6th Grade - Gateway 3

The instructional materials for Ready Grade 6 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

Students engage with problems and exercises through a consistent lesson structure. Use What You Know presents problems where students connect prior knowledge to the new concepts. Find Out More introduces the lesson followed by Model It, Picture It, Connect It, and Try It. This sequence repeats during Learn About as students engage with exercises during modeled and guided instruction. During Independent Practice, students complete problems and apply their learning. Additional problems and exercises are provided in the Practice and Problem Solving Book. For example:

• In Unit 3, Lesson 16, Find Out More, students explain the terms in an expression and reflect on the expression $$8x^3$$. During Model It, students practice writing expressions for mathematical statements. In Connect It, students analyze the terms within their expressions, and in Try It, students write equivalent expressions. During Independent Practice, students write and analyze expressions within the context of real-world situations and solve problems using their expressions.
• In the Practice and Problem Solving Book, Lesson 9, Add and Subtract decimals, students engage in exercises to practice solving addition and subtraction decimal situations and engage in problems as they complete multi-step problems and analyze the thinking of others.

The instructional materials for Ready Grade 6 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences. Students are presented with a problem at the beginning of each chapter to introduce new concepts and build upon prior knowledge.

The Ready lesson structure is consistent across the series. Within each unit, concept development is sequential. Understand lessons expose students to a concept for the first time and have a heavier emphasis on developing conceptual understanding. There are consistent sections within these lessons that scaffold student learning. Find Out More activities engage students in whole-class examples within a lesson. Learn About reinforces the mathematical concept of the lesson through pictures, models, examples of possible solutions, and independent practice. Practice provides ongoing practice of newly-learned mathematical concepts and skills as students explain and reflect.

The Ready progression chart shows what students learned in previous grades, what they will learn in their current grade, and how this will relate to future grades. In the Teacher Resource Book at the beginning of each lesson, the Learning Progression restates what had been taught in the previous grade, what is taught in the current grade, and what will be taught in the next grade. Lessons are designed using a scaffolded approach that begins with teacher-guided instruction, moves to working with partners, and concludes with completing problems independently. For example, in Unit 4 Geometry, the lessons are sequenced to build understanding of the area of two-dimensional figures to finding the surface area and volume of three dimensional figures. Lesson 22 builds understanding of the area of polygons. In Lesson 23, students build understanding of polygons in the coordinate plane, and in Lesson 24 they develop an understanding of nets and surface area. Lesson 25 finishes the sequence and the unit as students engage with volume. Each lesson includes opportunities for students to apply their knowledge about geometry to solve real-world problems.

The instructional materials reviewed for Ready Grade 6 meet expectations that there is a variety in what students produce.

Students respond and produce solutions in a variety of ways. Students provide evidence through drawings, representations, and written explanations. Students analyze and defend the work of others, and they justify their conclusions with verbal statements and mathematical reasoning.

In Picture It, Model It, Connect It, and Try It, students represent the problems in drawings and make connections between the drawings and the equations. In Pair/Share students discuss approaches to solving problems with another student, promoting students to justify their work and reason through the work of others. Question types vary and include multiple choice, true/false, draw a model, short answer, solve, explain, find the mistake, and multi-step performance tasks. For example:

• In the Practice and Problem Solving Book, Lesson 24 students find the surface area of a triangular prism using a diagram and net of a triangular prism in Problems 4 and 5, and in Problem 6 they evaluate an expression that represents the diagram to identify a mistake.
• In Lesson 13, Absolute Value and Ordering Numbers, Guided Practice, Pair/Share, Problem 15, students use a number line to discuss: “Are positive numbers always greater than negative numbers?” In Problem 16, students are presented with a table to discuss: “How do you compare negative numbers?”
• In Unit 2, Interim Assessment Performance Task, students are given a situation to solve a problem by first drawing and labeling a diagram, marking measurements appropriately using either fractions or decimals. They then Reflect on Mathematical Practices by explaining what models helped them solve the problem and whether it was easier to use fractions or decimals to be precise.

The instructional materials reviewed for Ready Grade 6 meet expectations for providing manipulatives that are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

In the Hands-On Activities found within each lesson, students use a variety of manipulatives including number cubes, coins, painter’s tape, index cards, and integer chips. For example:

• In Unit 2, Lesson 11, Hand-on Activity, students “use physical representations of multiples to find the least common multiple.” Students create sets of cards showing the first 10 multiples of their chosen numbers. Students then choose cards to find the least common multiple.
• In Unit 3, Lesson 17, Expression and Equations, Hands-on Activity, students use colored counters to represent the use of the distributive property to write expressions.
• In Unit 4, Lesson 22, Area of Polygons, Hands-on Activity, students use grid paper, rulers, and a pencil to find the area of a composite figure.

A detailed Manipulative List is included in the Program Implementation section. For Grade 6 these include centimeter cubes, two or more color counters, number cubes, algebra tiles, tangrams, pan balance, etc. These are used throughout the series to represent mathematical ideas and connect to written methods.

The instructional materials for Ready Grade 6 meet the expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. Support is provided in the following ways:

Step by Step in each lesson organizes content into chunks for student learning and includes guiding questions, key points, and teacher prompts. For example:

• Lesson 5, Solve Problems with Percents, Step by Step Mathematical Discourse 2 provides teachers with guidance and questions: “Look at the bar model. How does showing 800 as 10 parts with 80 in each help you understand the problem? Responses should include that using 10 parts makes it easier to figure 60% because the bar represents 100% and each part represents 10%, or 80 students. What other bar models could you use to find 60% of 800 easily? How many parts would you use? How many students would each represent?”
• In Lesson 24, Nets and Surface Area, Step by Step, Picture It, teachers are provided with guidance and questions to use: “What are the different shapes in this figure? How many of each face are there? Ask students to identify which faces have the same dimensions. Do you think you will use the same formula to find the areas of the faces of the triangle prism that you used for the rectangular prism? What formulas will you use to find the areas of the faces of the triangular prism?”

The Mathematical Discourse section in each lesson includes questions to engage students and advance their mathematical understanding. For example, in Lesson 17, Equivalent Expressions, the first Mathematical Discourse poses two questions: “1. Of the expressions you wrote on this page, which expression is easier for you to evaluate? Explain. 2. Explain why the commutative and associative properties for not apply to subtraction and division. How can you support your reasoning?” The second Mathematical Discourse also poses two questions: “1. Describe in your own words how the picture in Picture It shows 3(2+x). 2. How do the different ways to show the problem lead to different equivalent expressions?”

The instructional materials reviewed for Ready Grade 6 meet expectations that they contain a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials provide resources to support teacher planning.

• The Teacher Resource Book provides a separate pacing guide for the year, month, week, and day.
• The Unit Overview page includes lesson titles, page numbers, the primary and supporting standards, and content objectives. Two Common Core correlation charts, Ready Instruction Correlation and Interim Assessment Correlation, are included.
• The Cognitive Rigor and Ready Chart lists specific questions identified as DOK level 3.

The Teacher Resource Book contains components to assist with lesson delivery.

• At a Glance explains what students will be doing during each component of the lesson.
• Step by Step organizes the lesson into chunks and provides guiding questions.
• SMP Tips highlight specific Standards for Mathematical Practice.
• Mathematical Discourse includes questions to engage students and advance their learning. Possible answers and key ideas to listen for in student responses are included.
• Try It Solutions provide complete explanations and, in some cases, multiple solutions.
• Concept Extensions, ELL Support, and Visual Models provide support, suggestions, and strategies to engage students with activities that support varied abilities.
• Solutions in the Independent Practice section includes a correct response, at least one possible solution method, and the DOK level for the problems.
• Quick Check and Remediation includes an exit slip to monitor understanding. A chart includes error analysis and remediation suggestions.
• Hands-On Activity extends the concepts and skills, using manipulatives and a collaborative group approach.
• Challenge Activity extends the learning of those students who have mastered the skills and concepts.

The Teacher Toolbox found online contains the following technology components to assist with lesson delivery:

• Interactive Tutorials are referenced as part of Day 1 instruction for most lessons and provide interactive video clips for delivery of student mathematical learning.
• i-Ready Door 24 Plus is a free iPad app for fact-fluency practice but is not explicitly included in the Teacher Resource Book for instruction. i-Ready (available for additional purchase and used by most Ready users) is an Online Diagnostic and Instruction component.

The instructional materials reviewed for Ready Grade 6 meet expectations for containing a teacher edition in print and online that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their knowledge of the subject, as necessary.

In each lesson, information is provided for the teacher to understand and make connections between the mathematical content and practices, errors or misconceptions that may arise, and the rationale behind specific lesson parts. For example:

• In Unit 5, Lesson 26, the Learning Progression states: “In Grade 5, students made line plots and interpreted data. In this lesson students build on the knowledge and experiences developed in earlier grades to recognize statistical questions. Students write statistical questions and create models that represent the data from statistical questions. In Grade 7, students move from concentrating on analysis of data to production of data, understanding that good answers to statistical questions depend upon a good plan for collecting data relevant to the questions of interest.”
• In Lesson 9, the SMP TIP Attend to Precision states: “When you use the correct place-value names, such as 375 thousandths instead of point three-seven-five, you model to students that it is important to attend to precision.

Throughout Ready Grade 6 there is guidance for teachers that identifies and connects the underlying mathematics of a lesson. These are written in adult language.

The instructional materials reviewed for Ready Grade 6 meet expectations for containing a teacher edition (in print and in the online Teacher Toolbox) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

Each unit begins with a Lessons Progressions Chart. This chart lists lessons that students are building upon. These lessons come from previous grades and from Grade 6. For example, Unit 2, Lesson 5 builds upon Lessons 8 and 9 in Grade 5, connects to Grade 6, Lesson 10, and is related to Grade 7, Lessons 6 and 8. The chart also identifies the standard the lesson is preparing for: Standard 6.NS.3.

Each Lesson Overview includes a Learning Progression section. This section begins with an explanation of how the lesson builds on prior knowledge from Grade 5. The Learning Progression explains the lesson's overall connection to previous Grade 6 lessons and lessons in Grade 7, including the mathematical content of the lesson.

The instructional materials reviewed for Ready Grade 6 meet expectations for providing strategies for teachers to identify and address common student errors and misconceptions.

The Quick Check and Remediation at the end of a lesson presents a question to monitor understanding of the content of the lesson. This section includes a chart of incorrect answers, common errors, and remediation suggestions.

Lesson Quizzes provide teachers with a Common Misconceptions and Errors section that describes common misconceptions and errors.

Within lessons themselves, directions instruct teachers to watch for specific errors and misconceptions, and suggestions are provided to address these. For example, in Unit 1, Lesson 4, the Error Alert states, “Students who wrote 30,000 multiplied by the unit rate instead of dividing by it.”

The instructional materials reviewed for Ready Grade 6 meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Over the course of each lesson, responsibility for the learning process transfers from the teacher to the student. Students move from scaffolded support within the Guided Practice to independent problem solving within the Independent Practice. With guidance from the Teacher Resource book, feedback is provided to students throughout the lessons from the teacher. Feedback can be given in the Mathematical Discourse section in each lesson. Written feedback can be provided through the rubrics and student quizzes.

The instructional materials reviewed for Ready Grade 6 meet the expectation for offering ongoing formative and summative assessments that clearly denote which standards are being emphasized.

• Standards are clearly noted within assessments found in the Mathematics Assessments Teachers Guide.
• An Interim Assessment, providing standards correlations for each item, is located in the Interim Assessment - Teacher section of the Mid-Unit and End-of-Unit Resources tab at the end of each unit.
• Unit Assessments provide standards correlations for each item. Unit Assessments and correlations are found in the Unit Assessment Answer Key - Teacher section of the Mid-Unit and End-of-Unit Resources tab at the end of each unit.
• Lesson quizzes and quick checks are provided for most lessons. These quizzes assess the specific standards being taught in the lesson.

The instructional materials reviewed for Ready Grade 6 meet expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

• Each lesson follows a gradual release model in which scaffolded support is withdrawn as students gain mastery. Each lesson consists of four components: Introduction, Modeled and/or Guided Instruction, Guided Practice, and Independent Practice.
• Lessons are sequenced to build conceptual understanding, using concrete and pictorial representations to more abstract representations.
• The marginal notes in the Teacher Resource book often suggest ways to support students as a whole and subgroups of students who might need extra support. Notes include sections on vocabulary, concept extensions, visual models, hands-on activities, real-world connections, and challenge activities.
• Some lessons contain a Differentiated Instruction page which contains an Intervention Activity, On-Level Activity, and a Challenge Activity.

The instructional materials reviewed for Ready Grade 6 meet expectations for providing teachers with strategies for meeting the needs of a range of learners.

The Teacher Resource Book contains the following support:

• Each lesson includes a section called Small Group Differentiation that consists of two subsections: Reteach and Teacher-led Activities. (Some lessons have both subsections while others have one.) Specific lessons from earlier in the material, as well as the previous grade-level material in the series, are identified and can be used to review or fill in gaps in student knowledge. Every lesson also contains a Personalized Learning-Independent section.
• The marginal notes in the Teacher Resource Book suggest ways to support students as a whole and provide specific strategies for subgroups of students who might need extra support. This includes sections on vocabulary, concept extensions, visual models, hands-on activities, real-world connections, and challenge activities.
• Some lessons contain a Differentiated Instruction page that includes an Intervention Activity, On-Level Activity, and a Challenge Activity.
• The student Practice and Problem Solving book includes three levels of problems (basic, medium, challenge) that include verbal, visual, and symbolic representations.

The instructional materials reviewed for Ready Grade 6 meet expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The lesson structure introduces students to different strategies and representations for solving problems, and the embedded tasks use multiple representations (drawings, charts, graphs, numbers, or words) and different solution strategies that provide multiple points of entry to tasks. For example:

• In Unit 2, Lesson 11, Common Factors and Multiples, students explore different ways to identify common factors (listing, table), identify the greatest common factor (GCF), and use factors and the distributive property to write addition expressions as a product. They find the least common multiple (LCM) using number lines and lists. During Independent Practice, Problem 5 states “Choose two numbers from this list: 3, 4, 6, 12. Explain the difference between finding the greatest common factor and the least common multiple of the two numbers.” In giving students a choice on the numbers to use, there are multiple entry points to explain the differences. In addition, students have multiple models to find both the GCF and LCM.

Performance Tasks on Interim Assessments and Practice and Problem Solving include opportunities to engage with tasks with multiple entry points. For example:

• The Unit 3 Performance Task uses a situation that includes a goal to hike two trails with a lunch break in between, the pace of four different hikers within a range, the length of the trails, and the fact that all trails can be accessed from the same picnic spot. Students need to make a plan for the day by determining which two trails to hike, the pace for the hikes, approximate start and end for each hike, the length of time it will take to hike each trail they choose using an equation they develop. This task includes multiple entry points as students can begin in any order. In addition, the task requires students to reflect on Mathematical Practices by explaining how their equations model the problem, its independent and dependent variables, and by explaining their reasoning on the pace they choose.

The instructional materials reviewed for Ready Grade 6 meet expectations for suggesting support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

• ELL strategies can be found throughout the curriculum to build language development and understanding. For example, the Unit 2, Lesson 13, Lesson Introductions includes four prompts for teachers to support English Language Learners including: “Tell students this page models the distance a number is from zero. Mention that this distance is also called the absolute value.”
• Performance tasks found at the end of every unit provide students with a graphic organizer to help build vocabulary.
• In each lesson, there are tips for using visual models in order to provide support, suggestions, and strategies to engage students with activities that support varied abilities.
• Some lessons include a Differentiated Instruction section that includes an Intervention Activity, an On-Level Activity, and a Challenge Activity.
• Small Group Differentiation guidance is also included in the Lesson Pacing Guide. For example, Unit 4, Lesson 18, Understand Solution to Equations identifies one lesson from Grade 5 and two lessons from Grade 6.

The instructional materials reviewed for Ready Grade 6 meet expectations for providing opportunities for advanced students to investigate mathematics content at greater depth.

Materials offer the following instructional support for advanced learners:

• Each lesson of the Teacher Resource Book includes a Challenge Activity that provides students who have mastered the concepts and skills of the lesson with a more sophisticated problem. For example in Unit 2, Lesson 16, Algebraic Expressions Challenge Activity, students are given a situation for which they need to create a model, write an expression, provide a table to evaluate their expressions, and then apply that knowledge to a second situation with an explanation of all the possibilities.
• Concept Extensions provide an opportunity for students to extend their learning at a greater depth. In Unit 2, Lesson 15, Numerical Expressions with Exponents, the Concept Extension states: “Model Exponents. Tell students you will model 24. Use counters to show $$2^4$$ or 2 x 2. Ask: What happens to the number of counters when I show the next power or 2 or $$2^3$$? (The number of counters is multiplied by 2.) Show the result. Repeat this question and process for $$2^4$$. Ask students to predict the number of counters for $$2^1$$. Have volunteers show the model and explain their reasoning. (Following the model backwards, the number of counters is divided by 2, so 4 divided by 2 is 2 counters, for $$2^1$$. Ask students to predict the number of counters for $$2^0$$.)

The instructional materials reviewed for Ready Grade 6 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics.

• The names of characters and the context of real-world situations in problems represent a variety of cultural groups.
• Interactive Tutorials found in the online Teacher Toolbox represent students of both genders and various ethnicities.
• Cartoon characters presented in the student edition represent students of both genders and various ethnicities.