3rd Grade - Gateway 3

The materials reviewed for enVision Mathematics Grade 3 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. 

The Teacher’s Edition Program Overview provides comprehensive guidance to assist teachers in presenting the student and ancillary materials. It contains four major components: Overview of enVision Mathematics, User’s Guide, Correlation and Content Guide.

• The Overview provides the table of contents for the course as well as a pacing guide for a traditional year long course as well as block/half year course. The authors provide the Program Goal and Organization, in addition to information about their attention to Focus, Coherence, Rigor, the Math Practices, and Assessment.

• The User’s Guide introduces the components of the program and then proceeds to illustrate how to use a ‘lesson’: Lesson Overview, Problem-Based Learning, Visual Learning, and Assess and Differentiate. In this section, there is additional information that addresses more specific areas such as STEM, Building Mathematical Literacy, Routines, and Supporting English Language Learners.

• The Correlation provides the correlation for the grade.

• The Content Guide portion directs teachers to resources such as the Big Ideas in Mathematics, Scope and Sequence, Glossary, and Index.

Within the Teacher’s Edition, each Lesson is presented in a consistent format that opens with a  Lesson Overview, followed by probing questions to provide multiple entry points to the content, error intervention, supports for English Language Learners, and ends with multiple Response to Intervention (RtI) differentiated instruction.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The Teacher’s Edition includes numerous brief annotations and suggestions at the topic and lesson level organized around multiple mathematics education strategies and initiatives, including the CCSSM Shifts in Instructional Practice (i.e., focus, coherence, rigor), CCSSM practices, STEM projects, and 3-ACT Math Tasks, and Problem-Based Learning. Examples of these annotations and suggestions from the Teacher’s Edition include:

• Topic 1, Lesson 1-4, Visual Learning Bridge, Teachers begin the Classroom Conversation by saying the following, “Why do you need 3 equal groups? [There are 3 friends who want to share the toys equally.]” 

• Topic 6, Lesson 6-6, Problem Solving, Problem 10, “Vocabulary Fill in the blanks. Mandy finds the ___ of this shape by dividing it into rectangles. Phil gets the same answer by counting ___.” The materials shows the outline of a shape on a unit square grid. Teacher guidance: “Vocabulary If students struggle to fill in the blanks, have them work with a partner and review the vocabulary terms for the topic by using the Topic 6 My Word Cards available on SavvasRealize.com.”

• Topic 15, Lesson 15-3, Problem Solving, Problem 11, “Reasoning Explain which of the shapes at the right you can cover with whole unit squares and not have any gaps or overlaps.” The materials show a parallelogram and a rectangle. Teacher guidance: “Reason Abstractly Help students to understand that to cover one of the shapes with unit squares and not have any gaps or overlaps, the unit squares need to fit tightly into all the corners of the shape. Can you fit unit squares tightly into all the corners of the shape on the left? Why or why not? [No; Sample answer: The corners are not right angles.]”

The materials reviewed for enVision Mathematics Grade 3 meet expectations for containing adult-level explanations and examples of the more complex grade concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. 

The materials provide professional development videos at two levels to help teachers improve their knowledge of the grade they are teaching.

• “Professional development topic videos are at SavvasRealize.com. In these Topic Overview Videos, an author highlights and gives helpful perspectives on important mathematics concepts and skills in the topic. The video is a quick, focused ‘Watch me first’ experience as you start your planning for the topic.

• Professional development lesson videos are at SavvasRealize.com. These Listen and Look for Lesson Videos provide important information about the lesson.

An example of the content of a Professional development video:

• Topic 10: Professional Development (topic) Video, “Multiplying by multiples of ten is an important component in computing products of multi-digit factors. Students can combine their knowledge of place-value patterns, properties of multiplication, and basic facts in several ways to develop strategies that lead to fluency in multiplying by multi-digit numbers … Students can use what they know about patterns and skip counting and place value to skip count by multiples of ten on an open number line ... This pattern is the result of the structure of our base-ten numeration system … the fact that each place-value position is ten times the position to its right … Using these types of patterns to find products of multiples of ten helps students build confidence and accuracy in multiple-digit computation.”

The Math Background: Coherence, Look Ahead section, provides adult-level explanations and examples of concepts beyond the current grade as it relates what students are learning currently to future learning.

An example of how the materials support teachers to develop their own knowledge beyond the current grade:

• Topic 14, Math Background: Coherence, Look Ahead, the materials state, “Grade 4 Time In Topic 10, students will use the four operations to solve word problems involving intervals of time, including problems with simple fractions.” An example word problem is given where teachers are tasked with figuring out how many hours a runner trains for a race. “Equivalence In Units of Measure In Topic 13, students will extend their understanding of customary and metric units of length, weight, mass, and capacity (liquid volume). They will learn the relative sizes of various measurement units and solve problems involving converting a measurement from a larger unit to a smaller one.” A word problem about making gallons of punch given a recipe is provided.

The materials reviewed for enVision Mathematics Grade 3 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Standards correlation information is indicated in the Teacher’s Edition Program Overview, the Topic Planner, the Lesson Overview, and throughout each lesson. Examples include:

• The Teacher’s Edition Program Overview, Grade 3 Correlation to Standards For Mathematical Content organizes standards by their Domain and  Major Cluster and indicates those lessons and activities within the Student’s Edition and Teacher’s Edition that align with the standard. Lessons and activities with the most in-depth coverage of a standard are distinguished by boldface. The Correlation document also includes the Mathematical Practices. Although the application of the mathematical practices can be found throughout the program, the document indicates examples of lessons and activities within the Student’s Edition and Teacher’s Edition that align with each math practice.

• The Teacher’s Edition Program Overview, Scope & Sequence organizes standards by their Domain, Major Cluster, and specific component. The document indicates those topics that align with the specific component of the standard.

• The Teacher’s Edition, Topic Planner indicates the standards and Mathematical Practices that align to each lesson.

The Teacher’s Edition, Math Background: Coherence provides information that summarizes the content connections across grades. Examples of where explanations of the role of the specific grade-level mathematics are present in the context of the series include:

• Topic 2, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Patterns, Skip Counting, and Equal Groups” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 2 connect to what students will learn later?” and provides a Grade 4 connection, “Multiplication with Greater Numbers In Topics 3 and 4, students will multiply greater numbers using strategies and properties.”

• Topic 6, Math Background: Coherence, the materials highlight four of the learnings within the topics: “Area as Covering, Relate Area to Multiplication and Addition, Distributive Property, and Area of Irregular Rectilinear Figures” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 6 connect to what students will learn later?” and provides a Grade 4 connection, “Area Formulas In Topic 13, students will apply the area formula for a rectangle to solve real-world and mathematical problems.”

• Topic 12, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Fractions, Fraction Representations, and Line Plots” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 12 connect to what students will learn later?” and provides Grade 4 connections, “Fraction Equivalence and Comparison In Topic 8, students will generate equivalent fractions by multiplying or dividing the numerator and denominator by the same nonzero number. They will also, compare fractions with different numerators and different denominators. Operations with Fractions In Topic 9, students will add and subtract fractions and mixed numbers with like denominators. In Topic 10, students will multiply a whole number and a fraction. Solve Problems Involving Fractions and Line Plots In Topic 11, students will solve problems involving line plots that display measurements in fractions of a unit (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}).”

The materials reviewed for enVision Mathematics Grade 3 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. The Teacher’s Edition Program Overview provides detailed explanations behind the instructional approaches of the program and cites research-based strategies for the layout of the program. Unless otherwise noted all examples are found in the Teacher’s Edition Program Overview.

Examples where materials explain the instructional approaches of the program and describe research-based strategies include:

• The Program Goal section states the following: “The major goal in developing enVision Mathematics was to create a program for which we can promise student success and higher achievement. We have achieved this goal. We know this for two reasons. 1. EFFICACY RESEARCH First, the development of enVision Mathematics started with a curriculum that research has shown to be highly effective: the original enVisionMATH program (PRES Associates, 2009; What Works Clearinghouse, 2013). 2. RESEARCH PRINCIPLES FOR TEACHING WITH UNDERSTANDING The second reason we can promise success is that enVision Mathematics fully embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.”

• The Instructional Model section states the following: “There has been more research in the past fifteen years showing the effectiveness of problem-based teaching and learning, part of the core instructional approach used in enVision Mathematics, than any other area of teaching and learning mathematics (see e.g., Lester and Charles, 2003). Furthermore, rigor in mathematics curriculum and instruction begins with problem-based teaching and learning. … there are two key steps to the core instructional model in enVision Mathematics. STEP 1 PROBLEM-BASED LEARNING Introduce concepts and procedures with a problem-solving experience. Research shows that conceptual understanding is developed when new mathematics is introduced in the context of solving a real problem in which ideas related to the new content are embedded (Kapur, 2010; Lester and Charles, 2003; Scott, 2014)... STEP 2 VISUAL LEARNING Make the important mathematics explicit with enhanced direct instruction connected to Step 1. The important mathematics is the new concept or procedure students should understand (Hiebert, 2003; Rasmussen, Yackel, and King, 2003). Quite often the important mathematics will come naturally from the classroom discussion around students’ thinking and solutions from the Solve and Share task…”

• Other research includes the following:

  • Hiebert, J.; T. Carpenter; E. Fennema; K. Fuson; D. Wearne; H. Murray; A. Olivier; and P.Human. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann, 1997.

  • Hiebert, J. (2003). Signposts for teaching mathematics through problem solving. In F. Lester, Jr. and R. Charles, eds. Teaching mathematics through problem solving: Grades Pre-K–6 (pp. 53–61). Reston, VA: National Council of Teachers of Mathematics.

• Throughout the Teacher’s Edition Program Overview references to research-based strategies are cited with some reference pages included at the end of some authors' work.

The materials reviewed for enVision Mathematics Grade 3 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

In the online Teacher Resources for each grade, a Materials List is provided in table format identifying the required materials and the topic(s) where they will be used. Additionally, the materials needed for each lesson can be found in the Topic Planner and the Lesson Overview. Example includes:

• Topic 1, Topic Planner, Lesson 1-2, Materials, “Number lines  (or TT 7), Colored pencils”

• Topic 6, Lesson 6-1, Lesson Resources, Materials, “Two-color tiles (or Teaching Tool 8), Area of shapes (or Teaching Tool 12), Centimeter grid paper (or Teaching Tool 13)”

• Teacher Resources, Grade 3: Materials List, the table indicates that Topic 14 will require the following materials: “1-liter bottles, 1-liter beaker, Blank Clock Faces (Teaching Tool 20), ...”

The materials reviewed for enVision Mathematics Grade 3 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. 

The assessment system provides multiple opportunities to determine student’s learning throughout the lessons and topics. Answer keys and scoring guides are provided. In addition, teachers are given recommendations for Math Diagnosis and Intervention System (MDIS) lessons based on student scores. If assessments are given on the digital platform, students are automatically placed into intervention based on their responses.

Examples include:

• Topic 3, Lesson 3-2, Independent Practice, Evaluate, Quick Check, Problem 11, “A check mark indicates items for prescribing differentiation on the next page. Items 11 and 18: each 1 point. Item 17: up to 3 points.” For example, Directions: “Leveled Practice Multiply. You may use counters or pictures to help. 11. 7 \times 3 = ___” The following page, Step 3: Assess and Differentiate states, “Use the Quick Check on the previous page to prescribe differentiated instruction. I Intervention 0-3 points, O On-Level 4 points, A Advanced 5 points.” The materials provide follow-up activities—to be assigned at the teacher’s discretion—to students at each indicated level: Intervention Activity I, Technology Center I O A, Reteach to Build Understanding I,  Build Mathematical Literacy I O, Enrichment O A, Activity Centers I O A, and Additional Practice Leveled Assignment I Items 1-5, 11-12, 15, 17, 19-21, O Items 3-4, 6-7, 12-13, 15-16, 18-21 and A Items 8-14, 16, 18-21.

• Topic 5, Topic Assessment, Problem 4, “Find the product. 5 \times 7” Students choose amongst answer options (A) 28 (B) 30 (C) 35 (D) 42. Item Analysis for Diagnosis and Intervention indicates: DOK 1; MDIS B52, and B53; Standard 3.OA.C.7. Scoring Guide indicates: 4 1 point “Correct choice selected.”

• Topic 8, Topic Performance Task, Problem 5, “Vacation Trip Mia is planning a vacation in Orlando, FL. The Mia’s Route table shows her route and the miles she will drive. … Mia has to book a hotel and buy theme park tickets. The Hotel Prices and Theme Park Prices tables show the total prices for Mia’s stay. The Mia’s Options list shows two plans that Mia can choose from. … 5. One theme park has a special offer. For each ticket Mia buys, she gets another ticket free. Shade the squares in the table at the right to show this pattern. Explain why the pattern is true.” The materials show a 5 by 5 addition table. Item Analysis for Diagnosis and Intervention indicates: DOK 3, MDIS C23, Standard 3.OA.D.9, MP.8. Scoring Guide indicates: 2 points “Pattern identified with explanation” and 1 point “Pattern identified without explanation.”

• Topics 1-12, Cumulative/Benchmark Assessment, Problem 15, “Lenny’s goal is to collect 500 tabs from cans to recycle. One month he collects 217 tabs and the next month he collects 186 tabs. How many more tabs does Lenny need to collect? Write equations to represent the number of tabs he still needs to collect.” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS E3, Standard 3.OA.D.8. Scoring Guide indicates: 2 points “Correct answer and equations” and 1 point “Correct answer or equations.”

The materials reviewed for enVision Mathematics Grade 3 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The materials provide formative and summative assessments throughout the grade as print and digital resources. As detailed in the Assessment Sourcebook, the formative assessments—observational tools, Convince Me!, Guided Practice, and Quick Checks—occur during and/or at the end of a lesson. The summative assessments—Topic Assessment, Topic Performance Task, and Cumulative/Benchmark Assessments—occur at the end of a topic, group of topics, and at the end of the year.  The four Cumulative/Benchmark Assessments address Topics 1-4, 1-8, 1-11, and 1-16. 

• Observational Assessment Tools “Use Realize Scout Observational Assessment and/or the Solve & Share Observation Tool blackline master.”

• Convince Me! “Assess students’ understanding of concepts and skills presented in each example; results can be used to modify instruction as needed.”

• Guided Practice “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to review or revisit content.”

• Quick Check “Assess students’ conceptual understanding and procedural fluency with lesson content; results can be used to prescribe differentiated instruction.”

• Topic Assessment “Assess students’ conceptual understanding and procedural fluency with topic content.” Additional Topic Assessments are available with ExamView.

• Topic Performance Task “Assess students’ ability to apply concepts learned and proficiency with math practices.

• Cumulative/Benchmark Assessments “Assess students’ understanding of and proficiency with concepts and skills taught throughout the school year.”

The formative and summative assessments allow students to demonstrate their conceptual understanding, procedural fluency, and ability to make application through a variety of item types. Examples include: 

• Order; Categorize

• Matching

• Graphing

• Yes or No; True or False

• Number line

• True or False

• Multiple choice

• Fill-in-the-blank

• Technology-enhanced responses (e.g., drag and drop)

• Constructed response (i.e., short and extended responses)

The materials reviewed for enVision Mathematics Grade 3 meet expectations for providing strategies and support for students in special populations to support their regular and active participation in learning grade-level mathematics. 

The materials provide strategies and support for students in special populations via its 3-tier Response to Intervention (RtI) Differentiated Instruction plan.

• Tier 1 offers Ongoing Intervention: “During the core lesson, monitor progress, reteach as needed, and extend students’ thinking.” 

  • Types of support include:

    • Guiding Questions -  In the Teacher’s Edition Guiding questions are used to monitor understanding during instruction. Online Guiding Questions Guiding questions are also in the online Visual Learning Animation Plus.

    • Preventing Misconceptions -  This feature in the Teacher’s Edition is embedded in the guiding questions.

    • Error Intervention: If… then… - This feature in the Teacher’s Edition is provided during Guided Practice. It spotlights common errors and gives suggestions for addressing them. 

    • Reteaching - Reteaching sets are at the end of the topic in the Student’s Edition. They provide additional examples, reminders, and practice. Use these sets as needed before students do the Independent Practice. 

    • Higher Order Thinking - These problems require students to think more deeply about the rich, conceptual knowledge developed in the lesson.

    • Practice Buddy Online - Online interactive practice is provided for most lessons.

• Tier 2 offers Strategic Intervention: “At the end of the lesson, assess to identify students’ strengths and needs and then provide appropriate support.” The Quick Check (either in print or online) is used to prescribe differentiated instruction for Tier 2 interventions based on the following scale: I = Intervention 0-3 points, O = On-Level 4 points and A = Advanced 5 points. 

  • Types of support include:

    • Intervention Activity (I) - Teachers work with struggling students. 

    • Technology Center Activities (I, O, A) - Digital Math Tools Activities reinforce the lesson content or previously taught content using a suite of digital math tools. Online Games practice the lesson content or previously taught content.

    • Reteach to Build Understanding (I) - This is a page of guided reteaching.

    • Build Mathematical Literacy (I, O) - Help students read math problems.

    • Enrichment (O, A) - Enhances students’ thinking.

    • Activity Centers (I, O, A) - Pick a Project lets students choose from a variety of engaging, rich projects. enVision STEM Activity is related to the topic science theme introduced at the start of the topic. Problem-Solving Leveled Reading Mat is used with a lesson-specific activity.

    • Additional Practice (I, O, A) - Use the leveled assignment to provide differentiated practice.

• Tier 3 offers Intensive Intervention: “As needed, provide more instruction that is on or below grade level for students who are struggling.”

  • Math Diagnosis and Intervention System (MDIS)

    • Diagnosis Use the diagnostic test in the system. Also, use the item analysis charts given with program assessments at the start of a grade or topic, or a the end of a topic, group of topics, or the year.

    • Intervention Lessons These two-page lessons include guided instruction followed by practice. The system includes lessons below, on, and above grade level, separated into five booklets.

    • Teacher Supports Teacher Notes provide the support needed to conduct a short lesson. The Lesson focuses on vocabulary, concept development, and practice. The Teacher’s Guide contains individual and class record forms, correlations to Student’s Edition lessons, and correlation of the Common Core State Standards to MDIS.

Examples of the materials providing strategies and support for students in special populations include: 

• Topic 1, Lesson 1-1, RtI 1, “Reteaching Assign Reteaching Set A on p.31.” Set A on p. 31 states the following, “Remember that you can use addition or multiplication to join equal groups. Complete each equation. Use counters or draw a picture to help. 1. 2+2+2 = 3\times ___ … 3. 8+__+__=3__\times 8” Set A also provides an MDIS lesson, B43, for additional support and the standard that correspond to the set, 3.OA.1.

• Topic 6, Lesson 6-6, RtI 2, “Use the QUICK CHECK on the previous page to prescribe differentiated instruction. Activity Centers (I, O, A), Problem-Solving Leveled Reading Mats Have students read the Problem Solving Leveled Reading Mat for Topic 6 and then complete Problem-Solving Reading Activity 6-6. The reading is leveled on the two sides of the mat.  See the Problem-Solving Leveled Reading Activity Guide for other suggestions on how to use this mat.”

The materials reviewed for enVision Mathematics Grade 3 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.

The Teacher’s Edition Program Overview, Supporting English Language Learners section, list the following strategies and supports: 

• “Lesson Language Objective for each lesson indicates a way that students can demonstrate their understanding of the math content through language modalities.

• Two ELL suggestions for every lesson are provided in the Teacher’s Edition. One suggestion is used with Solve & Share and the other is used with the Visual Learning Bridge.

• Levels of English language proficiency are indicated, and they align with the following levels identified in WIDA (World-Class Instructional Design and Assessment): Entering, Emerging, Developing, Expanding, Bridging.

• ELL consultants, Janice Corona from Dallas, Texas, and Jim Cummins from Toronto, Canada, ensured quality ELL instruction.

• Language Support Handbook provides topic and lesson instructional support that promotes language development. Includes teaching support for Academic Vocabulary, Lesson Self-Assessment Recording Sheets, and more.

• Visual Learning Animation Plus provides motion and sound to help lower language barriers to learning.

• Visual Learning Bridge often has visual models to help give meaning to math language. Instruction is stepped out to visually organize important ideas.

• Animated Glossary is always available to students and teachers while using digital resources. The glossary is in English and Spanish.

• Pictures with a purpose appear in lesson practice to help communicate information related to math concepts or to real-world problems. You many want to display the Interactive Student Edition pages so you can point to specific pictures or words on the pages when discussing the practice”

Examples where the materials provide strategies and supports for students who read, write, and/or speak in a language other than English include:

• Topic 3, Lesson 3-2, English Language Learners (Use with the Solve & Share), “Developing Write ‘How many rows are there?’ and ‘How many pictures are in each row?’ Ask students to read the questions to each other and complete this sentence ‘There are ___.’ ” This strategy/support falls under the Speaking category.

• Topic 7, Lesson 7-3, English Language Learners (Use with the Solve & Share), “Bridging Ask one student to choose new values for each number of pages read. Instruct the other students to take notes as they listen and to draw a new bar graph to show the revised data.” This strategy/support falls under the Writing category.

• Topic 11, Lesson 11-4, English Language Learners (Use with the Visual Learning Bridge), “Expanding Read Box A. What reasoning did Danielle use to determine whether Gina could buy the computer program? Pair students. Have partners read Box C and reread Box A. Have students explain to each other how the reasoning differs in the two boxes.” This strategy/support falls under the Listening category.

A general support that the materials provide for students who read, write, and/or speak in a language other than English and Spanish include PDFs that may be downloaded and translated to meet individual student needs.

The materials for enVision Mathematics Grade 3 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. 

In general, the manipulatives are visual images printed in the materials or virtual manipulatives found in the online materials. On occasion, students are prompted to use tools such as counters, cubes, place value blocks, ten frames, a ruler, a protractor, and grid paper. If and when the materials prompt students to use particular manipulatives, they are used appropriately. Examples of the overall use of manipulatives throughout the grade include:

• Teacher’s Edition Program Overview, Program Components indicates that “Manipulative Kits” accompany Teacher Resource Masters (online and in print). 

• Teacher’s Edition Program Overview, Using a Lesson, Assess and Differentiate, Quick-and-Easy Centers Kit for Differentiated Instruction includes “Holds mats, pages, and manipulatives for the Technology Center (Digital Math Tools Activities) and for the Activity Centers.”

• Teacher’s Edition Program Overview, Routines, Quick and Easy Implementation, “Accessible Available in both English and Spanish, the routines require little preparation and few or no physical materials. When needed, common manipulatives are used to reinforce hands-on experiences.”

• Teacher’s Edition Program Overview, Math Practices, MP.5, states, “Students become fluent in the use of a wide assortment of tools ranging from physical objects, including manipulatives, rulers, protractors, and even pencil and paper, to digital tools, such as Online Math Tools and computers.”

Examples of how manipulatives, both virtual and physical, are representations of the mathematical objects they represent and, when appropriate to written methods, include:

• Topic 4, Lesson 4-1, Solve & Share, students use two-color counters to make arrays and show how they know products and quotients. “Use 24 counters to make arrays with equal rows. Write multiplication and division equations to describe your arrays.” Teacher guidance: “BEFORE 1. Introduce the Solve & Share Problem Provide 24 two-color counters (or Teaching Tool 9) to each student. ... DURING 3. Observe Students at Work To support productive struggle, observe and, if needed, ask guiding questions that elicit thinking. How do students use facts to represent rows of equal groups of counters? Students might write the multiplication facts with a product of 24. If needed, ask How will you find the number of counters in each row?” 

• Topic 6, Lesson 6-1, Problem Solving, Problem 9, students use two-color square tiles (or Teaching Tool 8) to represent a specified region and decide if they agree with the area calculation of another person.. “Critique Reasoning Janet covers the red square with square tiles. She says, ‘I covered this shape with 12 unit squares, so I know it has an area of 12 square units.’ Do you agree with Janet? Explain.” The materials show the outline of a red square on a blue tile background.

• Topic 13, Lesson 13-1, Independent Practice, Problem 7, students use fraction strips to find equivalent fractions. “Find each equivalent fraction. Use fraction strips or draw area models to help. 7. \frac{2}{6}=\frac{ }{3}”