanations of the mathematical concepts in each lesson are provided. For example, in Unit 2,  Math Background, Understanding Content Across Grades, insights are provided for prior knowledge, current lesson, and future learning in starting Lesson 7:

• Prior Knowledge, “Insights on: Adding and Subtracting Decimals. Students learn to add and subtract decimals using the same variety of models and strategies they used to add and subtract whole numbers. Common Error - When students add and subtract decimals using the standard algorithm, they may line the numbers up by the end digits, rather than by place- value. A place-value chart is an excellent tool to help students focus on the value of each digit.” This information is accompanied by example problems worked out using a number line and another using a place value chart. 

• Current Lesson, insights are provided on adding, subtracting, and multiplying decimals. For example: “Multiplying Decimals - When multiplying decimals, students use both decimal and fractional forms of the factors to make sense of the placement of the decimal point in the product.” This information is accompanied by an example problem worked out using the relationship between decimals and fractions to understand decimal multiplication.

• Future Learning, insights are provided on understanding addition with positive and negative numbers. For example, “Students use integer chips to observe a key difference between adding two numbers with different signs and adding two numbers with the same sign. When the addends have different signs, they can cancel out zero pairs. When the addends have the same sign, all the chips are of the same type, so there are no zero pairs.” This is accompanied by an example problem solved using integer chips.

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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. 

• The Correlations Document describes lesson correlation to the CCSSM through multiple lenses. The document identifies the major and supporting areas of focus within the CCSSM, and the lessons address those standards. There is a table correlating each lesson with the standards covered, designating standards as “Focus”, “Developing”, or “Applied” within each lesson. The Correlations Document also identifies the Standards of Mathematical Practice that are included in each lesson. One table is organized by MP while the other is organized by lesson. The Unit Review Correlation identifies the associated standard and lesson to each problem within the Unit Review, along with their Depth of Knowledge level. 

• The Program Overview provides teachers and explanations for how the standards are addressed in each unit. One section identified is the coherence section titled “Lesson Progression.” 

• At the beginning of each Unit, “Lesson Progression” shows how each standard connects to and builds upon the previous grade levels. Each standard is identified in each lesson. It is arranged in a flow chart and connects lessons to future grade levels. 

• In the lesson overview, prior knowledge is identified, so teachers know what standards are linked to prior work. Future grade level content is also identified.

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

• Grade Level Support, “Learning Progression” identifies prerequisite skills for each lesson and their related standards for the two prior grade levels, when applicable, in a flow chart. For example, the materials identify a prerequisite skill for Unit 2 as, “Multiply with fractions and divide with unit fractions.” It identifies two connected standards from prior grades, Multiply Fractions by Whole Numbers, 4.NF.4, 4.NF.4c, and designated essential skills, Multiply Fractions in Word Problems, 5.NF.6, and Divide Unit Fractions in Word Problems, 5.NF.7, 5.NF.7c.

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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. Examples include:

• The Teacher’s Guide and the Program Implementation area in the digital platform contains a section “Understanding the Try-Discuss-Connect Routine.” This routine is embedded throughout the program. This document explains how the routine is used. “Ready Classroom Mathematics empowers all students to own their learning through a discourse-based instructional routine. Lessons are divided into Explore, Develop, and Refine sessions and are taught over the course of a week. In Explore and Develop sessions teachers facilitate mathematical discourse through a Try-Discuss-Connect instructional routine.” 

• “Using a Session” in the Teacher’s Guide describes the planning and support features within the Teacher’s Guide. This includes each component of the lesson and teacher’s guide and describes why it is important in the lesson. For example, “SMPs are infused throughout the instructional model. Deepen Understanding is a consistent opportunity to build understanding of a key lesson concept by extending mathematical discourse. The content connects a particular aspect of lesson learning to an SMP, showing how it might look in the classroom.”

• Integrating Language and Mathematics identifies and explains the six language routines embedded within the curriculum. It identifies each routine, why a teacher may use it, the process and what part of the Try-Discuss-Connect Routine it can be used within. For example, for Say It Another Way, “What: A routine to help students paraphrase as a way to process a word problem or other written text and confirm understanding. Why: Paraphrasing helps students figure out whether they have understood something they have read or heard...How: Students read or listen to a word problem or other written text. One student paraphrases the text. Other students give a thumbs-up to show that the paraphrase is accurate and complete.”

Materials reference relevant research sources. Examples include: 

• Boaler, (2016), Mathematical Mindsets

• Council of the Great City Schools, (2016), A Framework for Re-Envisioning Mathematics Instruction for English Language Learners

• Kersaint, (2016), Orchestrating Mathematical Discourse to Enhance Student Learning

• National Council of Teachers of Mathematics, (2010), Teaching and Learning Mathematics

• National Council of Teachers of Mathematics, (2014), Principals to Action

• National Council of Teachers of Mathematics, (2014), Using Research to Improve Instruction

• Richhart, (2009), Creating Cultures of Thinking

Materials include research-based strategies. Examples include: 

• “Collaborative learning (partner or small group) encourages students to present and defend their ideas, make sense of and critique the ideas of others, and refine and amend their approaches.” Examples include, “Ready Classroom Mathematics lessons provide multiple opportunities for collaborative learning, such as Discuss It prompts where students explain and justify their strategies to each other and Consider This prompts where students compare problem-solving approaches, solutions, and reasoning.” The research included to support this is, “Research tells us that when students work collaboratively, which also gives them opportunities to see and understand mathematics connections, equitable outcomes result.” (Boaler, 2016)

• Professional Development, contains an adapted excerpt from Reimagining the Mathematics Classroom, co authored by Dr. Mark Ellis for teachers. The excerpt explains “funds of knowledge” to teachers and how they can apply this knowledge using the materials. “Connect to Culture in the Teacher’s Guide for each lesson offers suggestions for tapping into students’ funds of knowledge and connecting the knowledge to Try It and other problems.”

The materials reviewed for i-Ready Mathematics Grade 6 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The Teacher’s Guide includes an Activity Sheet in the Table of Contents which provides a list of printable tools and resources. “Dot Paper, Frayer Model 2, Fraction Bars are available to print and copy for each student.” Materials include a Manipulatives List by Lesson for each grade level. For example: 

Unit 3, Lesson 12: 6 unit tiles per pair and 7 two-color counters per pair.

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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 mathematical practices across the series. 

The formative and summative assessments include a variety of item types to measure grade-level standards. For example:

• Fill-in-the-blank

• Multiple select

• Matching

• Graphing

• Constructed response (short and extended responses)

• Technology-enhanced items, e.g., drag and drop, drop-down menus, matching 

Assessments are provided as a PDF or online for teachers that can be provided to students in either format.

The materials reviewed for i-Ready Classroom Mathematics Grade 6 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

• At the end of the Lesson Quiz in the Teacher’s edition, there is a section for differentiation that provides suggestions for Reteach (Tools for Instruction), Reinforce (Math Center Activity), and Extend (Enrichment Activity). Reteach, “Students who require additional support for prerequisite or on-level skills will benefit from activities that provide targeted skills instruction.” Reinforce, “Students who require practice to reinforce concepts and skills and deepen understanding will benefit from small group collaborative games and activities (available on-level, below-level, and above-level versions).” Extend, “Students who have achieved proficiency with concepts and skills and are ready for additional challenges will benefit from group collaborative games and activities that extend understanding.” The digital platform contains these activities for each lesson.

• In Refine lessons, the teacher’s edition provides suggestions to Group & Differentiate, “Identify groupings for differentiation based on the Start and problems 1-3. A recommended sequence of activities for each group is suggested below. Use the resources on the next page to differentiate and close the lesson.” Resources are suggested for groups Approaching Proficiency, Meeting Proficiency, and Extending Beyond Proficiency. The resources are found in the digital platform (Reteach, Reinforce, Extend). The following pages also contain descriptions of additional activities in the teacher’s edition for Reteach, Reinforce, and Extend.

The materials reviewed for i-Ready Classroom Mathematics Grade 6 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Refine sessions provide recommendations for students that demonstrate understanding “Extending Beyond Proficiency” to engage in problems for reinforcement and a challenge. The number of problems is the same as students who are considered to be “Meeting Proficiency”. Additional Enrichment Activities can be found online in the Small Group Differentiation Extend section.

In Explore and Develop sessions, the materials contain a Deepen Understanding section to extend understanding of the lesson’s key concepts through the use of discourse with students. The section contains teacher prompts and suggestions for what ideas to look for from students. Each Deepen Understanding is labeled with an embedded mathematical practice. Examples include: 

• Unit 1, Lesson 2, Enrichment Activity Building Shapes, students are provided with a challenge question at the beginning and multiple opportunities to draw and explain their answer. “How can you build different polygons with given areas using triangles, rectangles, and parallelograms?”

• Unit 5, Lesson 21, Session 3, Deepen Understanding provides teachers with prompts to support students in noticing how representations differ based on an equation. “Prompt students to compare differences between a hanger diagram for an addition equation and one for a multiplication equation...Ask: In what situations could a hanger diagram be used to represent addition or multiplication? When could it only represent addition? Listen for: Multiplication is repeated addition. If the variable addend repeats, then you can use a multiplication equation. If not, then you have to use an addition equation.”

The materials reviewed for i-Ready Classroom Mathematics Grade 6 meet expectations for providing strategies and support 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:

• Each Lesson Session includes differentiated support for various levels of English proficiency with level 1-3, levels 2-4, and levels 3-5 identified. Support for Academic language is used during the “Try-Discuss-Connect Language” routines in each lesson. 

• In the Program Overview, language expectations charts are provided that describe the language English Learners can understand and produce in connection with students’ levels of English proficiency. Teachers can use the examples to help meet the needs of English Learners. 

• Each Unit Overview connects with one of the CCSS addressed in the unit and shows an example of how language expectations can help to differentiate instruction to meet the needs of English learners. 

• In the Program Overview, there is an Integrate Language and Mathematics section. “Scaffolded language support for a specific problem is outlined. These suggestions for scaffolding and amplifying language can be applied to other problems as well.” 

• Language objectives are included and students are expected to understand and produce language as they work on lesson objectives. Graphic organizers are used to help students access prior knowledge and vocabulary they build on in the lesson.

• Discourse cards are available in the Teacher Digital Experience under the Ready Classroom Mathematics Toolbox. These cards provide sentence starters and questions to help students engage in conversations with their partners, small groups or the whole class. 

• All classroom materials are available in Spanish.

• Multilingual Glossary is available in Arabic, Chinese, Haitian Creole, Portuguese, Russian, Tagalog, Urdu, and Vietnamese. There is a Bilingual glossary in the student textbook that includes mathematics vocabulary in English and Spanish.

The materials reviewed for i-Ready Classroom Mathematics Grade 6 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.

Digital tools are available for students. These tools include Counters and Connecting Cubes, Base-Ten Blocks, Number Line, Multiplication Models, Perimeter and Area, and Fraction Models. Geometry, Scientific Calculator and Graphing Calculator are also included but cannot be reviewed as these tools are powered by Desmos. Support videos are available for each of the digital tools, explaining how they may be used and their functions. For example:

• Grade 6 Standard Manipulative Kit includes algebra tiles, plastic rulers, centimeter cubes, base ten blocks, number cubes, rainbow color tiles, two color counters and connecting cubes. A la carte items are available. The materials state that these items may only be used once, may be common to classrooms, or print options are available. A la carte items include fraction bars, tangrams, geoboards, geosolids and rainbow fraction circle set. 

• Visual models such as number lines, graphs, or bars, are also available but cannot be manipulated.

The “Try-Discuss-Connect” routine embedded throughout every lesson provides students the opportunity to connect and transition from the use of manipulatives to written methods. Inside of the digital platform, Program Implementation, “Try-Discuss-Connect Routine Resources, Understanding the Try-Discuss-Connect Instructional Routine”, the guide describes how the routine helps students transition from manipulatives to written methods. In the Try It activity, “students have access to a variety of tools and manipulatives to use to represent the problem situation. During the Discuss It activity, “Students present and explain their solution methods and listen to and critique the reasoning of others, models and representations.” “The class then looks at the strategies highlighted in the Picture It and Model It, and students make connections between strategies, their own strategies, and the strategies discussed as a class.” During the Connect It activity, “Students apply their thinking during the lesson to new problems.” The routine integrates the CRA model in the:

• Try It, “Students use concrete, representational, or abstract strategies to solve the problem, based on their understanding of the problem or mathematical concept.

• Discuss It, “Students who use more concrete approaches begin to make connections to representational or abstract approaches as they engage in partner discussions.”

• Connect It, “Through the Connect It questions, students connect concrete and representational approaches to more abstract understanding as they formalize their connections.”