2nd Grade - Gateway 3

The materials reviewed for enVision Mathematics Grade 2 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 1-6, Independent Practice, Problem 5, “Subtract. Complete the addition fact that can help you.” Students fill in the blanks.  8-1 = ___ ; 1 + ___ = 8. Teacher guidance: “Encourage students to think about how subtraction is related to addition as they work through the problems. If needed, have them practice writing related addition and subtraction equations.”

• Topic 3, Lesson 3-1, Visual Learning Bridge, Essential Question, “Ask How can you use patterns on a hundred chart to help you add numbers mentally?” Teachers begin the Classroom Conversation by saying the following, “Use Appropriate Tools Strategically What tool do you see here? [Part of a hundred chart] Why do we start at 54? [It is the greater of the two addends.] How many tens are in 18? [One] How can you add 1 ten on the hundreds chart? [Move down one row] How can you add 8 ones? [Move right 6 spaces to 70 and then move 2 more to 72.]”

• Topic 6, Lesson 6-3, Problem Solving, Problem 13, “Higher Order Thinking Write a subtraction story using two 2-digit numbers. Then solve the problem in your story.” Teacher guidance: “Higher Order Thinking After students have written their stories, ask them to explain if their stories compare two numbers, show a take-away situation, or describe a part-part-whole problem. Showcase each of those different types of subtraction problems as students share their stories.” 

The materials reviewed for enVision Mathematics Grade 2 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 2: Professional Development (topic) Video, “The Understanding of equal groups prepares students for the concept of multiplication and division by understanding the concept of even and odd and arrays … Students should build on their experiences with doubles by decomposing or breaking apart a number into two equal addends into equal groups. … When students have an understanding of equal groups it supports their development of multiplication and division.”

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 3 One-and Two-Step Word Problems In Topic 11, students will use letters to represent unknown quantities when solving one- and two-step word problems. Solve Word Problems Involving Time, Mass, and Liquid Volume In Topic 14, students will use number lines and bar diagrams to represent word problems involving other measurable attributes: time, liquid volume, and mass.”  An example word problem with the solution is shown. “Perimeter In Topic 16, students will solve problems involving perimeter.” An example word problem is shown, with a diagram of a floor of a room, teachers are tasked with writing an equation to find the missing length of the room. 

The materials reviewed for enVision Mathematics Grade 2 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 2 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 3, Math Background: Coherence, the materials highlight two of the learnings within the topics: “Decompose Tens and Ones, and Mental Math” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 3 connect to what students will learn later?” and provides a Grade 3 connection, “Add and Subtract Within 1,000 In Topic 8, students will use place-value understanding, properties of operations, and the relationship between addition and subtraction to add and subtract within 1,000. In Topic 9, students will fluently add and subtract within 1,000.”

• Topic 8, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Solve Problems with Coins and Bills, Tell Time, and Count Money and Tell Time” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 8 connect to what students will learn later?” and provides a Grade 3 connection, “Time to the Minute and Elasped Time In Topic 14, students will learn to tell time to the nearest minute. They will also learn to measure elapsed time and to solve word problems involving time. To find time to the nearest minute, students count by 5s and then 1s. The clock shows 10:27.”

• Topic 12, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Estimate and Measure Length, Use Different Length Units, and Addition, Subtraction, and Length” 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 a Grade 3 connection, “Measure to the Nearest Fourth Inch and Half Inch In Topic 12, students will extend their understanding of measuring lengths in customary units to measuring to the nearest fourth inch or half inch.”

The materials reviewed for enVision Mathematics Grade 2 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

The materials reviewed for enVision Mathematics Grade 2 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, “Counters (or Teaching Tool 6), Connecting cubes (or Teaching Tool 5)”

• Topic 6, Lesson 6-1, Lesson Resources, Materials, “Place-value blocks (or Teaching Tool 19), Place-Value Mat A (or Teaching Tool 26)”

• Teacher Resources, Grade 2: Materials List, the table indicates that Topic 12 will require the following materials: “1-inch squares (or Teaching Tool 39), Buttons or counters (or Teaching Tool 6), Centimeter rulers and meter sticks (or Teaching Tool 40), ...”

The materials reviewed for enVision Mathematics Grade 2 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 2, Lesson 2-4, Independent Practice, Evaluate, Quick Check, Problem 4, “A check mark indicates items for prescribing differentiation on the next page. Items 4 and 9: up to 2 points each; Item 10: 1 point.” For example, Directions: “Draw an array to show each problem. Use repeated addition to solve. 4. Kristin makes an array with her books. She places them in 5 rows with 4 books in each row. How many books does Kristin have in all?” Students complete the equation ___ + ___ + ___ + ___ + ___  = ___ books. 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 1-3, 5, O Items 1-5, and A Items 1-5.

• Topic 6, Topic Assessment Masters, Problem 4, “Dan has 58 shells. Kelsey has 6 fewer shells than Dan. Kelsey gives 8 shells to her friend. Part A Which pair of equations should be used to find how many shells Kelsey has now? (A) 58 - 6 = 52, 52 - 8 = 44 (B) 58 - 6 = 52, 52 + 8 = 60 (C) 58 + 6 = 64, 64 - 8 = 56 (D) 58 + 6 = 64, 64 + 8 = 72. Part B How many shells does Kelsey have now? ___ shells” Item Analysis for Diagnosis and Intervention indicates: DOK 2; MDIS C26, E2, and E3; Standard 2.OA.A.1.” Scoring Guide indicates: “4A 1 point, Correct choice selected; 4B 1 point, Correct solution.”

• Topic 9, Topic Performance Task, Problem 5, “Reading Record These students love to read! These books show the number of pages each student has read so far this year.” The materials show Ken’s book 512 pages, Luisa’s book 493 pages, Ruth’s book 427 pages, and Tim’s book 378 pages. “Diane reads the same number of minutes each week. How many minutes does she read after 4 weeks? After 5 weeks? Skip count on the number line above to find the answers.” Students fill in the blanks: “After 4 weeks: ___ minutes; After 5 weeks: ___ minutes.” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS A40, Standard 2.NBT.A.2, 2,NBT.B.8, MP7. Scoring Guide indicates: 2 points “Two correct answers” and 1 point “One correct answer.”

• Topics 1-12, Cumulative/Benchmark Assessment, Problem 12, “What is the value of the 2 in the number 283? What is the expanded form of 283?” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS A33 and A34, Standard 2.NBT.A.1, 2.NBT.A.3.1. Scoring Guide indicates: “2 points “Correct value of digit in number AND correct expanded form” and 1 point “Correct value of digit in number OR correct expanded form.”

The materials reviewed for enVision Mathematics Grade 2 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 1-15. 

• 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 2 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 2, Lesson 2-2, RtI 1, “Guided Practice Error Intervention Item 2 If students have difficulty writing the equation, then remind them to add the number of squares in each row to find the sum. If needed, guide them as they count the squares, then count aloud with them to get the sum.”

• Topic 6, Lesson 6-3, RtI 2, “Use the QUICK CHECK on the previous page to prescribe differentiated instruction. Technology Center (I, O, A), Math Games A link to the Robo-Launch—Add and Subtract 2-Digit Numbers math game to use with this lesson is provided at SavvasRealize.com.”

The materials reviewed for enVision Mathematics Grade 2 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), “Entering Read the problem with students. What are you asked to add? Have students listen as you define the following terms: addends, open number line, equation, sum. Ask them to repeat definitions to a partner.” This strategy/support falls under the Listening category. 

• Topic 7, Lesson 7-3, English Language Learners (Use with the Solve & Share), “Reading Read the problem aloud. Then have students answer the questions to confirm understanding.” The teacher then have the choice between Entering, Emerging or Expanding, strategies and supports.

• Topic 11, Lesson 11-4, English Language Learners (Use with the Visual Learning Bridge), “Developing Have students discuss with a partner the steps taken to solve the problem. Then partners should work together to write and explanation of their solution. Encourage student to use the words first, next, and last in their explanations.” This strategy/support falls under the Writing 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 2 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 2, Lesson 2-2, Solve & Share, students use connecting cubes (or Teaching Tool 5) to represent students and show how they know that the possible number of students in a class is an odd number. “The students in Ms. Jenn’s class work in pairs. One student does not have a partner. How many students could be in Ms. Jenn’s class? Use cubes to show your thinking. Draw a picture of your work.” Teacher guidance: “BEFORE 1. Introduce the Solve & Share Problem Provide each student with 30 connecting cubes… DURING 3. Observe Students at Work To support productive struggle, observe and, if needed, ask guiding questions that elicit thinking. How do students use the connecting cubes to represent the problem? Students might make groups of two cubes, then add one more. If needed, ask How can you show a pair of students with the connecting cubes?” How do students determine the number of students in the class? Students might count by 2s and then 1 more. Or they might add. If needed, ask What do your cubes show? How can you find the total number of students?”

• Topic 8, Lesson 8-3, Independent Practice, Problem 6, students use bills (or Teaching Tool 31) to count money and show that they know the total value. “Ms. Wills has these dollar bills. Count on to find the total value.” The materials show three $100 bills. 

• Topic 12, Lesson 12-2, Independent Practice, Problem 3, students use inch rulers (or Teaching Tool 38) to measure the length of a backpack and show that they know the length in inches.” Estimate the height or length of each real object. Then use a ruler to measure. Compare your estimate and measurement.” The materials show a backpack and prompt students to measure its width; then they fill in the blanks “about ___ inches.” one for an estimate and one for an actual measurement. Teacher guidance: “Item 3 What do you do if your measurement is greater than 12 inches? How do you find the total length?”