3rd-5th Grade - Gateway 1

The materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

According to the Implementation Handbook, the assessment system provides a variety of opportunities for students to independently demonstrate their mastery of grade-level standards through formal summative assessments which include, Chapter Test, Big Idea Task, Multi-Chapter Test, Quarter DAP (Diagnostic Adaptive Progression - online only), End-of-Course Test, Post-Course DAP, and Alternative Chapter Assessment. These assessments vary in format, providing different ways to accurately describe student performance at a particular point-in-time. All assessments are aligned with grade-level content standards as represented in the “Assessment Correlation (by Standard)” and an “Assessment Correlation (by Course).” Both include a breakdown identifying content standards for each assessment item.

The materials in Grades 3–4 are divided into 14 chapters, each containing Mid-Chapter Tests and Chapter Tests. Grade 5 is divided into 15 chapters, each containing Mid-Chapter Tests and Chapter Tests. The materials also include four Multi-Chapter Tests, a Prerequisite Skills Test, and an End-of-Course Test. Examples include:

• Grade 3, Chapter 7, Chapter Test, Exercise 4 states,  “Select all numbers that round to 250, when rounded to the nearest ten.” Answer choices include: a. 244, b. 245, c. 243, d. 247, e. 256. (3.NBT.1)

• Grade 4, Chapter 2, Chapter Test, Exercise 6 states, “Find the sum. 1,932+547=\square. Use estimation to check the reasonableness of your answer. Round to the hundreds place. \square+500=\square.” (4.NBT.4)

• Grade 5, Multi Chapter Test 3, Exercise 5 states,“You have 2 cups of yogurt. A serving of yogurt is \frac{1}{3} cup. How many servings did you have?” (5.NF.7c)

The materials reviewed for Math & YOU Grades 3 though Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

CCSS standards are identified on the digital assessments for each item on the following formal assessments: Chapter Performance Task, Big Idea Task, Chapter Test, Multi-Chapter Test, and End-of-Course Test. The print materials do not always identify the CCSS or Practices; however, the problems on the print assessments are identical to the problems in the digital assessments. Two correlation resources demonstrate assessment items alignment to the standards. The Assessment Correlation (by Course) is organized by assessment and identifies the standards addressed on each assessment. The Assessment Correlation (by Standard) lists every assessment item in which a specific standard is addressed.

The Digital Teaching Experience and the Teacher Toolkit: Course Essentials identify the Standards for Mathematical Practice (SMPs) for Big Idea Tasks and Chapter Performance Tasks; however, the materials do not identify the SMPs consistently for each item on other formal assessments.

Examples include: 

• Grade 3, Chapter 3, Chapter Test, Exercise 7 states, “Each day you stretch for 2 minutes and run for 8 minutes. At the end of the third day, how much more time was spent running than stretching?” (3.OA.3)

• Grade 4, Multi Chapter Test 1, Exercise 13 states, “You find 134,251 – 112,048 = 22,203. Which expression can you use to check your answer? A 22,203 + 112,048, B 22,203 + 134,251, C 134,251 + 112,048, D 134,203 + 22,203.” (4.NBT.3, 4.NBT.4)

• Grade 5, Chapter 7, Big Ideas Task, Exercise 2 states, “a. Write a real-life word problem that uses an estimate of the quotient of 154.82 and 4.78. Find and interpret the estimate. Is your estimate an overestimate or underestimate? Explain how you know. b. Use two different strategies to find 154.82\div4.78. Explain how the strategies are similar and different. Then describe a situation where one strategy may be more appropriate to use than the other.” (5.NBT.4, 5.NBT.7, MP1, MP4, MP5).

The materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Students are exposed to a variety of question types, modalities, and complexity levels to develop and demonstrate their understanding of course content. The modalities and question types provide different contexts and settings, ensuring students have opportunities to demonstrate their understanding through tasks that require them to reason and communicate in a variety of ways. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, short answer, and extended response.

Formative assessments occur at the lesson level through structures such as the Prerequisite Skills Test, Chapter Performance Task, In-Class Practice, and Connecting Big Ideas. These assessments provide opportunities for students to demonstrate their understanding of grade-level content standards through a variety of item types, including Drag and Drop, Fill in the Blank, Matching, Multi-Select, Response Matrix, Short Response, and Single Select.

Examples include: 

• Grade 3, Chapter 4 Test, Exercise 4 states, “You arrange 24 counters into 8 equal rows. How many counters are in each row? 8 rows of ___ counters 24\div8= ___.” The materials assess the full intent of 3.OA.3 as students use division within 100 to solve a real-world problem involving equal groups, represent the situation with an equation containing a symbol for the unknown, and interpret the quotient in the context of the problem.

• Grade 4, Chapter 12, Big Idea Tasks, Exercise 3 states, “a. Two adjacent angles form an acute angle. One angle measure is 42°. Your friend says that the unknown angle measure is less than 48°. Is your friend correct? If yes, use mathematics to explain your friend’s reasoning. If not, make a convincing argument to help your friend understand the error in the reasoning. b. Three adjacent angles form a right angle. All three angles have different measures. Write three possible angle measures. Use a tool to draw your angles. Explain your choice of tool. c. Two adjacent angles form an obtuse angle. One angle measure is 52°. What is the least measure of the unknown angle? What is the greatest measure of the unknown angle? Explain why. d. Three adjacent angles form a straight angle. Two of the angle measures are 36° and 49°. Classify the third angle. Explain how you know.” The materials assess the full intent of MP3 as students construct viable arguments and critique the reasoning of others by analyzing mathematical relationships among angle measures, justifying their conclusions with evidence, identifying and explaining errors in reasoning, and using precise mathematical language and tools to communicate and defend their thinking.

• Grade 5, Chapter 11 Test, Exercise 2 states, “Find the equivalent length. 7 cm = ___ mm.” Exercise 8, “A water bottle contains 24 fluid ounces of water. How many cups are in 3 water bottles?” The materials assess the full intent of 5.MD.1 as students convert among different-sized units within a single measurement system and apply those conversions to solve multi-step, real-world problems involving length and liquid volume.

The materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide students with consistent opportunities to engage in the full intent of all Grade 3-5 standards. Each lesson begins with an opening activity, Dig In or Motivate, followed by student-centered explorations in Investigate. Learning continues with Key Concept and concludes with opportunities to make real-world connections in Connect to Real Life. The Student Practice workbooks offer additional opportunities for students to reinforce the knowledge and skills developed through each lesson.

Three correlation resources demonstrate alignment to the standards and show that students engage with the full intent of the standards throughout the course. The Standards Correlation (by Course) is organized by course component and identifies the standards addressed in each lesson. The Standards Correlation (by Standard) lists every lesson in which a specific standard is addressed. The Standards-Based Practice Correlation connects each Standards-Based Practice activity in the Practice Workbook to a content standard and identifies lessons where that standard is reinforced. Across the program, students have multiple opportunities to independently demonstrate their understanding of the full intent of the standards.

Examples include:

• Grade 3, Chapter 1, Lessons 1, 3, and 4, engages students with the full intent of 3.OA.1 (Interpret products of whole numbers, e.g., interpret 5\times7 as the total number of objects in 5 groups of 7 objects each.) Students represent and interpret multiplication as equal groups and arrays by drawing models and writing corresponding addition and multiplication equations to show the total number of objects. Lesson 1, Practice, Exercise 1 states, “Draw equal groups. Then complete the equations. 2 groups of 8, ___ + ___ = ___, ___ \times ___ = ___.” Students write addition and multiplication equations to represent objects in groups. Lesson 3, Practice, Exercise 3 states, “Draw an array to multiply. 5\times6= ___” Students draw an array to model the multiplication equation. Lesson 4, Investigate states, “Write a multiplication equation for the array. ___\times___ = ___.” A visual array displays five rows of three dots. Students use an array to write the corresponding multiplication equation.

• Grade 4, Chapter 7, Lessons 1 and 2, engages students with the full intent of 4.NF.1 (Explain why a fraction a/b is equivalent to a fraction (n\times a)/(n\times b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.) Students apply visual models, number lines, and operations of multiplication and division to reason about and determine equivalent fractions. Lesson 1, Teacher Edition, Nick’s Notes Investigate states, “How can you visually determine which fractions are the same as \frac{1}{2}” Lesson 2, In Class Practice, Exercise 1 states, “Use a number line to find an equivalent fraction.” Students label a number line with fifths and tenths to find equivalent fractions. 

• Grade 5, Chapter 2, Lesson 3, engages students with the full intent of 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.) Students translate between verbal statements and numerical expressions, demonstrating understanding of the relationship between words and symbolic notation. In-Class Practice, Exercise 7 states, “Write the expression in words. 26-9\times0.” In Class Practice, Exercise 11 states, “Write the words as an expression. Then use a property of addition to write an equivalent expression. Add 9 to the sum of 21 and 6.”

The materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade as included in the following grade-level breakdowns. 

Grade 3:

• The approximate number of chapters devoted to major work of the grade (including assessments and related supporting work) is 9.5 out of 13, approximately 73%.

• The approximate number of lessons devoted to major work of the grade is 63 out of 87, approximately 72%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 120 out of 161, approximately 75%.

Grade 4: 

• The approximate number of chapters devoted to major work of the grade (including assessments and related supporting work) is 9 out of 13, approximately 69%.

• The approximate number of lessons devoted to major work of the grade is 62 out of 89, approximately 70%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 114 out of 163, approximately 70%. 

Grade 5: 

• The approximate number of chapters devoted to major work of the grade (including assessments and related supporting work) is 10 out of 14, approximately 71%.

• The approximate number of lessons devoted to major work of the grade is 64 out of 87, approximately 74%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 122 out of 165, approximately 74%. 

An instructional day analysis across Grades 3 through Grade 5 is most representative of the instructional materials as the days include major work, supporting work connected to major work, and the assessments embedded within each chapter. Approximately 75% of the materials in Grade 3, 70% of the materials in Grade 4, and 74% of the materials in Grade 5 focus on major work of the grade

The materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so that supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers within the Standards Correlation (by Course). 

An example of a connection in Grade 3 includes:

• Chapter 12, Lesson 3, Student Edition, connects the supporting work of 3.MD.3 (Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs) to the major work of 3.OA.8 (Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). In-Class Practice Exercise states, “7. Make a picture graph for the data above. Which type of graph do you prefer to represent this data?” A bar graph above shows the favorite seasons of students. “8. You plan to plant 4 types of seeds in your garden. How many seeds are you planning to plant in all?” A bar graph shows the number of each type of seed planted in the garden. “9. You get 10 more bean seeds. Will you plant more beans or corn?” 

An example of a connection in Grade 4 includes:

• Chapter 8, Lesson 10, Student Edition, connects the supporting work of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale) to the major work of 4.NF.3c (Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction). Students use a problem solving plan to help solve word problems involving mixed numbers and fractions with like denominators. In-Class Practice Exercise 2 states, “A family spent 2\frac{2}{4} hours traveling to a theme park, 7\frac{1}{4} hours at the theme park, and 2\frac{3}{4} hours traveling home. How much more time did the family spend at the theme park than traveling?”

An example of a connection in Grade 5 includes:

• Chapter 11, Lesson 6, Student Edition, connects the supporting work of 5.MD.2 (Make a line plot to display a data set of measurements in fractions of a unit (\frac{1}{2}, \frac{1}{4}, \frac{1}{8}).) Use operations on fractions for this grade to solve problems involving information presented in line plots) to the major work of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers). Students read a table showing fractional amounts of berries with different denominators. They create a line plot to represent the data and then calculate the total cups of berries needed if they make one of each smoothie. Practice Exercise, “2. The table shows the amounts of berries required to make 10 different smoothie recipes. Make a line plot of the data.” Students are given a table titled “Berries (cup)” that lists the fractions: \frac{3}{4},\frac{1}{2}, \frac{1}{8}, \frac{1}{2}, \frac{3}{4}, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{3}{4}, \frac{1}{4}. “What is the most common amount of berries required? How many times as many recipes require \frac{3}{4} cup as \frac{1}{4} cup of berries? 3. How many total cups of berries are needed to make one of each smoothie?”

The instructional materials for Math & YOU Grades 3 through Grade 5 meet expectations for including problems and activities that connect two or more clusters in a domain, or two or more domains in a grade. 

Connections among the major work of the grade are present throughout the materials where appropriate. These connections are listed for teachers Standards Correlation (by Course) within the Digital Teaching Experience under Teacher Toolkit: Course Essentials and may appear in one or more phases of a typical lesson: Example, In-Class Practice, and Practice Exercises. 

An example of a connection in Grade 3 includes:

• Chapter 6, Lesson 3, Student Edition, connects the major work of 3.OA.B (Understand properties of multiplication and the relationship between multiplication and division) to the major work of 3.MD.C (Geometric measurement: understand concepts of area and relate area to multiplication and to addition). Students use the given width and area of a pool to determine its length by applying multiplication or division. In-Class Practice Exercise 14 states, “A public pool is 8 meters wide and has an area of 72 square meters. What is the length of the pool?”

An example of a connection in Grade 4 includes:

• Chapter 10, Lesson 5, Student Edition, connects the major work of 4.NF.A (Extend understanding of fraction equivalence and ordering) to the major work of 4.NF.C (Understand decimal notation for fractions, and compare decimal fractions). Students use addition and multiplication to solve a problem involving the number of steps a doctor takes in one week. Practice Exercise 15 states, “Which gecko is longer? Explain.” Two geckos are shown: a Leopard Gecko and an Electric Blue Gecko. The Leopard Gecko is labeled 0.05 m and 0.06 m. The Electric Blue Gecko is labeled \frac{7}{100}m.

An example of a connection in Grade 5 includes:

• Chapter 8, Lesson 4, Student Edition, connects the major work of 5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths) to the major work of 5.NF.A (Use equivalent fractions as a strategy to add and subtract fractions). Students add fractions with unlike denominators showing their understanding of place value less than one. In-Class Practice Exercise 8 states, “Find the sum. \frac{5}{12}+\frac{3}{5}=.” 

The instructional materials reviewed for Math & YOU Grades 3 through Grade 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

The materials provide multiple features to support coherence across grade levels. Co-Author Notes, Insight Videos, and Coherence details in chapters and lessons explain how current learning builds from prior learning and extends to future learning. The Mathematics of the Chapter Overview in each chapter provides coherence perspectives through “What we’re doing,” “Why we’re doing it,” and “Essential background” insights. Standards for Content and Mathematical Practice provides COHERENCE Throughout the Grades presents chapter charts that show learning progressions, available in both the Teaching Edition and the Digital Teaching Experience, with Common Core standard codes for reference to prior, current, and future learning. Each lesson includes a Coherence section in the overview that summarizes the lesson focus within the broader learning progression.

An example of a connection to future grades in Grade 3 includes:

• Chapter 9, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 3.NF.3 (Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size) to future work in Grade 4, where students utilize fraction equivalence to compare fractions using benchmarks and equivalent fractions (4.NF.1 & 4.NF.2). Mathematics of the Chapter, Nick’s Notes states, “Students will then learn to express whole numbers as fractions and recognize that fractions are equivalent to whole numbers. The chapter ends with students using models and reasoning to compare fractions that have the same numerator or the same denominator and then comparing fractions on a number line. Understanding fraction equivalence is essential to future success in mathematics. Through exploration, students will learn that they need to consider the number of equal parts of each whole and the size of the equal parts to compare fractions.”

An example of a connection to prior knowledge in Grade 3 includes:

• Chapter 1, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem) to the previous work of Grade 2, where students organized information in rows and columns along with repeated addition (2.G.2 & 2.OA.4). Mathematics of the Chapter, Nick’s Notes states, “This chapter develops an understanding of multiplication and division through multiple representations: equal groups, equal-sized groups, number lines, skip counting, and writing equations. This understanding builds the foundation for future use with multi-digit whole numbers, integers, fractions, and decimals. Students begin to build connections between prior knowledge of repeated addition and multiplication as they see how prior knowledge can be applied to a new concept. This interconnectedness of mathematics is important for understanding the progression of numbers and operations, rather than compartmentalizing mathematics as a set of non-related topics. Students build on their previous experiences with organizing equal groups into rows and columns of an array. Organizing objects into rows and columns helps students see patterns for products as well as spatially understand that a 3\times5 array has the same total objects as a 5\times3 array. A connection is made to the Commutative Property of Multiplication.” 

An example of a connection to future grades in Grade 4 includes:

• Chapter 7, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 4.NF.1 (Explain why a fraction a/b is equivalent to a fraction (n\times a)/(n\times b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions) to future work of Grade 5, where students solve word problems using benchmark fractions to determine the reasonableness of their answer (5.NF.2). Mathematics of the Chapter, Nick’s Notes states, “In this chapter, students will apply their understanding of numerators and denominators to compare fractions. Students will learn to find equivalent fractions and begin to work on operations with fractions. Models such as Fraction Strips, area models, and number lines provide students with a visual of the whole. As understanding of equivalent fractions develops, students will compare fractions without relying on models. Key terms in this chapter are equivalent fractions, common factor, and benchmark. Equivalent fractions can be found using area models and number lines. These models can be used to verify that the process of multiplying or dividing the numerator or denominator by the same number produces an equivalent fraction. In area models, the amount shaded in each model is the same. Additional lines are drawn to divide each original part into two or more equal parts. Students will compare fractions by using a benchmark and by using equivalent fractions. The process of comparing fractions provides an opportunity for students to develop an understanding of relationships among fractions. This deepens their understanding of what numerators and denominators tell about a fraction and what it means when two fractions have the same numerator or the same denominator. In previous grades, students located fractions on a number line, including unit fractions and fractions greater than 1. A common misconception about fractions is that the numerator and denominator are two values instead of one. By showing the position of a fraction on a number line, students realize that the fraction represents a single number. Another misconception is that the greater the denominator, the greater the fraction. Using visual models of fractions and mixed numbers, as well as Fraction Strips, help correct this common misconception. Students will build on an understanding of how multiplication can be used to create equivalent fractions. Likewise, the numerator and denominator can be divided by a common factor to find an equivalent fraction. Avoid emphasizing simplifying the fraction, since the focus of this chapter is on the process of using multiplication and division to find equivalent fractions and not simplification, which is addressed later.”

An example of a connection to prior knowledge in Grade 4 includes:

• Chapter 2, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm) to previous work of Grade 3, where students used different strategies to add and subtract within 1,000 (3.NBT.2). Mathematics of the Chapter, Nick’s Notes states, “Previous experiences with skip counting by tens and hundreds provide students with a foundation for the count on and count back strategies needed to understand how to find sums and differences of larger numbers. In Grade 3, students added and subtracted within 1,000 and gained fluency with strategies and algorithms based on place value, properties of operations, and the relationship between addition and subtraction.”

An example of a connection to future grades in Grade 5 includes:

• Chapter 3, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through The Grades, connects 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used) to the future work of Grade 6, where students use the standard algorithm to add and subtract mulit-digit decimals (6.NS.3). Mathematics of the Chapter, Nick’s Notes states, “This chapter focuses on understanding how place value is used to add and subtract decimals. Students will model decimal addition and subtraction using base ten blocks and make quick sketches of the models as a transition to understanding multiplication using the standard algorithm. They will check the reasonableness of their answers by using estimation. Students will recognize when an estimation is appropriate instead of the exact sum or difference by noticing words such as roughly, approximately, or about within the context. Students will utilize partial sums and partial differences when completing decimal equations. Mental math strategies are modeled and discussed, showing that an equation can be completed using more than one strategy. The problem-solving lesson involves money, where students will find the sum or total costs, difference between prices, and the amount of change received. Students will use rounding and compatible numbers as ways to estimate sums and differences. It is important to remind students that there is no one correct way to estimate, but answers should be reasonable and result in a sum or difference close to the actual value. The representation of decimals using base ten blocks is essential in developing an understanding of how to use place value in the addition and subtraction of decimals. By modeling with base ten blocks, students make the connection to adding and subtracting like place values of decimals. Decimal subtraction with regrouping is modeled first with base ten blocks before subtracting by place value.”

An example of a connection to prior knowledge in Grade 5 includes:

• Chapter 1, Teacher Edition, Standards for Content and Mathematical Practice, COHERENCE Through the Grades, connects 5.NBT.1 (Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and \frac{1}{10} of what it represents in the place to its left) to the previous work of Grade 4, where students learn the value of a digit in one place is 10 times the value of the digit in the place to the right of it. Mathematics of the Chapter, Nick’s Notes states, "In this chapter, students will focus on place value relationships, writing different forms of a number, finding powers of ten, comparing decimals, and rounding decimals. By using a place-value chart, students will apply patterns of ten to decimal relationships to read and write decimals to the thousandths in different forms. Various representations of numbers are written in standard form, word form, and expanded form. Students will use repeated reasoning to compare digits in one place to the next and understand that the patterns of 10 apply to both whole numbers and decimals. Students will use their understanding of place value and the number line to compare and round decimal numbers. In Grade 4, students used manipulatives and fraction equivalents to develop an understanding of tenths and hundredths. They used base ten blocks to represent and compare whole numbers and decimals. To represent decimals, a flat is defined as 1 whole, so a rod represents 1 tenth, and a unit represents 1 hundredth. This chapter introduces a thousandth, which can be represented as one part of 10 equal parts of a unit, so it is 1 of 1,000 equal parts of a whole. In Grade 4, students used number lines and place value charts to compare decimals to the hundredths place. In this chapter, students will use number lines to identify the halfway number and visualize rounding decimals to any place.”