## Math in Focus: Singapore Math

##### v1.5
###### Usability
Our Review Process

Title ISBN Edition Publisher Year
Teacher Assessment Guide Grade 3 9780358104964
Singapore Math Fact Fluency Grade 3 9780358105169
Student Edition Set Grade 3 9780358116790
Extra Practice and Homework Set Grade 3 9780358116899
CCSS Teacher Edition Set Grade 3 9780358116998
Teacher and Student Activity Cards Grade K 9780358104926
Teacher Assessment Guide Grade K 9780358104933
Singapore Math Fact Fluency Grade K 9780358105138
Student Edition Set Grade K 9780358116769
Extra Practice and Homework Set Grade K 9780358116868
CCSS Teacher Edition Set Grade K 9780358116967
Teacher Assessment Guide Grade 4 9780358104971
Singapore Math Fact Fluency Grade 4 9780358105176
Student Edition Set Grade 4 9780358116806
Extra Practice and Homework Set Grade 4 9780358116905
CCSS Teacher Edition Set Grade 4 9780358117001
Teacher Assessment Guide Grade 7 9780358105008
Student Edition Set Grade 7 9780358116837
Extra Practice and Homework Set Grade 7 9780358116936
CCSS Teacher Edition Set Grade 7 9780358117032
Teacher Assessment Guide Grade 8 9780358105015
Student Edition Set Grade 8 9780358116844
Extra Practice and Homework Set Grade 8 9780358116943
CCSS Teacher Edition Set Grade 8 9780358117049
Transition Guide Courses 1-3 9780358216643
Teacher Assessment Guide Grade 5 9780358104988
Singapore Math Fact Fluency Grade 5 9780358105183
Student Edition Set Grade 5 9780358116813
Extra Practice and Homework Set Grade 5 9780358116912
CCSS Teacher Edition Set Grade 5 9780358117018
Teacher Assessment Guide Grade 1 9780358104940
Singapore Math Fact Fluency Grade 1 9780358105145
Student Edition Set Grade 1 9780358116776
Extra Practice and Homework Set Grade 1 9780358116875
CCSS Teacher Edition Set Grade 1 9780358116974
Teacher Assessment Guide Grade 6 9780358104995
Singapore Math Fact Fluency Grade 6 9780358105190
Student Edition Set Grade 6 9780358116820
Extra Practice and Homework Set Grade 6 9780358116929
CCSS Teacher Edition Set Grade 6 9780358117025
Teacher Assessment Guide Grade 2 9780358104957
Singapore Math Fact Fluency Grade 2 9780358105152
Student Edition Set Grade 2 9780358116783
Extra Practice and Homework Set Grade 2 9780358116882
CCSS Teacher Edition Set Grade 2 9780358116981
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### Overall Summary

The materials reviewed for Math in Focus: Singapore Math Grade 4 partially meet expectations for Alignment to the CCSSM. In Gateway 1, the materials partially meet expectations for focus and coherence, and in Gateway 2, the materials do not meet expectations for rigor and practice-content connections.

###### Alignment
Does Not Meet Expectations
Not Rated

### Focus & Coherence

The materials reviewed for Math in Focus: Singapore Math Grade 4 partially meet expectations for focus and coherence. For focus, the materials partially meet expectations for assessing grade-level content and providing all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials meet expectations for coherence and consistency with the CCSSM.

##### Gateway 1
Partially Meets Expectations

#### Criterion 1.1: Focus

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 partially meet expectations for focus as they assess grade-level content and partially provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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Materials assess the grade-level content and, if applicable, content from earlier grades.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Summative assessments provided by the materials include Chapter Tests, Cumulative Reviews, and Benchmark Assessments and are available in print and digitally. According to the Preface of the Math in Focus: Assessment Guide, "Assessments are flexible, teachers are free to decide how to use them with their students. ... Recommended scoring rubrics are also provided for some short answer and all constructed response items to aid teachers in their marking." The following evidence is based upon the provided assessments and acknowledges the flexibility teachers have in administering them in order to understand their students' learning.

The provided assessments, found in the Assessment Guide Teacher Edition, assess grade-level standards. Examples include:

• In Chapter Test 1, Section B, Item 8 (Paper) states, “Store A earns $10,347. Store A earns$1,255 more than Store B. How much do the two stores earn in all? Show your work and write your answer in the space below.” (4.NBT.4)

• In Chapter Test 2, Section B, Item 9 (Online) states, “Mr. Jones has 3,250 grams of blueberries. His neighbor gives him another 2,750 grams of blueberries. He uses 5,250 grams of blueberries to make some jelly. He then divides the remaining blueberries equally among his 3 children. How many grams of blueberries does each child receive? Show your work and write your answer in the space below.” (4.OA.3)

• In Chapter Test 3, Section A, Item 4 (Online) states, “Which pairs of fractions show a correct comparison? Choose the two correct answers.  A) \frac{2}{3}>\frac{1}{4} B) \frac{3}{5}<\frac{3}{8} C) \frac{4}{5}>\frac{3}{4} D) \frac{3}{8}>{2}{3} E) \frac{1}{10}>\frac{7}{10}.” (4.NF.2)

• In Chapter Test 6, Section B, Item 7 (Paper) states, “Michael bent a 40-inch wire into a square. What is the area inside the square? Show your work and write your answer in the space below.” A picture of a square is shown. (4.MD.3)

• In Chapter Test 8, Section C, Item 11 (Paper) states, “Jason says that the shape below is a rectangle. Do you agree with Jason? Explain.” (4.G.2)

The provided assessments also assess above-grade assessment items that could be removed or modified without impacting the structure or intent of the materials. Examples include:

• In Chapter 4 Test, Section A, Item 5 (Online) states, “A number has two decimal places. It is 1 when rounded to the nearest whole number.  What could the number be? Choose the three correct answers. A) 1.19, B) 1.24, C) 1.48, D) 1.59, E) 1.67, F) 1.88.” This item assesses 5.NBT.4 (Use place value understanding to round decimals to any place).

• In the Mid-Year Benchmark Assessment, Section A, Item 9 (Online) states, “What is the product of 753 and 19? A) 7,110, B) 7,530, C) 13,887, D) 14,307.” This item assesses 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm).

• In Chapter Test 6, Section B, Item 6 (Paper) states, “Find the area of the shaded parts of the figure. Show your work and write your answer in the space below.” (An image of a rectangle with labeled sides of 8m and 10m is provided. There is an unshaded parallelogram cutting the rectangle into 2 trapezoids.) This item assesses 6.G.1 (Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems).

• In the End-of-Year Benchmark Assessment, Section A, Item 6 (Paper) states, “What is the missing number? 20 + 3 + 0.05 = ___  A) 2.35, B) 20.35, C) 23.05, D) 23.5.” This item assesses 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).

##### Indicator {{'1b' | indicatorName}}

Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 partially meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Materials provide opportunities for students to engage in grade-level problems during Engage, Learn, Think, Try, Activity, and Independent Practice portions of the lesson. Engage activities present an inquiry task that encourages mathematical connections. Learn activities are teacher-facilitated inquiry problems that explore new concepts. Think activities provide problems that stimulate critical thinking and creative solutions. Try activities are guided practice opportunities to reinforce new learning. Activity problems reinforce learning concepts while students work with a partner or small group. Independent Practice problems help students consolidate their learning and provide teachers information to form small group differentiation learning groups.

The materials provide students extensive work with grade-level problems to meet the full intent of some grade-level standards. Examples include:

• In Section 2.2, Quotients and Remainders, students find quotients and remainders in division problems. In Engage, Problem 1, page 121, students use counters to solve equal sharing problems. The problem states, “Share 11 red counters with your partner. How did you do it? Draw a sketch to explain your thinking.” In Try, Problem 1, page 122, students work in pairs to find quotients and remainders. The problem states, ”Write each division quotient and remainder. 21 ÷ 8 = __ R __ , Quotient = __ , Remainder = __ .” In Independent Practice, Problem 1, page 123, students use pictures to divide. The problem states, “Divide 18 keychains equally among 7 children. How many keychains are left? 18 ÷ 7 + __, Quotient = __, Remainder = __ , So 18 ÷ 7 = __. Each child receives keychains. There are _ keychains left.” Eighteen keychains are shown. Students engage with extensive work to meet the full intent of 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models).

• In Section 3.1 Equivalent Fractions, students use multiplication and division to find equivalent fractions and write a fraction in simplest form. In Engage, Problem A, page 231, students use models to show equivalent fractions. The problem states, “Use fraction circles or fraction strips to show $$\frac{1}{2}$$ in two ways.” In Learn, Problem 3, page 232, students use division to find equivalent fractions. The problem states, “You can also find equivalent fractions by simplifying fractions. You simplify a fraction by dividing the numerator and denominator by the same number.” Two sets of visual fraction models are shown. In Hands-On Activity, Problem 1, page 233, students use strips of paper to find equivalent fractions. The problem states, “Your teacher will give you two strips of paper. Use them to form a pair of equivalent fractions. Paste the strips in the box.” Two fraction strips are shown. In Try, Problem 7, page 235, students write fractions in simplest form. The problem states, “Write each fraction in simplest form, $$\frac{8}{12}=\frac{\Box}{3}$$.” In Independent Practice, Problem 7, page 238, states, “ Find the next eight equivalent fractions, $$\frac{1}{5}$$ = __ = __ = __ = __ = __ = __ = __ = __ .” Students engage with the extensive work to meet the full intent of 4.NF.1 (Explain what a fraction $$\frac{a}{b}$$ is equivalent to a fraction $$\frac{n×a}{n×b}$$ 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).

• In Section 4.5, Fractions and Decimals, students express a fraction as a decimal and a decimal as a fraction. In Engage, Problem A, page 383, students use a number line to express a fraction as a decimal. The problem states, “Draw a number line with endpoints 0 and 1. Mark a point ot show $$\frac{1}{2}$$ on your number line. What decimal does the point represent? Explain your thinking to your partner.” In Learn, Problem 3, page 384, students express fractions as decimals. The problem states, “Express $$3\frac{4}{5}$$ as a decimal.” In Try, Problem 1, page 386, students express decimals as fractions on a number line. The problem states, “Express each decimal as a fraction or mixed number in simplest form, 0.4 = ___.” In Learn, Problem 1, page 387, students add tenths and hundredths by converting fractions to decimals. The problem states, “A table is $$\frac{7}{10}$$ meter long. A bench is $$\frac{81}{100}$$ meter long. Find the total length of the table and the bench. Express your answer as a decimal.” Students engage with extensive work to meet the full intent of 4.NF.6 (Use decimal notation for fractions with denominators 10 or 100).

• In Section 7.4, Finding Unknown Angles, students solve real-world problems by using addition or subtraction to find unknown angle measures. In Try, Problem 1, page 216, states, “The measure of ∠XOY is 125o and the measure of ∠YOZ is $$15\degree$$. Find the measure of ∠XOZ.” In Engage, page 219, students “Fold a square piece of paper along the diagonal. What can you say about the new angles formed in the corners? What are the measures of the new angles? How do you know?” In Try, Problem 2, page 220, states, “Ms. Mitchell has a square piece of cloth. She wants to cut the cloth as shown to make a pattern. What is the measure of ∠y?” In Independent Practice, Problem 4, page 222, students find missing angle measures. The problem states, “Silvana buys a paper fan during a carnival. Find the measure of ∠LTN.” Students engage with extensive work to meet the full intent of 4.MD.7 (Recognize angle measure as additive).

• In Section 8.1, Classifying Triangles, students classify triangles by their angle measures. In Hands-on Activity, Problem 2, page 261, students work in pairs to classify triangles. The problem states, “Sort the triangles by their angle measures and complete the table below. Write the letter of each triangle in the correct group.” In Try, Problem 1, page 262, states, “Which of these triangles is right, obtuse, or acute? Use a protractor to help you. Measure of ∠PRQ = __$$\degree$$, Measure of ∠PQR = __$$\degree$$, Measure of∠QRP = __$$\degree$$. Triangle PQR is an/an __ triangle.” In Independent Practice, Problem 2, page 264, states, “Circle the triangle that does not belong in each set. Explain.” Three acute triangles and one obtuse angle triangle are shown. Students engage with extensive work to meet the full intent of 4.G.2 (Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size and recognize right triangles as a category, and identify right triangles).

Materials do not provide students the opportunity to engage with the full intent of some grade-level standards. For example:

• Students are not provided the opportunity to engage with the full intent of 4.NF.3b (Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation). According to the the Common Core State Standards Correlations, page T73, 4.NF.3b is addressed in Section 3.4, Mixed Numbers; Section 3.5, Improper Fractions (page 267); and 3.6, Renaming Improper Fractions and Mixed Numbers (page 275). In, Section 3.5, Improper Fractions, Learn, Problem 4, page 269, states, “Write an improper fraction for the shaded parts. Express the fraction in simplest form.” In these two sections, students work with reducing fractions, mixed numbers, and simplifying improper fractions. However, students are not “decomposing a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation or justifying decompositions by using a visual fraction model.”

Materials do not provide extensive work with some grade-level standards. Examples include:

• The materials do not provide extensive work with 4.NBT.1(Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right). According to the Common Core State Standards Correlations, page T71, this standard is addressed in Section 1.1, Numbers to 100,000, page 11 and Section 1.2, Numbers to 1,000,000, page 19. In these sections, students work with place value, understanding the value of a digit and writing numbers in a variety of forms. Students do not have the opportunity to engage with extensive work of 4.NBT.1.

• The materials do not provide extensive work with 4.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction by a whole number). In Sections 3.8 and 3.9, 4.NF.4a, b, and c are addressed. In Section 3.8, Multiplying Fractions and Whole Numbers, students have the opportunity to multiply a whole number and a fraction, and relate the product to a multiple of a unit fraction. For example, in the Independent Practice, Problem 5 states, “$$3 × \frac{1}{6}$$.” Problem 6, “$$11 × \frac{2}{3}$$.” In Section 3.9, Real-World Problems: Fractions, students have the opportunity to solve real-world problems involving multiplying whole numbers and fractions. For example, in the Independent Practice, Problem 10 states, “Solve. Draw a bar model to help you. Ethan walks $$\frac{5}{6}$$ kilometer to th bus stop each morning. How far does he walk in 5 days?” Problem 11 states, “Solve. Draw a bar model to help you. Mr. Brown spends $$\frac{2}{3}$$ hour gardening every day. How long does he spend gardening in one week?” Per the Math in Focus 2020 Comprehensive Alignment to CCSS: Grade 4, these are the two lessons that address 4.NF.4a, b, c, and d. Therefore, students do not have the opportunity to engage in extensive work with 4.NF.4a, b, and c.

#### Criterion 1.2: Coherence

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for coherence. The majority of the materials, when implemented as designed, address the major clusters of the grade, and the materials have supporting content that enhances focus and coherence simultaneously by engaging students in the major work of the grade. The materials also include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. The materials partially have content from future grades that is identified and related to grade-level work and relate grade-level concepts explicitly to prior knowledge from earlier grades.

##### Indicator {{'1c' | indicatorName}}

When implemented as designed, the majority of the materials address the major clusters of each grade.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations that, when implemented as designed, the majority of the materials address the major cluster of each grade.

• There are 9 instructional chapters, of which 4.5 address major work of the grade, or supporting work connected to major work of the grade, approximately 50%.

• There are 82 sections (lessons), of which 46.5 address major work of the grade, or supporting work connected to major work of the grade, approximately 57%.

• There are 151 days of instruction, of which 102.5 days address major work of the grade, or supporting work connected to the major work of the grade, approximately 68%.

A day-level analysis is most representative of the instructional materials because the days include all instructional learning components. As a result, approximately 68% of the instructional materials focus on major work of the grade.

##### Indicator {{'1d' | indicatorName}}

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Examples include:

• Section 2.6, Multiples, connects the supporting work of 4.OA.4 (Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number…) to the major work of 4.NF.2 (Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as $$\frac{1}{2}$$. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, eg.., by using a visual fraction model). This connection is evident in Problems 1 and 2 on page 185. Problem 1 states, “Find the first common multiple of 2 and 3.” Problem 2 states, “How can you use the answer in Problem 1 to compare $$\frac{1}{2}$$ and $$\frac{2}{3}$$?”

• Section 3.9, Real-World Problems: Fractions, connects the supporting work of 4.MD.4 (Make a line plot to display a data set of measurements in fractions of a unit $$(\frac{1}{2}, \frac{1}{4}, \frac{1}{8})$$ Solve problems involving addition and subtraction of fractions by using information presented in line plots) with 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). In Independent Practice, Problem 3, page 316, students use data charts to solve problems involving addition and subtraction of fractions by answering the question. Problem 4 states, “Answer the question. Zoe planted some seeds. After two months, she recorded the heights of the plants.” Students are provided a pre-filled table to create a fraction line plot to answer Problems 7, 8, and 9a. Problem 7 states, “What is the difference in height between the shortest and tallest plants?” Problem 8 states, “What is the sum of the heights of the shortest and tallest plants?” Problem 9a states, “Three plants grew $$\frac{3}{4}$$ feet How much did they grow in all?”

• Section 5.1, Length in Customary Units, connects with supporting work of 4.MD.1 (Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit) to the major work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number…). In Try, Problem 1, page 10, states, “Practice measuring length in feet and inches. Write each length in inches. You can use a table to help. 3 ft = __$$\bigcirc$$__=____in.”

• Section 6.1, Area and Unknown Sides connects supporting work of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems) to the major work of 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors as students solve area problems that require division). In Independent Practice, Problem 10, page 128, states, “The area of a rectangular pool is 225 square meters. The width of the pool is 9 meters. a. Find the length of the pool. b. Find the perimeter of the pool.”

• Section 7.4, Finding Unknown Angles, connects supporting work of 4.MD.7 (Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems) to the major work of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). In Try, Problem 1, page 220, students “Practice solving real-world problems involving unknown angles. Find each unknown angle. Vijay used a rectangular piece of cardboard and a triangular piece of cardboard to make a model of a house. The measure of ∠PTQ is 35$$\degree$$. What is the measure of ∠PTS?”

##### Indicator {{'1e' | indicatorName}}

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

The instructional materials for Math in Focus: Singapore Math Grade 4 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Examples of connections between major work and major work and connections between supporting work and supporting include:

• Section 1.6, Real-World Problems: Addition and Subtraction, connects the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic) to the major work of 4.OA.A (Use the four operations with whole numbers to solve problems, as students use bar models to solve real-world addition and subtraction problems). In Independent Practice, Problem 1, page 73, states, “A construction company needed 15,010 tiles to completely cover the lobby of a building. Some tiles were laid on the first week and 7,823 tiles were laid on the second week. At the end of the second week, 1,950 tiles were left. How many tiles were laid on the first week?”

• Section 2.7, Real World Problems: The Four Operations, connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic, as students solve multi-step story problems involving the four operations). In Independent Practice, Problem 2, page 202, states, “Kimberly made some bookmarks to sell for charity. She kept 1,022 bookmarks aside and bundled the rest equally in packs of 3. She sold all the bookmark packs for $4 each, and earned$3,704 for the charity. How many bookmarks did Kimberly make?”

• Section 4.1, Understanding Tenths, connects the major work of 4.NF.A (Extend understanding of fraction equivalent and ordering) to the major work of 4.NF.C (Understand decimal notation for fractions and compare decimals fractions, as students explore tenths with fraction models). In Independent Practice, Problem 1, page 353, students “Write the decimal represented by the shaded part in each figure.” An image with a circle divided into tenths with three of the tenths shaded is provided.

• Section 4.5 Fractions and Decimals, connects the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers) to the major work of 4.NF.C (Understand decimal notation for fractions, and compare decimal fractions, as students add fractions and express the answer as a decimal). In Try, Problem 1, page 388, students “Add. Express each answer as a decimal. $$\frac{3}{10} + \frac{49}{100}$$.”

• Section 7.2, Drawing Angles to 180o, connects supporting work 4.G.A (Draw and identify lines and angles, and classify shapes by properties of their lines and angles) to supporting work 4.MD.C (Geometric measurement: understand concepts of angle and measure angles, as students draw and identify angles by understanding concepts of angle measurement). In Try, Problem 4, page 201, students “Join the marked endpoint of each ray to one of the dots to form an angle with the given value. Then, label the angle. Measure of ∠b = 80$$\degree$$.” The included drawing shows a horizontal ray with two points (one at the vertex, the other near the arrow). There are two dots located above the vertex. Students must determine which of the two dots will provide the correct angle measure when connected.

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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 reviewed for Math in Focus: Singapore Math Grade 4 partially 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.

Materials relate grade-level concepts to prior knowledge from earlier grades. Prior Knowledge highlights the concepts and skills students need before beginning a new chapter. The section What have students learned? states the learning objectives and prior knowledge relevant to each chapter. The Math Background identifies the key learning objectives and provides an overview of how prior work connects with grade level work. Examples include:

• In Teacher Edition, Chapter 3, Fractions and Mixed Numbers, Chapter Overview, Math Background, page 221A, connects prior work of 3.NF.A with grade-level work of 4.NF.B.3c. It states, “In earlier grades, students have learned to represent fractions pictorially, as a sum of unit fraction, and on a number line. Students have learned how to find equivalent fractions and compare and order fractions less than one. Students will be familiar with the term like fractions. In this chapter, they add and subtract unlike fractions with and without renaming. The unlike fractions at this grade level are restricted to denominators that are multiples of one of them (called related fractions), so that only one fraction needs to be renamed. They will also be introduced to the concept of fractions of a set, and will apply this knowledge to multiply a fraction by a whole number.”

• In Teacher Edition, Chapter 4, Decimals, Recall Prior Knowledge, page 342, connects prior work of 3.NF.A with grade-level work of 4.NF.6. It states, “In Grades 3 and 4, students learned to read, write, and identify fractions of a whole, interpret pictorial representations of mixed numbers, and use multiplication and division to find equivalent fractions. They used number lines and place value to round numbers to the nearest ten and learned to add like fractions.”

• In Teacher Edition, Chapter 5, Conversion of Measurements, Chapter Overview, Math Background, page 1A, connects prior work of 3.MD.2 to grade-level work of 4.MD.1. It states, “In earlier grades, students learned to measure length, weight, and capacity using metric and customary units of measurement. They learned to convert metric and customary units from larger to smaller units. Students estimated and measured length to the closest $$\frac{1}{2}$$ and $$\frac{1}{4}$$ inch. In this chapter, students will further their understanding of length, mass, weight, and volume. They will understand that, when converting from larger to smaller units, a greater number of smaller units is needed, so they must multiply. They will also learn how to use and convert metric measurements for length, mass, weight, and volume.”

• In Teacher Edition, Chapter 6, Area and Perimeter, Chapter Overview, Math Background, page 101A, connects prior work of 3.MD.C and 3.MD.D to grade-level work of 4.MD.3. It states, “In Grade 3, students gained an understanding of the concepts of perimeter and area. They learned to find the perimeter and area of squares and rectangles on grid paper. In this chapter, students learn to find the area and perimeter of figures using formulas.”

• In Teacher Edition, Chapter 8, Polygons and Symmetry, Chapter Overview, Math Background, page 255A, connects prior work of 3.G.1 to grade-level work of 4.G.2 and 4.G.3. It states, “In Grade 2, students explored plane shapes, learning to identify and describe the properties of both curved and straight-sided figures. In Grade 3, students build on their knowledge to classify shapes by identifying like properties. In this chapter, students will apply their knowledge of angles to define acute, obtuse, and right triangles. They will learn how to classify polygons based on whether the figures have sides of equal lengths or are parallel, for example.”

Within the Chapter Overview, Learning Continuum, materials relate grade-level concepts to upcoming learning but do not identify content from future grades. The section What will students learn next? states the learning objectives from the following chapter (or grade) to show the connection between the current chapter and what students will learn next. However, there is no specific correlation made to how the standards connect. The online materials do not include the standard notation. Examples include:

• In Teacher Edition, Chapter 2, Multiplication and Division, Learning Continuum, What will students learn next?, page 89G, states, “In Grade 5 Chapter 1, students will learn: Multiplying and dividing by 2-digit numbers fluently (5.NBT.5, 5.NBT.6), Real-world problems: four operations of whole numbers (5.OA.1). In Course 1 Chapter 1, students will learn: Common factors and multiples (6.NS.4).”

• In Teacher Edition, Chapter 4, Decimals, Learning Continuum, What will students learn next?, page 341E, states, “In grade 5, Chapter 5, students will learn: Four operations with decimals (5.NBT.2, 5.NBT.7).”

• In Teacher Edition, Chapter 6, Area and Perimeter, Learning Continuum, What will students learn next?, page 101E, states, “In Grade 5 Chapter 6, students will learn: Volume of rectangular prisms (5.MD.5). In Course 1 Chapter 10, student will learn: Area of triangles (6.G.1, 6.EE.2c), Area of parallelograms and trapezoids (6.G.1, 6.EE.2c), Area of other polygons (6.G.1, 6.EE.2c).”

• In Teacher Edition, Chapter 7, Angle and Line Segments, Learning Continuum, What will students learn next?, page 171E, states, “In grade 5, Chapter 8, students will learn: Classifying triangles (5.G.3).”

• In Teacher Edition, Chapter 9, Tables and Line Graphs, Learning Continuum, What will students learn next?, page 309E, states, “In Grade 5 Chapter 7, students will learn: Making and interpreting line graphs (5.MD.2), In Course 1 Chapter 12, students will learn: Collecting and tabulating data. (6.SP.1, 6.SP.5a, 6.SP.5b), Dot plots and number lines (6.SP.2, 6.SP.4), Histograms (6.SP.4).”

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In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.

The materials reviewed for Math in Focus: Singapore Math Grade 4 can be completed within a regular school year with little or no modifications to foster coherence between grades.

The recommended pacing information is found in the Teacher’s Edition and Chapter Planning Guide. The Chapter Planning Guide lists the Lesson Resources each section, which include the Student Edition, Extra Practice and Homework, Fact Fluency, as well as Reteach and Enrichment activities. Each section consists of one or more Engage-Learn-Try focus cycles followed by Independent Practice. Instructional pacing is provided in days, not minutes. For the purpose of this review, the Chapter Planning Guide provided by the Publisher in the Teacher's Edition was used. The Instructional Pathway, found in the Teacher Edition shows how each of the on-line and print resources can be used within each chapter. As designed, the instructional materials can be completed in 151 days.

• There are nine instructional chapters divided into sections. The pacing for each section ranges between one to three days, consisting of 103 instructional days.

• For each chapter, there is a range of one to two days for the Chapter Opener and Recall Prior Knowledge, totaling 13 days.

• For each Chapter, one day is spent on the Math Journal and Put On Your Thinking Cap, totaling 9 days.

• For each Chapter, two days are spent on the chapter’s closure, which consists of a Chapter Wrap-up, Chapter Review, Performance Task, and Project work, totaling 18 days.

• The cumulative review and benchmark assessments represent an additional 8 days.

• There are no additional days for each chapter’s reteach, extra practice, enrichment. These activities are included in the sections for each instructional chapter.

### Rigor & the Mathematical Practices

The materials reviewed for Math in Focus: Singapore Math Grade 4 do not meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. However, the materials do not make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

##### Gateway 2
Does Not Meet Expectations

#### Criterion 2.1: Rigor and Balance

Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for rigor. The materials give attention throughout the year to procedural skill and fluency, spend sufficient time working with engaging applications of mathematics, and do not always treat the three aspects of rigor together or separately. The materials partially develop conceptual understanding of key mathematical concepts.

##### Indicator {{'2a' | indicatorName}}

Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials reviewed for Math in Focus: Singapore Math Grade 4 partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials develop conceptual understanding throughout the grade level. Examples include:

• In Section 1.3, Comparing and Ordering Numbers, Engage, Problem 1, page 27, students use manipulatives to create visual representations of two numbers to develop conceptual understanding of greater than and less than. The problem states, “Use (place value chips) to show 4,517 and 4,537. Which is greater, 4,527 or 4,537?” Students develop conceptual understanding of 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, an expanded form) and 4.OA.5 (Generate a number or shape pattern that follows a given rule).

• In Section 2.1, Multiplying a 1-Digit or 2-Digit Number, Engage, page 101, students use concrete materials to multiply a whole number of up to four digits by a one-digit whole number. It states, “Use (place value chips) to model this problem. Dara has 2 boxes of shells. Each box contains 2,153 shells. How many shells does he have in all? Show two ways to find your answer. Explain your thinking to your partner.” Students develop conceptual understanding of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations).

• In Section 4.4, Rounding Decimals, Engage, page 375, students round and compare decimals to the nearest whole number. It states, “Santino and Lilian want to buy some wood pegs for a project. They need wood pegs of lengths 5.8 inches, 5.2 inches and 5.6 inches. A shop only sells wood pegs with lengths in whole numbers. Using rounding, Santino suggests buying two 5-inch pegs and two 6-inch pegs. Lillian suggests buying four 6-inch pegs. Who is correct? Draw a number line to explain your thinking.” Students develop conceptual understanding of 4.NF.7 (Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole).

Students have limited opportunities to independently demonstrate conceptual understanding. Examples include:

• In Section 2.7, Real-World Problems: The Four Operations, Independent Practice, Problem 2, page 202, students use bar models to solve multi-step problems using the four operations. Students “Solve. Draw a bar model to help you, Kimberly made some bookmarks to sell to charity. She kept 1,022 bookmarks aside and bundled the rest equally in packs of 3. She sold all the bookmark pacts for $4 each, and earned$3,704 for the charity. How many bookmarks did Kimberly make?” (4.OA.3)

• In Section 6.3, Real Problems: Area and Perimeter, Independent Practice, Problem 1, page 151, students solve a real-world problem involving the perimeter of a composite figure. The problem states, “The figure below shows a cattle yard on a farm. Find the length of the cattle fence around the perimeter of the yard.” (4.MD.3)

##### Indicator {{'2b' | indicatorName}}

Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill and fluency throughout the grade level. Examples include:

• In Section 1.4, Adding and Subtracting Multi-Digit Numbers, Engage, Problems a and b, page 41, students show and explain their strategies when describing a number as the sum of their place value. The materials state, “a) Use (place-value chips) to show the 74,528. Then, fill in each blank. 74,528 = 6 ten thousands ___ thousands 5 hundreds ___ ones. b). Fill in each blank. 68,364 = ___ten thousands 18 thousands ___ hundreds 26 tens 4 ones.” Students begin to develop procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 1.6, Real World Problems: Addition and Subtraction, Learn, Problem 2, page 63, students use bar models to show and explain their strategies for adding multi-digit whole numbers. The problem states, “Store A sold 16,245 baseball cards last year. Store A sold 5,648 more baseball cards than Store B. How many baseball cards did they sell in all?” Students begin to develop procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 2.5, Factors, Hands-on Activity, Problems 5 and 6, page 169, students work with factors and multiples. Problem 5 states, "Take 16 (counters) and arrange them in rows and columns.” Problem 6 states, “Use multiplication to list the different ways in the space below. Then, list the factors of 16. The factors are ___, ___, ___, ___, and ___.” Students begin to develop procedural skill and fluency of 4.OA.4 (Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors).

• In Section 3.2, Comparing and Ordering Fractions, Hands-on Activity, Activity 1, Problem 1, page 249, students compare numbers using bar models and benchmarks. Using models to order fractions, “Shade the correct number of parts for each fraction. Fractions represented are $$\frac{1}{4}, \frac{1}{12}, \frac{5}{6}$$. 2). Order the fractions from greatest to least: $$\frac{\Box}{greatest}$$, ____,$$\frac{\Box}{least}$$." In Independent Practice, Problem 1, page 251, states, "Use equivalent fractions to compare each pair of fractions. Which is less, $$\frac{3}{8}$$ or $$\frac{1}{2}$$?” Students develop procedural skill and fluency of 4.NF.2 (Compare two fractions with different numerators and denominators).

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• In Section 1.4, Adding and Subtracting Multi-Digit Numbers, Independent Practice, Problem 7, page 47, states, “Solve. Show your work 21,574 and 183,261.” Students independently demonstrate the procedural skill of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm).

• In Section 1.6, Real-World Problems: Addition and Subtraction, Independent Practice, Problem 3, page 75, states, “Last year, School A raised $24,950 for charity. School B raised$8,504 more than School A. School C raised $12,080 less than School B. How much did Schools B and C raise in all last year?” Students independently demonstrate the procedural skill of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). • In Section 4.5, Fractions and Decimals, Independent Practice, Problem 17, page 392, states, “Express each decimal as a fraction. Then, add. Express the answer as a decimal, 0.59 + 0.6.” Students demonstrate procedural skill and fluency of 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fraction with respective denominators 10 and 100) and 4.NF.6 (Use decimal notation for fractions with denominators 10 or 100). • In Section 9.1, Making and Interpreting a Table, Independent Practice, Problem 2, page 322, states, “Use the data in the table to answer each question. Each student in a class chose his or her favorite type of pie. The data is represented in the table below. c. How many more students prefer apple pies to lemon pies? d. How many students chose apple pie and strawberry pie as their favorites?” Students independently demonstrate procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). Grade 4 materials contain a separate Fact Fluency book so students can independently practice specific strategies to promote procedural skill and fluency. Examples include: • In Chapter 1, Fact Fluency, Working with Whole Numbers, Problems 1-5, page 5, students add and subtract multi-digit numbers with regrouping. The materials state, “Match each problem to its answer. 1) 14,100 - 18, 2) 13,999 + 3, 3) 14,764 - 735, 4) 13,900 + 192, 5) 15,082 - 1,027.” Answer choices: “14,002, 14,029, 14,092, 14,082, and 14,055.” Students independently practice procedural skill and fluency of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). • In Chapter 5, Fact Fluency, Conversion of Measurements, 5c Conversion of Length, Problems 1-3, page 65, students compute conversions of length. The materials state, “Fill in the blanks, ____ feet = 3 yards. Problem 2, ____feet = 2 miles. Problem 3, ____inches = 2 feet.” Students independently practice procedural skill and fluency of 4.MD.1 (Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; 1b, oz. l, ml; hr, min, sec). • In Chapter 6, Fact Fluency, Area and Perimeter, page 77, students calculate the area perimeter of a square and a rectangle given the length and width. The materials state, “Sketch and label the square or rectangle. Then, complete the table. The table has the following lengths/widths: “L: 5in, W: 4in; L: 4ft, W: 3ft; L: 8ft, W: 2ft; L: 9cm, W: 2cm; L: 6mm, W: 6mm.” Students independently practice procedural skill and fluency of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real-world and mathematical problems). • In Chapter 9, Fact Fluency, Tables, Fractions, and Decimals, 9e Real World Multiplication and Division Problems, page 102, students solve real-world multiplication and division problems. The materials state, “The Grade 4 students at an elementary school sold rolls of wrapping paper to raise money for a field trip. [A box showing Clue 1, Clue 2 and Clue 3 is shown.] Fill in the Blanks. Use the clues to help you. 1) Mr. Martinez’s class sold ____ rolls of wrapping paper. 2) Mr. Jones’s class sold ____ rolls of wrapping paper. 3) Mr. Thomas’ class sold ____ rolls of wrapping paper. 4) Each roll of wrapping paper costs$6. How much money did each class raise? Mr. Martinez’s class raised $____, Mr. Jones’s class raised$____, Mr. Thomas’ class raised $____,. 5) It costs$300 for the entire Grade 4 to go on the field trip. Did they raise enough money? Explain your thinking.” Students independently practice procedural skill and fluency of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison).

##### Indicator {{'2c' | indicatorName}}

Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Students have opportunities to independently demonstrate routine application of the mathematics. Examples include:

• In Section 1.6, Real-World Problems: Addition and Subtraction, Engage, page 61, students use a bar model to express a real-world problem. The materials state, “School X has 4,400 students enrolled. School Y has 250 more students enrolled than School X. School Z has 250 fewer students enrolled than School X. Draw a bar model to show the number of students in each school.” Students engage in the routine application of 4.OA.3 (Solve multistep word problems posed with whole numbers and have whole-number answers using the four operations, including problems in which remainders must be interpreted).

• In Section 2.7, Multiplication and Division, Try, Problem 1, page 197, students solve real-world problems using the bar model to represent the problem and identify the operation needed to solve it. The problem states, “Irene had $3756 to spend on office furniture. She bought a sofa for$1,195 and 6 chairs for $128 each. a) How much money did she spend altogether? b) How much money did she have left.” Students are provided with bar models for scaffolding. Students engage in the routine application of 4.OA.3 (Solve multistep word problems with whole numbers and have whole number answers using the four operations, including problems in which remainders must be interpreted). • In Section 5.3, Real-World Problems: Customary Units of Measure, Independent Practice, Problem 4, page 48, students solve real-world problems involving weight conversions. The problem states, “A family of elephants lives in a zoo. The female adult elephant gives birth to 2 baby elephants weighing 220 pounds each. The adult male elephant weighs 6 tons and the adult female weighs 4 tons. Find the total weight of the family of elephants in pounds.” Students independently engage in the routine application 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 a larger unit in terms of a smaller unit). • In Section 5.7, Time, Engage, page 74, students solve problems involving elapsed time. The materials state, “The clock shows half past 12 in the afternoon. What is the time shown on the clock 80 minutes later? What is another way to show the time? Share your ideas with your partner.” Students engage in the routine application 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). • In Chapter 7, Angles and Line Segments, Performance Task, Problem 3, page 251, students add missing pathways that include parallel and perpendicular line segments. The problem states, “A designer drew a model of some of the pathways in an apartment complex. Use your understanding of perpendicular and parallel lines to add the pathways that are missing in the drawing. [A drawing of a trapezoid is shown, with various points and lines.] a) Draw a pathway that is parallel to PR through point Q. b) Draw a pathway that is parallel.” Students engage in the routine application of 4.G.1 (Draw points, line segments, rays, angles, and perpendicular and parallel lines). Students have opportunities to independently demonstrate non-routine application of the mathematics. Examples include: • In Section 2.4, Real-World Problems: Multiplication and Division, Hands-on Activity, Problems 1 and 2, page 155, students work in pairs to solve a real-world problem involving multiplication. Problem 1 states, “Write a real-world problem for the bar model shown below.” Problem 2 states, “Ask your partner to solve the problem and explain how they check their answers.” A bar model showing 2,568 divided by 7 is shown. Students engage in the non-routine application of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison) and 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors). • In Chapter 3, Put on Your Thinking Cap, Problem 2, page 323, students use number sense to find the greatest difference between two mixed numbers, without repeating digits, The problem states, “Fill in each box with the digits 2, 3, 4, 6, 8, or 9 to form two mixed numbers that have the greatest possible difference. Each digit can be used only once. Then find the difference. Express the answer in simplest form. ___ ___ $$\frac{\Box}{10}$$ - ___ $$\frac{\Box}{10}$$ = ___.” Students independently engage in the non-routine application of 4.NF.3c (Add and subtract mixed numbers with like denominators). • In Section 6.1, Area and Unknown Sides, Engage, page 117, students solve mathematical problems involving area and perimeter. The materials state, “Draw four different rectangles on a square grid. Use a table to make a list of length and width of each rectangle. Find the area of each rectangle. Discuss the relationship between the length and the area with your partner. What pattern do you notice?” Students engage in the non-routine application of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems). • In Section 6.2, Composite Figures, Engage, page 129, students use geoboards to create two rectangles with different lengths but the same perimeters. The materials state, “Rectangle A is 4cm longer than Rectangle B but their perimeters are the same. How is this possible? Explore all possibilities using a geoboard. Explain how you arrive at your answer.” Students engage in the non-routine application of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real-world and mathematical problems). ##### Indicator {{'2d' | indicatorName}} The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. All three aspects of rigor are present independently throughout Grade 4. Examples where instructional materials attend to application, procedural skill and fluency, or conceptual understanding, include: • In Section 4.3, Comparing and Ordering Decimals, Hands-on Activity, Activity 1, Problems 1-3, page 367, students develop conceptual understanding of place value to compare decimals to hundredths. Problem 1 states, “Use (place value disk pictures) 1, 0.1, 0.01 to show 8.5 and 9.2. Then, mark 8.5 and 9.2 on the number line.” Problem 2 states, “Ask your partner to use ‘greater than’ or ‘less than’ to describe the decimals.” Problem 3 states, “Trade places. Repeat 1 and 2 with 13.28 and 13.47.” Number lines are provided for Problems 1 and 3. Students develop conceptual understanding of 4.NF.7 (Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole). • In Section 5.3, Real-World Problems: Customary Units of Measure, Independent Practice, Problem 2, page 47, students apply the four operations to solve real-world problems. The problem states, “A pound of ground turkey cost$6. A pound of ground chicken cost \$4. A chef brought 8 pounds of ground turkey and 4 pounds of ground chicken to mix and make hamburger patties. How much did the ground turkey and ground chicken cost altogether?” Students use the four operations to solve word problems, masses of objects and problems that require expressing measurements given in a larger unit in terms of a smaller unit. (4.MD.2)

• In Section 7.2, Drawing Angles to 180°, Hands-On Activity, Problem 1, page 198, students develop procedural skill and fluency as they use a ruler and a protractor to draw angles.  The problem states, “Use a ruler and protractor to draw each of the following angles on the next page. a) 50$$\degree$$, b) 35$$\degree$$, c) 90$$\degree$$, d) 140$$\degree$$.” Students develop procedural skill and fluency with 4.MD.6 (Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure).

Multiple aspects of rigor are engaged simultaneously to develop a student's mathematical understanding. Examples include:

• In Section 1.1, Numbers to 100,000, Engage, page 11, students develop conceptual understanding alongside procedural skill and fluency as they use concrete materials to express a 4-digit number in word and expanded form. The materials state, “Use (base-ten blocks) to show 3,245. Write the number in expanded form and word form.” Students read and write multi-digit whole numbers using number names and expanded form. (4.NBT.2)

• In Chapter 3, Extra Practice and Homework, Fractions and Mixed Numbers, Activity 9, Problem 3, page 117, students develop conceptual understanding alongside application as they solve real-world problems involving fractions. The problem states, “Solve. Show your work. Draw a bar model to help you. Express each answer in simplest form. Farrah and Andrea baked 1 kilogram of bread in all. Farrah baked $$\frac{3}{10}$$ kilogram of bread. How much more bread did Andrea bake than Farrah?” Students solve word problems involving addition and subtraction of fractions referring to the same whole and having the same denominator, by using visual fraction models and equations to represent the problem. (4.NF.3d)

• In Section 6.1, Area and Perimeter, Try, Problem 3, page 111, students develop procedural skill and fluency alongside application to find the perimeter of a rectangle within a real-world context. The problem states, “A rectangular pool has a length of 15 meters and a width of 8 meters. What is the perimeter of the pool? Students apply the area and perimeter formulas for rectangles in real-world and mathematical problems. (4.MD.3)

#### Criterion 2.2: Math Practices

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Math in Focus: Singapore Math Grade 4 do not meet expectations for practice-content connections. The materials support the intentional development of MP3 and partially support the intentional development of MP6. The materials do not support the intentional development of MPs 1, 2, 4, 5, 7, and 8.

##### Indicator {{'2e' | indicatorName}}

Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 do not meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials identify the Standards for Mathematical Practice as Mathematical Habits. The MPs are not consistently identified for teachers within the unit summary or specific lessons and are often misidentified. Student guidance is not provided, and teacher guidance is generic and repetitive. Some activities are scaffolded preventing intentional development of the full intent of the MPs.

MP1 is not intentionally developed to meet its full intent as students have limited opportunities to make sense of problems and persevere in solving them, or use a variety of strategies to solve problems. Examples include:

• Section 2.3, Dividing by a 1-Digit Number, Math Sharing, page 133, is identified as Mathematical Habit 1: Persevere in solving problems.  The materials state, “I divided 420 by 5 mentally in this way: Step 1.400 ÷ 5 = 80 Step 2. 20 ÷ 5 = 4 Step 3. 80 + 4 = 84. Share how you can find 220 ÷ 4 in the same way.” Students are given a similar problem to solve and instructed to solve it “the same way,” which does not intentionally develop MP1.

• Section 5.7, Time, identifies Mathematical Habit 1: Persevere in solving problems in this lesson. However, no evidence was found for students to engage with MP1. Students do not analyze and make sense of problems as all problems are heavily scaffolded by the teacher and within the student materials. Students do not use a variety of strategies to make sense of and solve the problems as they are similar in nature, and students are instructed to use timelines to solve the problems. No evidence was found for students to monitor and evaluate their progress, determine if their answers make sense, or reflect on and revise their problem solving strategy.

• Chapter 7, Angles and Line Segments, Put on Your Thinking Cap! Problem 1, page 240, is identified as Mathematical Habit 1: Persevere in solving problems. Student directions, “An acute angle is smaller than 90$$\degree$$. ⦟PQR is an acute angle. How many acute angles are there altogether in the following figure?” A diagram of four overlapping angles is shown. Teacher guidance includes, “guide students on applying the various heuristics using the problem solving heuristic poster.” The use of the four-step problem solving model as a guide to solve problems is generic and does not intentionally provide students with the ability to solve the problem or actively engage in solving problems by working to understand the questions.

The materials identify MP1 in the Common Core Pathway and Pacing. However, MP1 is not intentionally addressed within the lesson activities and problems. Examples include:

• Section 1.6, Real World Problems: Addition and Subtraction, page 61, is noted as addressing MP1 on page 1 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 2.2, Quotient and Remainder, page 121, is noted as addressing MP1 on page 1 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 5.6, Real-World Problems: Metric Units of Measure, page 63, is noted as addressing MP1 on page 2 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 6.3, Real World Problems: Area and Perimeter, page 141, is noted as addressing MP1 on page 3 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

MP2 is not intentionally developed to meet its full intent as students have limited opportunities to reason abstractly and quantitatively to solve problems. Examples include:

• Chapter 1, Working with Whole Numbers, Put On Your Thinking Cap!, Problem 3, page 79, is identified as Mathematical Habit 2: Use mathematical reasoning. The problem states, “Use the digits 0 to 9 to form two numbers that are 9,000 when rounded to the nearest thousand and have the greatest possible difference. Use each digit only once.” Students do not consider the units involved in the problem, attend to the meaning of quantities, or explain/discuss what the numbers or symbols in an expression/equation represent.

• Chapter 4, Decimals, Put on Your Thinking Cap!, Problem 4, page 395, is identified as Mathematical Habit 2: Use mathematical reasoning. The problem states, “The height of a tree is 3 meters when rounded to the nearest whole number. Which of the following could be the actual height of the tree? 2.39 m, 2.48 m, 3.25 m, 3.51 m.” There is a picture of a tree noting the height of the tree in meters with a question mark. Teacher guidance includes, “Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students to start by considering the span of numbers that can round to 3.” The teacher guidance is generic and repetitive, and the use of the four-step problem solving model as a guide to solve problems does not provide the opportunity for students to attend to the meaning of quantities or explain and discuss what the numbers or symbols represent.

• Chapter 7, Angles and Line Segments, Math Journal, Problem 1a, page 239, is identified as Mathematical Habit 2: Use mathematical reasoning. Students instructions include, “The steps for measuring these angles are not in order. Arrange the steps in order by using 1, 2, or 3 in each box. Obtuse angle. Step____ Place the center of the base line of the protractor at vertex B of the angle. Step____ Place the base line of the protractor on ray BA. Step____Read the outer scale at the point where ray BC crosses it. The reading is 116$$\degree$$. So, the angle measure is 116$$\degree$$.” An obtuse angle with points A, B, and C is shown. Teacher guidance states, “Review with students the various strategies for using a protractor in order to measure angles accurately. Encourage students to work independently. You may want to pose these questions to students who are struggling with using mathematical reasoning. What is the first step in measuring an angle? What is the baseline and how is it a guide for measuring the angle? How do we choose which scale to use? What is the center mark and where should we place it on the angle?” Students do not represent situations symbolically or understand the relationships between problem scenarios and mathematical representations.

• Chapter 8, Polygons and Symmetry, Math Journal, Problem 2, page 294, is identified as Mathematical Habit 2: Use mathematical reasoning. Student instructions state, “a) Is the following quadrilateral a symmetric shape? Explain your answer. b) How would you check for symmetry in a figure?” Teacher’s guidance states, “Walk around the class to provide students with support as needed. Use the questions to guide and prompt students’ thinking. What does ‘symmetry’ mean? What strategy or tools would you use to determine symmetry? Have you tried all possible placements for the line of symmetry? Is your answer correct? How do you know for sure?” Students do not represent situations symbolically or understand the relationships between problem scenarios and mathematical representations.

The materials identify MP2 in the Common Core Pathway and Pacing. However, MP2 is not intentionally addressed within the lesson activities and problems. Examples include:

• Section 1.5, Rounding and Estimating, page 49, is noted as addressing MP2 on page 1 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 4.5, Fractions and Decimals, page 383, is noted as addressing MP2 on page 2 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Chapter 7, Chapter Opener, page 171, is noted as addressing MP2 on page 3 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 9.3, Line Graphs, page 331, is noted as addressing MP2 on page 4 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

##### Indicator {{'2f' | indicatorName}}

Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• In Chapter 2, Multiplication and Division, Math Journal, page 207, students critique the reasoning of others when they identify and correct the mistake in another student’s strategy. The materials state, “Ms. Scott had 708 pennies. She had twice as many pennies as Mr. Perez. Ms. Young had 3 times as many pennies as Mr. Perez. How many pennies did Ms. Young have? Eric’s solution: 708 × 2 = 1,416; 1,416 × 3 = 4,248. Ms. Young had 4,248 pennies. a. Explain the mistake in Eric’s solution. b. Show how you would solve the problem.” Teacher notes include, “Students have an opportunity to perform error analysis to correct Eric’s thinking, and solve the problem for themselves.”

• In Section 3.1, Equivalent Fractions, Hands-on Activity, Page 233, Problem 2, students construct viable arguments to explain why fractions are equal. The problem states, “Explain why the fractions are equal. Show how to use one fraction to find the other.” Students are provided with two images each representing a fraction, $$\frac{1}{2}$$ and $$\frac{2}{4}$$. Teacher guidance includes, “In 2, students will defend and justify their thinking, as well as critique the reasoning of their classmates.” and “You may want to conclude the activity with the following question. How would you explain what you are doing to someone who is absent today?”

• In Section 3.2, Comparing and Ordering Fractions, Hands-on Activity, Problem 2, page 241, students construct viable arguments as they work in pairs to compare fractions and consider which method they prefer. The problem states, “For each pair of fractions, explain to your partner why you chose that method. Show how you used the method to compare the fractions. a) $$\frac{7}{8}$$ and $$\frac{1}{6}$$ , b) $$\frac{11}{12}$$ and $$\frac{2}{3}$$.” Teacher guiding questions include, “Which method will you choose for the first pair of fractions, and why? Which method did you choose for the second pair of fractions, and why? When do you choose one method over another?”

• In Chapter 3, Fractions and Mixed Numbers, Math Journal, Page 321, students critique the reasoning of others. The materials state, “A recipe for an apple pie uses $$\frac{3}{4}$$ cup of flour. Jackson wants to make 2 apple pies. He only has a $$\frac{1}{4}$$-cup measuring cup. He says he has to use the cup 6 times to get the amount of flour he needs. Show why his reasoning is correct.” Teacher notes include, “Students have an opportunity to prove Jackson’s thinking. Have them draw bar models in addition to recording the math sentence used to achieve the answer. You may want to pose these questions to students who are struggling with constructing sable arguments. What is the problem asking you to do?” and “What do you know that can help you solve this problem?”

• In Chapter 4, Decimals, Math Journal, page 393, students construct viable arguments as they compare decimals. The materials state, “Chris and Mary compare 0.23 and 0.3. Chris says 23 is greater than 3. So, 0.23 is greater than 0.3. Mary says 23 tenths is greater than 3 tenths. So, 0.23 is greater than 0.3. Do you agree? Why or why not? Explain.” Teacher guidance includes, “Is there a tool you can use to check Chris and Mary’s thinking? If so what is it? If they have trouble beginning remind them of the four-step problem-solving model used to work through the problems. You may want to encourage discussion by posing this question to students. What do you know?”

• In Chapter 5, Conversion of Measurements, Math Journal, Page 85, students construct viable arguments as they determine if equations associated with measurement conversions are correct. Teacher guidance includes, “review the various strategies to convert both metric and customary units of measure of length, mass, and volume.” The teacher “encourages students to work independently” and poses questions for students who are struggling to construct viable arguments. Teachers ask, “What is the relationship between the units of measure? How does this help us know which operation to use to convert between them? How do you know that you are correct? How can you prove your thinking? Look at the following conversions. Which is correct? Explain. A) 2 gal = 10 pt , B) 3 lb = 48 oz, C) 5 yd = 8 ft, D) 12 km = 12 m, E) 3 L = 3,000 mL.”

Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits identified as evidence of MP3 are not intentionally addressed in the activities and problems. Examples includes:

• Section 3.1, Equivalent Fractions, is noted as addressing MP3 on page 231 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 7.5, Drawing Perpendicular and Parallel Line Segments, is noted as addressing MP3 on page 223-238 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 8.3, Symmetric Shapes and Lines of Symmetry, is noted as addressing MP3 on page 273-284 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

##### Indicator {{'2g' | indicatorName}}

Materials support the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 do not meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials identify the Standards for Mathematical Practice as Mathematical Habits. The MPs are not consistently identified for teachers within the unit summary or specific lessons and are often misidentified. Student guidance is not provided, and teacher guidance is generic and repetitive. Some activities are scaffolded preventing intentional development of the full intent of the MPs.

The language of MP4 has been altered to state: “Use mathematical models” which is not the same as model with mathematics. Students have limited opportunities to model with mathematics as they are often given the model to use. Examples include:

• Section 2.4, Real-World Problems: Multiplication and Division, identifies MP4 as the Mathematical Habit: Use mathematical models. No evidence was found for students to model the situation with an appropriate representation and use an appropriate strategy as students are instructed to use the bar models to solve real-world multiplication and division problems. For example, Independent Practice, Problems 1-8, students are instructed to “Solve. Draw a bar model to help you.” Problem 3 states, “A factory produced 438 chairs in 3 days. How many chairs did the factory produce each day?”

• Chapter 2, Multiplication and Division, Put Your Thinking Cap On, Problem 3, page 209 identifies MP4 as the Mathematical Habit: Use Mathematical Models. The problem states, “A group of friends decided to collect pins. Valery had no pins at first. Aiden gave Valery some of his pins. He then had 3 times as many pins as Valery. After Aiden gave 24 pins to a friend and 72 pins to a neighbor, he had no pins left. How many pins did Aiden have at first?” Teacher guidance includes, “require students to use a before and after bar model with equal parts to solve the problem. Go through the problem using the four-step problem-solving model, as outlined in the teacher notes on page 208. Suggest to students to start by understanding the information given and identifying what the question is asking. Encourage different strategies to find and then check their solution.” Since teachers are directed to tell students to “use a before and after bar model with equal parts” and “Go through the problem using the four-step problem-solving model,” students do not model the situation with an appropriate representation and/or use an appropriate strategy to solve the problem.

• Chapter 4, Decimals, Sections 4.1 - 4.5, Fractions and Decimals, identify MP4: Use mathematical models as aligned to these lessons. However, students do not model the situation with an appropriate representation and use an appropriate strategy as all models and strategies are provided: number lines, shaded circles and squares, rulers, and place value chips and charts. For example, in Section 4.2, Understanding Hundredths, Hands-On Activity, Problems 1 and 2, students record measurements in decimals. Problem 1 states, “Record each measurement in decimals (up to 2 decimal places) on a number line. a. the length of your desk, in meters; b. the mass of ten mathematics textbooks, in kilograms.” Problem 2 states, “Write the measurements in the place-value chart.”

• Chapter 7, Angles and Line Segments, Put On Your Thinking Cap, Problem 3, Page 240 identifies MP4 as the Mathematical Habit: Use Mathematical Models. Students' instructions include, “A square piece of paper is folded as shown. Find⦟p.” Teacher guidance includes, “3 requires students to find an unknown angle in a diagram of a folded piece of paper. Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students that they write down the known angles and the unknown angles and form an equation.” Students do not model the situation with an appropriate representation and/or use an appropriate strategy to solve the problem.

Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits identified as evidence of MP4 are not intentionally addressed in the activities and problems. Examples include:

• Section 1.6, Real-World Problems: Addition and Subtraction, is noted as addressing MP4 on page 61-76 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 2.3, Dividing by a One-Digit Number, page 125, is noted as addressing MP4 on page 1 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Chapter 3, Fractions and Mixed Numbers, is noted as addressing MP4 on page 324 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 7.3, Turns and Angle Measures, page 205, is noted as addressing MP4 on page 3 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

MP5 is not intentionally developed to meet its full intent as students have limited opportunities to choose tools strategically in connection to grade-level content standards. Examples include:

• Section 2.7, Real-World Problems: The Four Operations, identifies MP5 as aligned to this lesson. Throughout the entire lesson, students are instructed to use the bar model to solve problems. For example, Try, Problems 1-4, student directions state, “Solve. Draw a bar model to help you.” Independent Practice, Problems 1-6, student directions state, “Solve. Draw a bar model to help you.” Students do not use appropriate tools strategically as they are repeatedly instructed to use the bar model.

• Section 4.4, Rounding Decimals, identifies MP5 as aligned to this lesson. Throughout the entire lesson, students are either instructed to draw a number line or are provided with number lines to round decimals to the nearest whole number or tenth. Engage, page 377, states, “Draw a number line with the endpoints 0.3 and 0.4. Which tenth is 0.34 nearer to? Extend the number line to 0.5. Mark all the points in the hundredths that are nearer to 0.4 than to 0.3 or 0.5.” In Learn, page 377, all numbers lines are provided. In Independent Practice, pages 381-382, all number lines are provided. Students do not use appropriate tools strategically as they are repeatedly instructed to use the provided number lines or create their own.

• In Section 5.4, Length in Metric Units, Math Sharing, page 54, students solve a problem involving distances. The materials state, “Name a place that you think is 1 kilometer away from your school. How can you check if your guess is correct? Share your ideas with your classmates.” Students do not choose appropriate tools and/or strategies that will help develop their mathematical knowledge.

Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits identified as evidence of MP5 are not intentionally addressed in the activities and problems. Examples include:

• Section 1.5, Rounding and Estimating, is noted as addressing MP5 on pages 49-51 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 2.3, Dividing by a 1-Digit Number, is noted as addressing MP5 on pages 125-152 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 2.4, Real-World Problems: Multiplication and Division, is noted as addressing MP5 on pages 153-166 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 6.1, Area and Unknown Sides is noted as addressing MP5 on pages 107-128 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

##### Indicator {{'2h' | indicatorName}}

Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 partially meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Examples of students attending to the specialized language of mathematics include:

• In Chapter 1, Working with Whole Numbers, Math Journal, page 77, students attend the specialized language of mathematics as they order numbers from least to greatest and explain their reasoning. The materials state, “Look at the set of numbers. Explain the steps you would take to order the numbers from least to greatest. Explain how you know you are correct.” Numbers:  4,509; 45; 45,009; 450.” Teacher guidance includes, “Take a look at these numbers. What is the question asking you to do? What do you remember from the chapter that can help you? How would you show someone how to compare these numbers? Draw a visual and then explain how to compare and order the numbers.”

• In Section 3.7, Adding and Subtracting Mixed Numbers, Math Sharing, page 288, students attend the specialized language of mathematics as they solve a problem involving addition of mixed numbers. The materials state, “Discuss with your partner different ways to mentally add the following. $$1\frac{3}{8} + 1\frac{7}{8} +2\frac{5}{8}$$ = ?” Teacher guidance includes, “Please take out your math journals and write a letter to your friend, showing in pictures and explaining with equations and words, how you can add mixed numbers with regrouping.”

• In Chapter 6, Area and Perimeter, Math Journal, Page 155, students attend to the specialized language of mathematics as they use area to determine the side length of a square. The materials state, “The area of a square is given. Kimberly says that to find the length of one side, she can divide the area by 4. Is Kimberly correct? If not, explain to Kimberly how to find the length of one side of the square.” Teacher guidance includes, “Students have the opportunity to exhibit that they understand the difference between finding the unknown side of a square using its area and using the perimeter. You may want to pose these questions to students who are struggling with using precise mathematical language. What is Kimberly thinking? How do you know that Kimberly is incorrect? Why? How could she correct her thinking.”

• In Chapter 8, Polygons and Symmetry, Math Journal, Problem 1, page 293, students attend to the specialized language of mathematics. The problem states, “Explain why all squares are rectangles, but not all rectangles are squares. Explain how rectangles are related to parallelograms.” Teacher guidance includes, “You may want to pose these questions to students who are struggling with using precise mathematical language and mathematical reasoning. What strategy would you use to determine all properties of squares and rectangles? How are they similar? How are they different? What is a parallelogram? What properties of a parallelogram does a rectangle share?”

Students have limited opportunities to attend to precision in connection to grade-level content. Examples include:

• In Chapter 6, Area and Perimeter, Put On Your Thinking Cap, Problem 1, page 156, students find lengths of a side of a square based on constraints. The problem states, “What is the length of one side of a square if its perimeter and area have the same numerical value?” Teacher guidance includes, “You may want to guide students on applying the various heuristics using the problem-solving heuristics posters. Refer students to the corresponding teacher resources for prompts and worked solutions.”

Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits are not intentionally addressed in the activities and problems. Examples include:

• Chapter 3, Chapter Opener, Fractions and Mixed Numbers, page 221 is noted as addressing MP6 on page 2 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 5.7, Time, page 71, is noted as addressing MP6 on page 2 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 7.3, Turns and Angle Measures, is noted as addressing MP6 on page 205-212 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 8.4, Making Symmetric Shapes and Patterns, page 285, is noted as addressing MP6 on page 3 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

##### Indicator {{'2i' | indicatorName}}

Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Math in Focus: Singapore Math Grade 4 do not meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. The materials identify the Standards for Mathematical Practice as Mathematical Habits. The MPs are not consistently identified for teachers within the unit summary or specific lessons and are often misidentified. Student guidance is not provided, and teacher guidance is generic and repetitive. Some activities are scaffolded preventing intentional development of the full intent of the MPs.

Students have limited opportunities to look for and make use of structure, look for and explain the structure within mathematical representations, or analyze a problem and look for more than one approach. Examples include:

• In Section 2.5, Factors, Math Sharing, page 172, students, “Find a common factor of 6 and 8, Then, find the simplest form of $$\frac{6}{8}$$. a) What number can be used to divide the numerator and denominator of $$\frac{6}{8}$$? b) Discuss with your partner what you notice about the numbers used in a.” Students do not look for patterns or structures to make generalizations and solve the problem, nor do they look at and decompose “complicated” things into “simpler” things.

• In Section 2.6, Multiples, Math Sharing, Page 185, students connect the multiples of 2 and 3 to fractions. The materials state, “1) Find the first common multiple of 2 and 3. 2) How can you use the answer in 1 to compare $$\frac{1}{2}$$ and $$\frac{2}{3}$$ ?” Teacher guidance includes: “Challenge students to use a set of fraction circles for halves, thirds and sixths to connect the multiples of 2 and 3 to fractions. Keep their exploration as concrete as possible. This should help them understand and explain. Select pairs to share their discussions.” Students cannot look for patterns or generalizations because of the simple numbers used in the problem. Teacher guidance is generic and does not allow students to independently develop MP7, look for and make use of structure.

• Chapter 4, Decimals, Put On Your Thinking Cap!, Problem 2, page 394 identifies MP7 as the Mathematical Habit: Make use of structure. Student instructions state, “Look at the hundred square grid. How many more parts must you shade to represent 0.6?” Teacher guidance states, “2 requires students to use visual models in problem solving. Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students to start by thinking about the ways they can make tenths.” Students do not look for and make use of structure within mathematical representations.

Materials identify the Mathematical Habits for MP7 in the Common Core Pathway and Pacing, Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. For example:

• Section 1.4, Adding and Subtracting Multi-Digit Numbers, is noted as addressing MP7 on pages 41-43 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 2.1, Multiplying by a One-Digit or a Two-Digit Number, page 101, is noted as addressing MP 7 on page 1 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 5.2, Weight and Volume in Customary Units, page 19, is noted as addressing MP 7 on page 2 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 6.2, Composite Figures, is noted as addressing MP7 on pages 129 -140 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

Students have limited opportunities to look for and express regularity in repeated reasoning in connection to grade-level content. Examples include:

• Chapter 3, Fractions and Mixed Numbers, Put On Your Thinking Cap! Problem 1, page 322 identifies MP8 as Mathematical Habit 8: Look for patterns. Student instructions state, “What fraction of the figure is shaded? Express your answer in simplest form.” A picture of a rectangle divided into eight sections is shown. Some of the figure is shaded orange. Teacher guidance states, “1 requires students to think about logical ways to work through problem solving with a plan. Step 1 Understand the problem. What do you know? What information do we find in the problem? Step 2 Think of a plan. Which heuristic will you use to solve the problem? Step 3 Carry out the plan. How will you apply the heuristic to the problem? Step 4 Check the answer. How do you know if you are correct? Is your answer reasonable?” Students do not look for and express regularity in repeated reasoning.

• In Chapter 5, Conversion of Measurements, Put On Your Thinking Cap!, Problem 2, page 87, students think about how to add or remove water from pails. The problem states, “Tyler has a 12-liter pail and a 5-liter pail. Explain how he can get the following amount of water using these pails. a) 2 liters b) 3 liters.” Teacher guidance states, “2 requires students to think logically and flexibly. Go through the problem using the four step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students that they start by thinking about multiples of two and three in ways that will help them add or remove water in order to solve.” This lesson’s teacher guidance does not include specific guiding questions to enable students to consistently look for patterns and express regularity in repeated reasoning (MP8).

• In Chapter 7, Angles and Segments, Put On Your Thinking Cap!, Problem 2, pages 240-240A, students use repeated reasoning and look for patterns as they use they apply their understanding of turns to find the number of $$\frac{1}{4}$$ turns that the hands of a clock move during a period of time. The problem states, “How many right angles does the hour hand of a clock move from 8 A.M. today to 2 A.M. tomorrow?” Teacher guidance states, “Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students that they start by drawing a clock face, and mark on the positions of the hands at 8 A.M. today and 2 A.M.” This lesson’s teacher guidance does not include specific guiding questions to enable students to consistently look for patterns and express regularity in repeated reasoning.

Materials identify focus Mathematical Habits for MP8 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples includes:

• Section 1.3, Comparing and Ordering Numbers, is noted as addressing MP8 on page 27 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

• Section 4.4, Rounding Decimals, page 375, is noted as addressing MP 8 on page 2 in the Grade 4 Math in Focus 2020 Common Core Pathway and Pacing.

• Section 8.4, Making Symmetric Shapes and Patterns, is noted as addressing MP8 on pages 285-292 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives.

### Usability

Not Rated

#### Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

##### Indicator {{'3a' | indicatorName}}

Materials provide 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.

##### Indicator {{'3b' | indicatorName}}

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

##### Indicator {{'3c' | indicatorName}}

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

##### Indicator {{'3d' | indicatorName}}

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

##### Indicator {{'3e' | indicatorName}}

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

##### Indicator {{'3f' | indicatorName}}

Materials provide a comprehensive list of supplies needed to support instructional activities.

##### Indicator {{'3g' | indicatorName}}

This is not an assessed indicator in Mathematics.

##### Indicator {{'3h' | indicatorName}}

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

##### Indicator {{'3i' | indicatorName}}

Assessment information is included in the materials to indicate which standards are assessed.

##### Indicator {{'3j' | indicatorName}}

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

##### Indicator {{'3k' | indicatorName}}

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

##### Indicator {{'3l' | indicatorName}}

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

#### Criterion 3.3: Student Supports

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

##### Indicator {{'3m' | indicatorName}}

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

##### Indicator {{'3n' | indicatorName}}

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

##### Indicator {{'3o' | indicatorName}}

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

##### Indicator {{'3p' | indicatorName}}

Materials provide opportunities for teachers to use a variety of grouping strategies.

##### Indicator {{'3q' | indicatorName}}

Materials provide 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.

##### Indicator {{'3r' | indicatorName}}

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

##### Indicator {{'3s' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

##### Indicator {{'3t' | indicatorName}}

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

##### Indicator {{'3u' | indicatorName}}

Materials provide supports for different reading levels to ensure accessibility for students.

##### Indicator {{'3v' | indicatorName}}

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

#### Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

##### Indicator {{'3w' | indicatorName}}

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

##### Indicator {{'3x' | indicatorName}}

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

##### Indicator {{'3y' | indicatorName}}

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

##### Indicator {{'3z' | indicatorName}}

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

## Report Overview

### Summary of Alignment & Usability for Math in Focus: Singapore Math | Math

#### Math K-2

The materials reviewed for Math in Focus: Singapore Math Grades K-2 do not meet expectations for Alignment to the CCSSM. In Gateway 1, the materials do not meet expectations for focus and partially meet expectations for coherence.

##### Kindergarten
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated

#### Math 3-5

The materials reviewed for Math in Focus: Singapore Math Grades 3-5 do not meet expectations for Alignment to the CCSSM. For Grade 4, the materials partially meet expectations for focus and coherence in Gateway 1 and do not meet expectations for rigor and practice-content connections in Gateway 2. For Grades 3 and 5, the materials do not meet expectations for focus and coherence in Gateway 1.

###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated

#### Math 6-8

The materials reviewed for Math in Focus: Singapore Math Grades 6-8 vary in meeting expectations for Alignment to the CCSSM. For Grades 6 and 7, the materials partially meet expectations for Alignment to the CCSSM as they meet expectations for Gateway 1 and do not meet expectations for Gateway 2. For Grade 8, the materials partially meet expectations for Gateway 1 and do not meet expectations for Gateway 2.

###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Partially Meets Expectations
Not Rated
###### Alignment
Does Not Meet Expectations
Not Rated

## Report for {{ report.grade.shortname }}

### Overall Summary

###### Alignment
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###### Usability
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### {{ gateway.title }}

##### Gateway {{ gateway.number }}
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