4th Grade - Gateway 3

The materials reviewed for enVision Mathematics Grade 4 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, Teachers begin the Classroom Conversation by saying the following “Where else have you seen numbers in the hundred thousands? [Sample answer: Cost of homes, attendance at amusement parks, money]” 

• Topic 7, Lesson 7-5, Problem Solving, Problem 24, “Isabel wrote this mystery problem. The quotient is a multiple of 6. The dividend is a multiple of 9. The divisor is a factor of 12. Write one possible equation to Isabel’s mystery problem.” Teacher guidance: “If students have difficulty knowing where to start, encourage them to make sequential lists of multiples of 6 and 9. Students should then list the factors of 12 and use their lists to find possible solutions.”

• Topic 12, Lesson 12-3, Guided Practice, Problem 1, “Cy says, ‘0.20 is greater than 0.2. Because 20 is greater than 2.’ Do you agree? Explain.” Teacher guidance: “Error Intervention If students are having difficulty comparing 0.20 and 0.2, then ask them to model 0.20 on one hundredths grid and 0.2 on another hundredths grid. Which model has more hundredths shaded? [Neither] Which decimal is greater? [Neither; the decimals are equivalent.]”

The materials reviewed for enVision Mathematics Grade 4 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 15: Professional Development (topic) Video: “Students learn about angle measure and how to use a protractor to measure angles. Step one in this process is for students not only to understand what an angle is, but what is meant by measuring an angle. Many students believe that the measure of an angle depends upon the length of the rays … Students develop basic understanding of angle measurement first by using fractions ot a circle … to find angle measure. They connect measuring angles to work with equivalent fractions. … Be sure students pay attention to where to place the vertex of the angle and how to line up one ray along the horizontal edge. … By the end of this topic, students have developed a conceptual understanding of angle measure, learned the skill of using a protractor to measure angles, and are able to apply this understanding and skill to solve problems involving angle measure.”

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 11, Math Background: Coherence, Look Ahead, the materials state, “Grade 5 Extend Work with Line Plots In Topic 10, students will extend their work with line plots. Data represented on line plots will be used to solve problems involving ordering fractions, adding and subtracting fractions and mixed numbers with unlike denominators, and multiplying with fractions and mixed numbers.” An example of a line plot is shown.

The materials reviewed for enVision Mathematics Grade 4 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 4 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 4, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Estimation, Models and the Distributive Property, and Problem Solving” with a description provided for each learning, including which lesson(s) cover the learnings. The “Look Ahead” section asks the question, “How does Topic 4 connect to what students will learn later?” and provides Grade 5 connections, “Develop Fluency with the Standard Multiplication Algorithm In Topic 3, students will develop fluency in using the standard algorithm for multiplying multi-digit whole numbers. Multiply Decimals In Topic 4, students will use models and strategies to multiply decimals to hundredths.”

• Topic 8, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Equivalent Fractions, Visual Models, and Word Problems Involving Fractions” 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 5 connection, “Fraction Computation In Topic 7, students will add and subtract fractions and mixed numbers with unlike denominators. In Topic 8, they will multiply fractions. In Topic 9, they will interpret a fraction as division, and they will divide unit fractions and whole numbers.”

• Topic 16, Math Background: Coherence, the materials highlight three of the learnings within the topics: “Use Line Reltionships in Classifying Quadrilaterals, Classify Triangles and Quadrilaterals, and Recognize and Draw Line-Symmetric 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 16 connect to what students will learn later?” and provides Grade 5 connections, “Understand the Coordinate Plane In Topic 14, students will learn that a coordinate plane is defined by perpendicular number lines called the x-axis and the y-axis. These lines intersect at the origin which has coordinates (0, 0). Classify Two-Dimensional Figures In Topic 16, students will classify triangles and quadrilaterls based on their properties. They will learn that an attribute belonging to a category of plane figures also belongs to all subcategories. They further learn to classify and organize two-dimensional figures into Venn diagrams based on the attributes of the figures.”

The materials reviewed for enVision Mathematics Grade 4 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 4 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, “Place-Value Charts (or TT 3)”

• Topic 7, Lesson 7-1, Lesson Resources, Materials, “Centimeter grid paper (or Teaching Tool 9), 2-color square counters (or Teaching Tool 16)”

• Teacher Resources, Grade 4: Materials List, the table indicates that Topic 12 will require the following materials: “1-inch grid paper, 2-color counters (or Teaching Tool 15), Decimal models (or Teaching Tool 7), ...”

The materials reviewed for enVision Mathematics Grade 4 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 4, Lesson 4-2, Independent Practice, Evaluate, Quick Check, Problem 11, “Check mark indicates items for prescribing differentiation on the next page. Item 5: each 1 point. Items 12 and 13: up to 3 points.” For example, Directions “During a basketball game, 75 cups of fruit punch were sold. Each cup holds 20 fluid ounces. How many total fluid ounces of fruit punch were sold?” 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 1-4, 7-9, 12, O Items 1-2, 4-5, 7-8, 11-12, and A Items 1-2, 5-6, 9-12.

• Topic 7, Topic Assessment, Problem 4, “Determine if each number is prime or composite. Then write all the factors for each number. 19, 33” Item Analysis for Diagnosis and Intervention indicates: DOK 2; MDIS G57; Standard 4.OA.B.4.” Scoring Guide indicates: 4 2 point “Correct answer and factors” and 1 point “Correct answer or factors.”

• Topic 10, Topic Performance Task, Problem 1A, “School Mural Paul has permission to paint a 20-panel mural for his school. Part of the mural is shown in the Painting a Mural table. Paul decides he needs help. The Helpers table shows how much several of his friends can paint each day and how many days a week they can paint.” The materials show a mural with the caption, “Paul paints \frac{9}{10} panel a day” and a table that lists three helpers and how many panels they can paint in a day and in a week. “1. The students want to find how long it will take to paint the mural if each works on a different part of the panels a different number of days a week. Part A How many panels can Leeza paint in a week? Use fraction strips to explain.” Item Analysis for Diagnosis and Intervention indicates: DOK 3, MDIS H47, Standard 4.NF.B.3a, MP.4. Scoring Guide indicates: 2 points “Correct answer and explanation” and 1 point “Correct answer or explanation.”

• Topics 1-16, Cumulative/Benchmark Assessment, Problem 12, “Marco has 2 pieces of rope that are each 8 yards long. How many feet of rope does Marco have? Explain.” Item Analysis for Diagnosis and Intervention indicates: DOK 2, MDIS I32, Standard 4.MD.A.1 and 4.MD.A.2”  Scoring Guide indicates: 2 points “Correct answer and explanation” and 1 point “Correct answer or explanation.”

The materials reviewed for enVision Mathematics Grade 4 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 4 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 1 If students try to solve using front-end estimation, then ask if rounding to the nearest thousand, which place do you look at? If rounding to the nearest hundred, which place do you look at?”

• Topic 6, Lesson 6-1, 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 6-1. 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 4 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 Pair students and have them answer these two questions on paper: ‘What is a week?’ ‘How many weeks are in the problem?’ ” This strategy/support falls under the Writing category.

• Topic 7, Lesson 7-3, English Language Learners (Use with the Solve & Share), “Speaking Write ‘array’ on the board. Use self-stick notes to create an array with 2 rows and 4 columns on the board.” The teacher then have the choice between Emerging, Developing 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.”

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 4 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 4-4, Independent Practice, Problem 6, students use an array to represent multiplication of two two-digit numbers and show the product. “Use the array drawn on a grid to find each product. 6. 18 \times 18.” The materials show an array on grid paper consisting of a 10 \times 10 pink region, a 10 \times 8 purple region, an 8 \times 10 blue region, and an 8 \times 8 green region.

• Topic 9, Lesson 9-3, Solve & Share, students use fraction strips to represent ingredients for nachos and tacos and find how much shredded cheese is used. “Jonas is making nachos and tacos for a family party. He uses \frac{2}{5} bag of shredded cheese for the nachos and \frac{1}{5} bag for the tacos. How much of the bag of shredded cheese does Jonas use? Solve this problem any way you choose.” Teacher guidance: “BEFORE 1. Pose the Solve & Share Problem You may wish to provide fraction strips (or Teaching Tool 13).  ... DURING 3. Observe Students at Work To support productive struggle, observe and, if needed, ask guiding questions that elicit thinking. What tools do students use to solve the problem? Students might use fraction strips or number lines to model the addition. If needed, ask: How can you use fraction strips or a number line to represent \frac{2}{5} and \frac{1}{5}? How do students find the sum of \frac{2}{5} and \frac{1}{5}? Students might draw 3 one-fifth fraction strips. If needed ask, How can you use your representation to add the fractions?”

• Topic 15, Lesson 15-3, Independent Practice, Problem 8, students use pattern blocks to represent an angle and show its measure. “Find the measure of each angle. Use pattern blocks to help.” The materials show a 120° angle.