Kindergarten - Gateway 3

The materials reviewed for enVision Mathematics Kindergarten 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 1-1, Visual Learning Bridge, Essential Question, “Ask How can you count objects?” Teachers begin the Classroom Conversation by doing the following: “Model telling a story about the picture and then have volunteers tell their own stories to the class. [I see worms wiggling up from the ground. How many worms are there?]” 

• Topic 4, Lesson 4-2, Independent Practice, Problem 5, Directions, “Higher Order Thinking Have students count the flowers in the group, draw a group of flowers that is less than the group shown, and then write the numbers to tell how many.” Teacher guidance: “Make Sense and Persevere Have students plan both their work and how they will explain it. Ask: How will you know that you have drawn a group that has fewer objects? Which group will have extra objects?[The top group will have extra objects]”

• Topic 10, Lesson 10-1, Guided Practice, Problems 4 and 5, Directions, “draw blocks to match the equation. Then have them tell how the picture and equation show 10 ones and some more ones.” Teacher guidance: “Remind students to fill the ten-frame first. Explain that even though the entire ten-frame will be filled, it is important to fill it in the proper order: left to right and top to bottom. Model to students as needed.”

The materials reviewed for enVision Mathematics Kindergarten 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 1: Professional Development (topic) Video, “Number is the most fundamental concept in all of mathematics. … The first critical skill that four or five year olds develop is rational counting. This means  they can take a set of objects, like counting cubes, move them one by one from an uncounted pile to a counted pile and as they move each one say the next number name. … In order to do this accurately, a child must be able to 

1. Be able to recite the number names in order up to the target number.

2. Establish a one-to-one correspondence between the action of moving one counter and saying one number name.

3. Understand that order doesn’t matter. The cubes can be counted in any order and the number remains the same.

4. Understand that the last number named is how many are in the set.

At yet a slightly more advanced level, a fifth understanding that students develop about this process is that

5. Even if the objects are rearranged spatially, their number “remains the same”.

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 4, Math Background: Coherence, Look Ahead, the materials state, “Grade 1 Comparison Situations In Topic 1, students will understand that subtraction can be used to determine how many more or how many fewer. Compare Two-Digit Numbers In Topic 9, students will compare two-digit numbers. They will also learn the symbols for ‘greater than’ and ‘less than’ and use these to write inequallities.” An example of an inequality is shown.

The materials reviewed for enVision Mathematics Kindergarten 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 K 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 highlights three of the learnings within the topics: “Counting Sequence, See Quantities as They Relate to 5 and 10, and Counting All and Counting Out” 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 Grade 1 connections, “Relate Counting to Addition In Topics 2, 3, and 4 students will use counting on and counting back as strategies for adding and subtracting within 20. Extend the Counting Sequence to 120 In Topic 7, students will count on from any number to 120, and read and write numbers to 120.” 

• Topic 7, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Representations, Solve Word Problems, and Fluency Development” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 7 connect to what students will learn later?” and provides a Grade 1 connection, “Subtract Within 20  In Topics 1, 2, 4, and 5, students will work on subtraction within 20. The emphasis will be on strategies and building fluency, Students will also subtract to compare. Grade 1 students will be expected to become fluent with subtracting within 10.”

• Topic 11, Math Background: Coherence, the materials highlight two of the learnings within the topics: “Extend the Count Sequence to Greater Numbers, and See Quantities as They Relate to 10” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 11 connect to what students will learn later?” and provides Grade 1 connections, “Extend the Counting Sequence to 120 In Topic 7, students will begin at any number less than 120, count to 120, and read and write numbers to 120. Count by Tens and Ones In Topic 8, students will count with groups of tens and leftover ones. Count On to Add and Subtract In Topics 2, 3, and 4, students will count on and count back to add and subtract with 20. In Topics 10 and 11, students will count on and count back to add and subtract 2-digit numbers.”

The materials reviewed for enVision Mathematics Kindergarten 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 Kindergarten 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), Crayons”

• Topic 6, Lesson 6-1, Lesson Resources, Materials, “Two-color counters (or Teaching Tool 6)”

• Teacher Resources, Grade K: Materials List, the table indicates that Topic 12 will require the following materials: “2-D and 3-D Shapes (or Teaching Tool 37), 3-D Shapes and Real-Life Objects (or Teaching Tool 39), Circles (or Teaching Tool 32), ...”

The materials reviewed for enVision Mathematics Kindergarten 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 1, Lesson 1-1, Independent Practice, Evaluate, Quick Check, Problems 7-9, “Check mark indicates items for prescribing differentiation on the next page. Items 7 and 8: each 1 point.  Item 9: up to 3 points.” “Directions 7-8 Have students color a box as they count each box to show how many.  9 Higher Order Thinking Have students draw 1, 2, or 3 nests, and then color a box as they draw each nest to show how many.” 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, O Items 1-2, 4-6, and A Items 3-7.

• Topic 7, Topic Assessment, Problem 4, “Directions Have students listen to the story and then complete the sentence to tell how many are left. Kyle sees 10 turtles at the zoo. 2 turtles crawl away. Write a number sentence that tells how many are left.” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS B8, Standards K.OA.A.1 and K.OA.A.2. Scoring Guide indicates: 1 point “Correct numbers are written.”

• Topic 10, Topic Performance Task, Problem 5, “Directions Say: Mason put his striped marbles in a five-frame. Then he buys ten more striped marbles. Have students write the number to tell how many striped marbles Mason had at first, and then color the part of the number chart to show how many striped marbles he has now. Then have them write an equation and ask them to explain how the picture and equation show 10 ones and some more ones.” Item Analysis for Diagnosis and Intervention indicates: DOK 3, MDIS A10, Standards K.NBT.A.1 and MP.7. Scoring Guide indicates: 3 points “Number 4 is written, 14 is marked in chart, and correct equation written for 14”; 2 points “Number 4 is written and 14 is marked in chart, or written equation matches 14 marked in chart”; 1 point “Number 4 is written, or 14 is marked in chart, or correct equation written for 14.” 

• Topics 1-14, Cumulative/Benchmark Assessment, Problem 9, “Directions Have students draw a circle below the book and a triangle beside the book.” Item Analysis for Diagnosis and Intervention indicates: DOK 3, MDIS D28, Standard K.G.A.1. Scoring Guide indicates: 2 points, “Draws a circle and a triangle in the correct positions relative to the book”; 1 point “Draws either a circle or a triangle in the correct position relative to the book.”

The materials reviewed for enVision Mathematics Kindergarten 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 1-14. 

• 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 Kindergarten 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, “Prevent Misconceptions Students may think the balls cannot be compared as they are not lined up. Show how the balls are matched 1 to 1 like in Boxes A and B, and there is still 1 soccer ball that is left over.”

• Topic 6, Lesson 6-6, RtI 2, “Use the QUICK CHECK on the previous page to prescribe differentiated instruction. Intervention Activity (I), Picture a Story, Materials Crayons, 

  • Tell this Put Together addition story: 2 students are hopping. 3 students are skipping. How many students are there in all? 

  • As you tell the story again, draw a group of 2 stick figures and say the number 2. Then draw a group of 3 stick figures and say the number 3. Have one student draw a circle around all the stick figures. Ask another student to tell how many figures there are in all. 

Tell a different number story and ask partners to draw a picture to solve it. Help students retell the story using in all.”

The materials reviewed for enVision Mathematics Kindergarten 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 Have students compare work with a partner and count the counters. Ask students to point to each counter as they count. Then have them tell the total number.” This strategy/support falls under the Speaking category. 

• Topic 7, Lesson 7-3, English Language Learners (Use with the Solve & Share), “Speaking Read the directions to students. Ask: How many lady bugs does Marta see at the start? What does it mean that some ladybugs then crawl away? Discuss a range of responses. For example, students may answer with actions, single words, sentences, or with reference to the picture on the page. Establish that some crawling away means some ladybugs have moved away from the group.” The teacher then has the choice between Entering, Emerging or Expanding, strategies and supports.

• Topic 11, Lesson 11-4, English Language Learners (Use with the Visual Learning Bridge), “Bridging Ask students to tell the number in the yellow box. Say: This is the number you will start count at. Ask students to tell the number in the red box. Say: This is the number you will stop counting at. Have students count each number. Ask students to explain the steps they took.” 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 Kindergarten 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 2-5, Problem Solving, Performance Task, Problem 5, students use manipulatives to represent stickers in the problem and determine a number greater than two. “Directions Read the problem aloud. Then have students use multiple problem-solving methods to solve the problem. 5. Model Use cubes, draw a picture, or use numbers to show how many stickers Marta has and Emily could have.” Teacher guidance: “5. Model with Math Students have been using connecting cubes as tools for modeling in this lesson. Students can be encouraged to use alternative tools for models, such as counters or other small objects.”

• Topic 6, Lesson 6-6, Solve & Share, students use cubes to represent the name rags in the problem and show how they know that the total is five. “Directions Say: Daniel’s teacher is making name tags for her students. She makes 3 name tags for boys. She makes 2 more for girls. Now she has 5 name tags. How does Daniel’s teacher know that she has made 5 name tags? Explain and then show how you know.” Teacher guidance: “BEFORE 1. Introduce the Solve & Share Problem Give each student 3 cubes of one color and 2 cubes of another color. .. DURING 3. Observe Students at Work To support productive struggle, observe and, if needed, ask guiding questions that elicit thinking. How do students represent the name tags to show how they know? Students might use cubes or draw a picture. Or they might use numbers. If needed, ask What tools do you have to help? What can you write?”

• Topic 8, Interactive Math Story, During the Story, students use manipulatives to represent flowers in the problem and show that they know the total is three. “Sue gives 2 of her flowers to Ruby. Sue and Ruby are good pals! 1 + 2 = ___ Now Ruby has ___ flowers.” Teacher guidance: “SPEAK … Have students use plastic spoons, pencils, or another easily countable manipulative as flowers.”