## Alignment: Overall Summary

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

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## Gateway 1:

### Focus & Coherence

0
7
12
14
13
12-14
Meets Expectations
8-11
Partially Meets Expectations
0-7
Does Not Meet Expectations

## Gateway 2:

### Rigor & Mathematical Practices

0
10
16
18
10
16-18
Meets Expectations
11-15
Partially Meets Expectations
0-10
Does Not Meet Expectations

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## Gateway 3:

### Usability

0
17
24
27
N/A
24-27
Meets Expectations
18-23
Partially Meets Expectations
0-17
Does Not Meet Expectations

## The Report

- Collapsed Version + Full Length Version

## Focus & Coherence

#### Meets Expectations

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Gateway One Details

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

### Criterion 1a - 1b

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

6/6
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Criterion Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

### Indicator 1a

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

2/2
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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 10 states, “What is the value of 12^3 - 4^2 × 9^2?” (6.EE.1)

• In Cumulative Review 1, Section A, Item 8 states, “What is the value of 975 ÷ 0.78? A. 12,500, B. 1,250, C. 125, D. 12.5.” (6.NS.2)

• In the Mid-Year Benchmark Assessment, Section B, Item 30 states, “A florist arranged red, pink, and white roses together into bouquets of the same size. The ratio of the number of red roses to the number of pink roses to the number of white roses is the same for all the bouquets. The table shows the different number of red, pink, and white roses in bouquets. Write each missing value in the table.” (6.RP.1)

• In Chapter Test 11, Section A, Item 2 states, “A cube has edges measuring 8 centimeters each. What is its surface area in square centimeters? A.192, B.256, C.384, D. 512.” (6.G.4)

• In Chapter Test 13, Section B, Item 7 states, “The data set shows the masses in grams of eight cell phones. 165, 93, 140, 185, 174, 143, 191, 122 Find the mean absolute deviation of the data set. Write your answer in the space below.” (6.SP.5c)

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 Test 1, Section C, Item 11 states, “Ann says the square root of 15^2 + 6^3 × 2^2 is 42. She explains that to find the answer, she first adds 152 and 63. Then, she multiplies the sum by 2^2. Finally she finds the square root of the product. Explain why Ann’s reasoning is incorrect. Determine the square root of 15^2 + 6^3 × 2^2.” This item assesses 8.EE.2 (Evaluate square roots of small perfect squares and cube roots of small perfect cubes).

• In Cumulative Review 1, Section C, Item 21, Part B states, “This question has three parts. The temperature at which a substance melts is called its melting point. The table shows the melting points of some elements. Name a pair of elements such that their melting points differ by 10°C.” A table with the list of elements is provided including Hydrogen, Oxygen, Nitrogen, Neon, and Flouride, with the respective melting points ($$\degree$$C) of −259, −219, −210, −249, −220.” This item assesses 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers).

### Indicator 1b

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

4/4
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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 one or more Focus Cycles of Engage, Learn, Try activities and opportunities for Independent Practice which provide students extensive work with grade-level problems to meet the full intent of grade-level standards. Examples include:

• In Section 3.1, Dividing Fractions, students divide a fraction, whole number, or mixed number by a fraction or mixed number. In the Engage activity on page 93, students use spatial reasoning, manipulatives and drawings to build the conceptual understanding of dividing proper fractions by unit fractions. The activity states, “Show how you find the number of eighths in \frac{3}{4} using manipulatives or a picture. Now, find the number of thirds in \frac{5}{6}. Draw a picture to explain your thinking.” In the Learn activity, Problem 1, page 93, students divide a proper fraction by a unit fraction. The problem states, “Vijay had \frac{2}{3} of a pizza. He cut it into equal slices. Each slice was \frac{1}{6} of the whole pizza. How many equal slices did Vijay cut?” In the Try activity, Problem 1, page 98, students practice dividing a proper fraction by a proper fraction. The problem states, “David had 89 of a fruit tart and some plates. He put \frac{2}{9} of the fruit tart on each plate. How many plates did he have?” In Independent Practice, Problem 17, students practice dividing proper fractions by improper fractions. The problem states, “ \frac{1}{9} ÷ \frac{14}{3}”. Students engage with extensive work and full intent of 6.NS.1 (Apply and extend previous understandings of multiplication and division to divide fractions).

• In Section 4.2, Equivalent Ratios, students write, simplify, and compare ratios. In the Engage activity on page 187, students discuss strategies for generating equivalent ratios. The activity states, “Look at the ratio 18:24. Can both terms be divided by 3? Can both terms be divided by 5? Make a list of numbers that both terms can be divided by. What do you notice about the numbers in the list? Now divide each term in the ratio 18:24 by the numbers in the list and write each result as a ratio. What can you say about the ratios?” In the Learn activity, Problem 1b, page 192, students work with descriptions of ratios to find quantities. The problem states, “To make 1 portion of dough, Sophia mixes 5 cups of flour with every 3 cups of water. Sophia wants to make 5 portions of dough. How many cups of flour and how many cups of water does she need?” In the Try activity, Problem 1, page 191, students practice working with tables of ratios. The problem states, “Mr. Smith uses the following table to prepare four mixtures of cement and sand using identical pails.” The table includes information about the number of pails of cement and sand for four different mixtures, and students are asked to identify the ratios of cement:sand, cement: sand (simplest form) and number of pails of cement/number of pails of sand as a fraction. In Independent Practice, Problem 39b, students practice working with descriptions of ratios to find quantities. The problem states, “Daniel uses 5 fluid ounces of lemonade concentrate for every 9 fluid ounces of orange juice concentrate to make a serving of fruit punch. If Daniel wants to make 4 servings of fruit punch, how many fluid ounces of lemonade concentrate and how many fluid ounces of orange juice concentrate does he need?” Students engage with extensive work and full intent of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 7.1, Using Letters to Represent Numbers, students use letters to represent unknown numbers and write algebraic expressions. In the Engage activity on page 5, students use a variable to represent an unknown or missing value with a variable in an addition scenario. The activity states, “Lily buys some plants. Her friend gives her 2 more plants. How many plants does Lily have now? Explain your thinking. How do you express the number of plants Lily has now? Discuss.” In Learn activity, Problem 2b, page 7, students write algebraic expressions with subtraction. The problem states, “Subtract 3 from a. a - 3.” In the Try activity, Problem 4, page 8, students practice writing algebraic expressions with subtraction, “10 less than h.” In Independent Practice, Problem 11a, students practice writing algebraic expressions with the four operations. The problem states, “Bianca is now x years old. Her father is 24 years older than her. Find her father’s age in terms of x.” Students engage with extensive work and full intent of 6.EE.2 (Write, read, and evaluate expressions in which letters stand for numbers).

• In Section 11.3, Volume of Rectangular Prisms, students find the volume of rectangular prisms. In the Engage activity, page 237, students calculate the volume of a rectangular prism with a fractional side length. The activity states, “Explain the steps you would take to find the volume of a rectangular prism that measures 4 centimeters by 6 centimeters by 3\frac{1}{2} centimeters. Compare your method to your partner’s.” In the Learn Activity, Problem 3, page 238, students calculate the volume of a rectangular prism. The problem states, “A rectangular prism measures 8.4 centimeters by 5.5 centimeters by 9 centimeters. What is its volume?” In the Try activity, Problem 1, page 240, students practice finding the volume of a rectangular prism. The problem states, “Find the volume of each rectangular prism. Length = 6 in. Width = 5\frac{1}{4} in. Height = 12 in.” In Independent Practice, Problem 8, students practice finding the number of cubes required to fill a rectangular box completely. The problem states, “Cameron fills the rectangular box on the right with \frac{1}{2}-inch cubes completely. How many \frac{1}{2}-inch cubes does he need?” The box shown on the right has dimensions of 15 in., 9 in., and 6 in. Students engage with extensive work and full intent of 6.G.2 (Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism).

### Criterion 1c - 1g

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

7/8
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Criterion Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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

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

2/2
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade.

• There are 13 chapters, of which 8 address major work of the grade, or supporting work connected to major work of the grade, approximately 62%.

• There are 53 sections (lessons), of which 35.5 address major work of the grade, or supporting work connected to major work of the grade, approximately 67%.

• There are 158 days of instruction, of which 104.5 days address major work of the grade, or supporting work connected to the major work of the grade, approximately 66%.

A day-level analysis is most representative of the instructional materials because the days include all instructional learning components. As a result, approximately 66% of the instructional materials focus on major work of the grade.

### Indicator 1d

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

2/2
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Digital materials provide “Math in Focus 2020 Comprehensive Alignment to CCSS: Course 1” located under Discover, Planning. This document identifies the standards taught in each chapter’s section allowing connections between supporting and major work to be seen. Examples include:

• In Section 1.2, Common Factors and Multiples, Try, Problem 3, page 15, students use the greatest common factor with the distributive property which connects the supporting work of 6.NS.4 (Find the greatest common factor of two whole numbers less than or equal to 100) with the major work of 6.EE.3 (Apply the properties of operations to generate equivalent equations). Students solve, “Express the sum of each pair of numbers as a product of the greatest common factor of the numbers and another sum. 60 + 85.”

• In Section 5.3, Distance and Speed, Try, Problem 2, page 263, students divide fractions to find unit rate which connects the supporting work of 6.NS.2 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions) with the major work of 6.RP.2 (Understand the concept of unit rate). Students solve, “A scooter travels $$\frac{1}{4}$$ mile in $$\frac{1}{2}$$ minute. Find the speed of the scooter in miles per minute.”

• In Section 7.4, Try, Problem 8, page 32, Expanding and Factoring Algebraic Expressions, students expand and factor simple algebraic expressions which connects the supporting work of 6.NS.4 (Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor) with the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions). Students solve, “Factor each expression. 12t - 8.”

• In Section 8.1, Solving Algebraic Equations, Independent Practice, Problem 21, page 66, students solve equations with one variable which connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) with the major work of 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers). Students solve, “9.9 = x + 5.4.”

• In the Chapter 9 Review, The Coordinate Plane, Problem 20, page 149, students plot points on the coordinate plane which connects the supporting work of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices) with the major work of 6.NS.6b (Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane). Students solve, “Simone knows how to plot Point A at (-4, 3) on a coordinate plane. She needs to plot Point B at (-4, -3). Where is point B located on the coordinate plane in relation to Point A?”

### Indicator 1e

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

2/2
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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.

Digital materials provide “Math in Focus 2020 Comprehensive Alignment to CCSS: Course 1” found under Discover, Planning. This document identifies the standards taught in each chapter’s section showing connections between supporting to supporting work and major to major work.

There are connections from supporting work to supporting work throughout the grade-level materials, when appropriate. Examples include:

• In Section 10.1, Areas of Triangles, Independent Practice, Problem 8, students connect the supporting work of 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume) to the supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples). Students solve, “Find the area of each triangle.” A triangle with a height of 6 cm and base of 15.5 cm is pictured.

• In Section 12.2, Dot Plots, Activity, Problem 3, page 277, students connect the supporting work of 6.SP.A (Develop understanding of statistical variability) to the supporting work of 6.SP.B (Summarize and describe distributions). Students toss two dice 20 times and record their results then, “Draw a dot plot to represent the data.”

• In Section 13.4, Interpreting Quartiles and Interquartile Ranges, Independent Practice, Problem 4, students connect the supporting work of 6.NS.B (Compute fluently with the multi-digit numbers and find common factors and multiples) to the supporting work of 6.SP.B (Summarize and describe distributions). Students solve, “Find the median, the lower quartile, the upper quartile, and the interquartile range of each data set. Height, in feet, of 14 lemon trees: 12.4, 11.8, 14.6, 13, 11.2, 15, 13.4, 11, 12.8, 13, 11.2, 14.4, 12, and 13.2?”

There are connections from major work to major work throughout the grade-level materials, when appropriate. Examples include:

• In Section 5.3, Distance and Speed, Try, Problem 1, page 265, connects the major work of 6.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions) to the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems). Students find the distance given the speed and time using the formula distance = speedtime. Students solve, “A train can travel 120 kilometers per hour. How far can the train travel in 5 minutes?”

• In Section 8.1, Solving Algebraic Equations, Independent Practice, Problem 19 connects the major work of 6.EE.B reason about and solve one-variable equations and inequalities to the major work of 6.NS.A, apply and extend previous understandings of multiplication and division to divide fractions by fractions. Students solve, “Solve each equation using the concept of balancing. Express each answer in simplest form. $$\frac{8}{9} =\frac{1}{3}$$f

• In Section 8.2, Writing Linear Equations, Independent Practice, Problem 7, connects the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the major work of 6.EE.C (Represent and analyze quantitative relationships between dependent and independent variables). Students solve, “Use graph paper. Solve. There are x sparrows in a tree. There are 50 sparrows on the ground beneath the tree. Let y represent the total number of sparrows in the tree and on the ground. a. Express y in terms of x. b. Make a table to show the relationship between y and x. Use values of x = 10, 20, 30, 40, and 50 in your table. c. Graph the relationship between y and x on a coordinate plane.”

### Indicator 1f

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.

1/2
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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. Recall Prior Knowledge and Math Background highlights the concepts and skills students need before beginning a new chapter. What have students learned? states the learning objectives and prior knowledge relevant to each chapter. The Scope and Sequence shows the progressions of standards from Course 1 through Course 3. The online materials do not include the standard notation. Examples include:

• In Chapter 2, Learning Continuum, What have students learned? states, “In Grade 4 Chapter 1, students have learned: Comparing and ordering numbers. (4.NBT.2) In Grade 5 Chapter 2, students have learned: Adding unlike fractions and mixed numbers. (5.NF.1) Subtracting unlike fractions and mixed numbers. (5.NF.1) In Grade 5 Chapter 4, students have learned: Understanding thousandths. (5.NBT.b, 5.NBT.3a) Comparing, ordering, and rounding decimals. (5.NBT.3b, 5.NBT.4)”

• In Chapter 5, Chapter Overview, Math Background states, “Students have learned how to multiply and divide whole numbers and fractions in Grades 4 and 5. They have also learned how to solve real-world problems using bar models. They have learned how to write ratios to compare two quantities and express ratios in simplest form in Chapter 4.”

• In Chapter 7, Recall Prior Knowledge states, “Students learned to use bar models to interpret and represent the part-who concept in addition and subtraction in Grade 2. In Grade 3, they learned to add and subtract up to 4-digit numbers without regrouping, use known multiplication facts to find other multiplication facts, write a multiplication equation and a related division equation, and use bar models to solve real-world multiplication and division problems. In Chapter 1, students learned to find the common factors and greatest common factor of two whole numbers.”

• In Chapter 8, Recall Prior Knowledge states, “Students learned to use fact families to solve problems in Grade 1, write multiplication and a related division in Grade 3, and plot points on a coordinate plane in Grade 5. In Chapter 2, students learned to represent, compare, and order positive and negative numbers on a number line. In Chapter 7, they learned to recognize and write simple algebraic expressions as well as evaluate algebraic expressions for given values of the variable.”

• In Chapter 12, Learning Continuum, What have students learned? states, “In Grade 3 Chapter 12, students have learned: Polygons. (3.G.1) In Grade 4 Chapter 6, students have learned: Area and unknown sides. (4.MD.3) In Grade 5 Chapter 7, students have learned: Making and Interpreting line plots. (5.MD.2)”

Materials provide grade-level standards of upcoming learning to future grades with no explanation of the relationship to grade-level content. What will students learn next? states the learning objectives from the following chapters of future courses to show the connection between the current chapter and what students will learn next. The Scope and Sequence shows the progressions of standards from Course 1 through Course 3. Examples include:

• In Chapter 1, Learning Continuum, What will students learn next? states, “In Course 3 Chapter 2, students will learn: Exponential notation. (8.EE.1), Squares, square roots, cubes, and cube roots. (8.EE.2)”

• In Chapter 3, Learning Continuum, What will students learn next? states, “In Course 2 Chapter 1, students will learn: Writing rational numbers as decimals. (7.NS.2d), Operations with decimals. (7.NS.1d, 7.NS.2c)”

• In Chapter 6, Learning Continuum, What will students learn next? states, “In Course 2 Chapter 4 students will learn: Percent increase and decrease. (7.RP.3), Real-world problems: percent increase and decrease. (7.RP.3)”

• In Chapter 9, Learning Continuum, What will students learn next? states, “In Course 3 Chapter 9, students will learn Translations. (8.G.3), Reflections. (8.G.3)”

• In Chapter 10, Learning Continuum, What will students learn next? states, “In Course 2 Chapter 7, students will learn: Constructing Triangles. (7.G.2), Scale Drawings and Areas. (7.G.1)”

### Indicator 1g

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

Narrative Evidence Only
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 can be completed within a regular school year with little to no modification to foster coherence between grades.

Recommended pacing information is found in the Teacher’s Edition, Chapter Planning Guide and online under Discover, Planning, Common Core Pathway and Pacing Course 1. Each section consists of one or more Engage-Learn-Try focus cycles followed by Independent Practice. Instructional minutes are not provided. As designed, the instructional materials can be completed in 158 days.

• There are 13 instructional chapters divided into sections of 119 instructional days.

• There is one day for each chapter’s instructional beginning consisting of Chapter Opener and Recall Prior Knowledge, for a total of 13 additional days.

• There is one day for each chapter’s closure consisting of Chapter Wrap-Up, Chapter Review, Performance Task, and Project work, for a total of 13 additional days.

• There is one day for each chapter’s Assessment, for a total of 13 additional days.

The online Common Core Pathway and Pacing Course 1 states the instructional materials can be completed in 161 days, three instructional days added to Chapter 12. For the purpose of this review the Chapter Planning Guide provided by the publisher in the Teacher’s Edition was used.

## Rigor & the Mathematical Practices

#### Does Not Meet Expectations

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Gateway Two Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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).

### Criterion 2a - 2d

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.

7/8
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Criterion Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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

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

1/2
+
-
Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Students have opportunities to develop conceptual understanding of mathematical concepts during the Engage and Learn portions of the lessons. Examples include:

• In Section 2.2, Number Lines and Negative Numbers, Engage, page 59, students extend their understanding of number lines to include negative numbers. The materials state, “On a number line, locate 3, 5, and 6. Discuss with your partner how you would represent -3, -5, and -6 on the same number line. How many strategies can you think of?” Students develop conceptual understanding of 6.NS.6 (Understand a rational number as a point on the number line).

• In Section 4.1, Comparing Two Quantities, Activity, Problems 1-3, page 176, students use bar models, fractions, and ratios to compare quantities. The materials state, “1. Use cubes to show the statement: The number of blue cubes is $$\frac{3}{8}$$ of the number of red cubes. 2. Draw a bar model to represent the statement in 1. 3. Rewrite the statement using a ratio.” Students develop conceptual understanding of 6.RP.1 (Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities).

• In Section 5.1, Rates and Unit Rate, Learn, Problem 1, page 238, students use bar models to identify a unit rate. The problem states, “A printer prints 750 pages in 10 minutes (a bar model showing 750 pages in 10 minute increments is provided). 10 min → 750, 1 min → $$\frac{750}{10}$$ = 75 The printer can print 75 pages every minute. The printer prints at a rate of 75 pages per minute.” Students develop conceptual understanding of 6.RP.2 (Understand the concept of a unit rate $$\frac{a}{b}$$ associated with a ratio a:b with b $$\ne$$ 0, and use rate language in the context of a ratio relationship).

• In Section 7.3, Simplifying Algebraic Expressions, Engage, page 19, students use cubes and bar models to combine like terms. The materials state, “1. Use cubes to show 2 groups of 3. How many cubes are there? 2. Maria bought 2 equal bags of apples. Draw a bar model to represent the situation. How would you express the total number of apples in the bags? Explain.” Students develop conceptual understanding of 6.EE.3 (Apply the properties of operations to generate equivalent expressions).

• In Section 7.5, Real-World Problems: Algebraic Expressions, Engage, page 35, students use bar models to solve real-world problems involving algebraic expressions. The materials state, “Zachary has x pens. Sofia has twice as many pens as Zachary. Sofia gives her sister 1 pen. Draw a bar model to find the number of pens Sofia has more than Zachary. Compare your bar model to your partner’s.” Students develop conceptual understanding of 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem).

Students have opportunities to demonstrate conceptual understanding through Try activities, which are guided practice opportunities to reinforce new learning. The Independent Practice does not continue the development of conceptual understanding, therefore students do not have opportunities to independently demonstrate conceptual understanding.

### Indicator 2b

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

2/2
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Students have opportunities to develop procedural skill and fluency during the Engage and Learn portions of the lessons. Examples include:

• In Section 1.3, Squares and Cubes, Learn, Problem 2, page 23, students use the idea of area of squares to find the square of a whole number. The problem states, “Find the square of 5. $$5^2$$ = 5 × 5 = 25. The square of 5 is 25.” Students develop procedural skill and fluency of 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents).

• In Section 3.3, Adding and Subtracting Decimals Fluently, Engage, page 121, students recall prior knowledge of decimal addition. The materials state, “Recall and Discuss what you learned about place value and decimal addition. Show how you find the sum of 4.25 and 5.798. Show your method.” Students develop procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• In Section 3.5, Dividing Decimals Fluently, Learn, page 139, Method 2, students evaluate division expressions by expressing them as a fraction, then divide. The materials state, “Express the quotient as a fraction then divide. 0.56 ÷ 0.04 = $$\frac{0.56}{0.04}$$ = $$\frac{56}{4}$$ = 14. Express the quotient as fraction. Multiply both the numerator and denominator by 100 to make the divisor a whole number. Simplify.” Students develop procedural skill and fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm).

• In Section 6.1, Understanding Percent, Learn, Problem 2, page 308, students express a part of a whole as a fraction and a percent. The problem states, “72 out of 100 cats are long-haired cats. What percent of the cats are long-haired cats? 72 out of 100 → $$\frac{72}{100}$$ (Express the fraction as a percent.) 72% of the cats are long-haired cats.” Students develop procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100).

• Lesson 9.2, Lengths of Line Segments, Engage, page 123, students find lengths of line segments on the x-axis and the y-axis. The materials state, “Plot Points A(2, 0), B(4, 0), C(0, 4) on the coordinate plane below. How many units along the x-axis is Point B from Point A? How many units along the y-axis is Point D from Point C?” Students develop procedural skill and fluency of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate).

Students have opportunities to independently practice procedural skill and fluency during the Independent Practice portion of the lesson. Examples include:

• In Section 3.4, Multiplying Decimal Fluently, Independent Practice, Problem 25, page 134, students multiply a multi-digit decimal by a number with one decimal place in vertical form. The materials state, “Write in vertical form. Then, multiply. 1.2 × 0.6”. Students independently practice procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• In Section 6.1, Understanding Percent, Independent Practice, Problem 4, page 313, students express each percent as a decimal. The materials state, “There are 750 spectators in the stadium, of which 420 are adults and the rest are children. a. What percent of the spectators are adults? b. What percent of the spectators are children?” Students independently practice procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100).

• In Lesson 9.2, Lengths of Line Segments, Independent Practice, Problem 4, page 133, students find the lengths of line segments on the coordinate plane. The materials state, “Use graph paper. Plot each pair of points on a coordinate plane. Connect the points to form a line segment and find its length. G(-6, -3) and H(-6, -8).” Students independently practice procedural skill and fluency of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate).

• In Section 13.4, Interpreting Quartiles and Interquartile Range, Independent Practice, Problem 2, page 345, students find the median, lower quartile, upper quartile, and interquartile range of each data set. The problem states, “Find the median, the lower quartile, the upper quartile, and the interquartile range of each data set. Scores of nine football players in a season: 33, 42, 31, 27, 47, 23, 40, 45, and 43.” Students independently practice procedural skill and fluency of 6.SP.5c (Giving quartile measures of center and variability).

Course 1 materials contain a separate Fact Fluency book so students can independently practice specific strategies to promote procedural skill and fluency. Examples include:

• In Fact Fluency, Chapter 2, Whole Numbers: Multiplication and Division, Problem 8, page 8, students use the standard algorithm to divide “454 ÷ 2 = ____.” Students independently practice procedural skill and fluency of 6.NS.2 (Fluently divide multi-digit numbers using the standard algorithm).

• In Fact Fluency, Chapter 3, Fractions and Decimals, Problem 9, page 21, students divide multi-digit decimals, “0.042 ÷ 0.03.” Students independently practice procedural skill and fluency of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation).

• In Fact Fluency, Chapter 8, Equations and Inequalities, Problem 4, page 6, students solve algebraic inequalities by graphing. The problem states, “Graph the inequality on a number line to show the possible solutions. Then, test a solution. p -1.5.” Students independently practice procedural skill and fluency of 6.EE.8 (Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams).

• In Fact Fluency, Chapter 11, Surface Area and Volume of Solids, Problem 6, page 85, students use order of operations to solve, “$$2^2$$ × 9 ÷ 3 - 3 = _____.” Students independently practice procedural skill and fluency of 6.EE.1 (Write and evaluate numerical expressions involving whole-number exponents).

### Indicator 2c

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

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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Students have opportunities throughout the materials to engage in routine application of mathematics. Examples include:

• In Section 3.6, Real-World Problems: Decimals, Independent Practice, Problem 5, Page 148, students divide decimals to solve real-world problems. The problem states, “Maya buys 6.93 pounds of raisins to make some loaves of raisin bread. Each loaf requires 0.33 pound of raisins. How many loaves of bread can she make?” Students independently engage in routine application of 6.NS.3 (Fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm).

• In Section 4.3, Real-World Problems: Ratios, Independent Practice, Problem 7, page 212, students solve real-world problems involving two sets of ratios. The problem states, “The ratio of the number of mystery books to the number of science fiction books in a bookcase is 4 : 3. The ratio of the number of science fiction books to the number of biographies is 4 : 5. If there are 48 science fiction books, find the total number of books in the bookcase.” Students independently engage in routine application of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 5.5, Real-World Problems: Speed and Average Speed, Learn, Problem 1, page 277, students draw and use diagrams to solve real-world problems involving speed, distance, and time. The problem states, “Adam cycled from Town X to Town Y. He started his journey from Town X at 7:45 A.M. and ended at Town Y at 9:15 A.M. He cycled at an average speed of 14 kilometers per hour. What distance did Adam cycle?” Students engage in routine application of 6.RP.3 (Students use ratio and rate reasoning to solve real-world and mathematical problems).

• In Section 7.5, Real-World Problems: Algebraic Expressions, Learn, Problem 1, page 35, students solve real-world problems involving algebraic expressions. The problem states, “Emma has y books. Matthew has 3 times as many books as Emma. Matthew buys another 7 books. How many more books does Matthew have than Emma? Give your answer in terms of y in the simplest form. If Emma has 25 books, how many more books does Matthew have than Emma?” Students engage in routine application of 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem).

Students have opportunities throughout the materials to engage in non-routine application of mathematics. Examples include:

• In Chapter 4, Performance Task, Problem 3, page 228, students adjust their models and thinking to respond to changes in the given information. The problem states, “The sixth-grade raised $$\frac{2}{3}$$ as much money as the seventh-graders at the school fair in the morning. In the afternoon, the sixth-graders continued to raise more money. In the end, they raised twice as much money as the seventh- graders. Both the sixth-graders and seventh-graders raised $324 in the end. How much more did the sixth-graders raise in the afternoon? Show and explain your work.” Students independently engage in non-routine application of 6.RP.3 (Use ratio and rate reasoning to solve real-world problems). • In Chapter 10, Put On Your thinking Cap! Problem 1, page 202, students find the area of a figure. The problem states, “Figure ABCD is made up of Square PQRS and four identical triangles. The area of Triangle APD is 49 square feet. The lengths of $$\bar{AP}$$ and $$\bar{PD}$$ are in the ratio 1 : 2. Find the area of Figure ABCD.” A diagram is provided. Students independently engage in non-routine application of 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). • In Section 11.3, Volume of Rectangular Prisms, Independent Practice, Problem 6, page 241, students find the length of an edge of a cube given its volume. The problem states, “Solve. A cube has a volume of 125 cubic inches. Find the length of its edge.” Students engage in non-routine application of 6.EE.2c (Evaluate expressions at specific values of their variables). • In Section 13.1 Mean, Engage, page 318, students explore what they know about mean and how it relates to total and missing values. The materials state, “The mean of four numbers is 32. The mean of two of the numbers is 28. Discuss the steps you would follow to find the mean of the other two numbers.” Students engage in non-routine application of 6.SP.3 (Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number). ### Indicator 2d 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. 2/2 + - Indicator Rating Details The materials reviewed for Math in Focus: Singapore Math Course 1 meet expectations in 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 the materials. For example: • In Section 3.4, Multiplying Decimals Fluently, Math Sharing, page 132, students use a concrete approach to multiply decimals. The materials state, “The model on the right shows 0.2 × 0.6 = 0.12. 1. (Find two other decimals that give a product of 0.12. 2.) Find two decimals that give a product of 0.36. Share with your classmates how you found the decimals.” Students develop conceptual understanding of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation). • In Section 5.2, Real-World Problems: Rates and Unit Rates, Activity, page 252, students compare unit rates in a real-world situation. The materials state, “1) Press your fingers firmly on your wrist to feel your pulse. Count the number of times your heart beats in 15 seconds while your partner uses a stopwatch to take the timing. 2) Find your resting heart rate per minute. Compare your unit heart rate to your partner’s unit heart rate. What do you notice? 3) Find your heart rate per minute after doing 30 jumping jacks. Compare your unit heart rate to your partner’s unit heart rate. What do you notice?” Students engage in application of 6.RP.3 (Use ratio and rate reasoning to solve real-world and mathematical problems). • In Section 6.4, Real-World Problems: Percent, Chapter Review, page 343, students practice converting between fractions, decimals, and percents. Problem 1 states, “Express each percent as a fraction in simplest form. 46%” Problem 4 states, “Express each percent as decimal 34%.” Problem 7 states, “Express each fraction as a percent $$\frac{17}{20}$$.” Problem 10 states, “Express each decimal as a percent 0.02.” Students engage in procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100). Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example: • In Section 3.2, Real-World Problems: Fractions, Learn,Problem 1, page 103, students divide a whole number by a fraction to solve real-world problems. The problem states, “A chef cooks 12 pounds of pasta each day. She uses $$\frac{3}{16}$$ pounds of pasta for each serving she prepares. How many servings of pasta does she prepare each day?” Students build procedural skill and fluency and apply the mathematics of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions). • In Section 6.3, Percent of a Quantity, Try, Problem 1, page 326, students practice finding the whole given a quantity and its percent. The problem states, “27% of the students in a school are in Grade 6. There are 540 Grade 6 students. How many students are there in the school?” A tape diagram to assist with filling in missing values is provided. Students develop conceptual understanding and build procedural skill and fluency of 6.RP.3c (Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent). • In Section 8.4, Solutions of Simple Inequalities, Independent Practice, Problem 27, page 92, students use inequalities to represent real-world situations. The problem states, “In the inequality q 24.3, q, represents the possible weights, in pounds, of a package. a) Is 24.4 a possible value of q? Explain. b) Is $$20\frac{7}{10}$$ a possible value of q? Explain. c) Use a number line to represent the solution set of the inequality. State the greatest possible weight of the package.” Students develop conceptual understanding and apply the mathematics of 6.EE.5 (Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true using substitution to determine whether a given number in a specified set makes an equation or inequality true). ### Criterion 2e - 2i Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs). 3/10 + - Criterion Rating Details The materials reviewed for Math in Focus: Singapore Math Course 1 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 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. 0/2 + - Indicator Rating Details The materials reviewed for Math in Focus: Singapore Math Course 1 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. Students have limited opportunities to make sense of problems and persevere in solving them in connection to grade-level content, identified as mathematical habits in the materials. Student materials do not provide guidance, teacher guidance is repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP1. Examples include: • In Chapter 1, Put On Your Thinking Cap! Problem 1, page 32, students solve real-world problems finding the greatest common factor of two whole numbers. The problem states, “Mr. Williams wants to create a rectangular patio in his backyard using square tiles. He does not want to cut any tiles. His patio measures 144 inches by 108 inches. Find the fewest square tiles he can use. (Hint: First, find the the largest size tile he can use.)” Teacher guidance states, “You may want to guide students on applying the various heuristics using the problem-solving heuristics poster. Refer students to the corresponding teacher resources for prompts and worked out solutions.” Teacher guidance gives a generic reference to have students use the heuristics poster which is repeated throughout the materials. • In Chapter 3, Fractions and Decimals, Put On Your Thinking Cap! Problem 1, page 151, students solve real-world problems involving decimals and fractions. The problem states, “Aidan, Grace, and Julia raised a sum of money for a charity. Aidan raised 0.7 of the sum of the money. Grace and Julia raised the rest of the money. If Grace raised $$\frac{5}{12}$$ of the money raised by both her and Julia, and Julia raised$847, how much did Aidan raise?” Teacher guidance states, “You may want to guide students on applying the various heuristics using the problem-solving heuristics poster. Refer students to the corresponding teacher resources for prompts and worked out solutions.” Teacher guidance gives a generic reference to have students use the heuristics poster which is repeated throughout the materials.

• In Chapter 8, Equations and Inequalities, Put On Your Thinking Cap! Problem 2, page 99, students solve real-world problems using equations and inequalities. The problem states, “The price of a box of Brand A crackers was x percent more than the price of a box of Brand B. Sydney paid $55.20 for some boxes of Brand A crackers. If she bought the same number of boxes of Brand B crackers instead, she would pay$7.20 less. What was the value of x?” Teacher guidance states, “Go through the problem using the four-step problem-solving method. Challenge students to describe how this question is similar to those they encountered earlier in the chapter and identify how it is different.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials.

Materials identify focus Mathematical Habits for MP1 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:

• Section 3.2, Real-World Problems: Fractions, is noted as addressing MP1 on pages 103-120 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make sense of problems and persevere in solving them are provided.

• Section 8.3, Real-World Problems: Equations, is noted as addressing MP1 on pages 77-82 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make sense of problems and persevere in solving them are provided.

Students have limited opportunities to reason abstractly and quantitatively in connection to grade-level content, identified as mathematical habits in the materials. Student guidance is not provided in the materials, teacher guidance is repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP2. Examples include:

• In Section 2.2, Negative Numbers, Independent Practice, Problem 36, page 67, students answer a question about opposites. The problem states, “Is the opposite of a number always negative? Explain your answer.” Teacher guidance states “assesses students' understanding of opposites.” This problem is not aligned with MP2, as students do not explain what the numbers or symbols in an expression or equation represent. Nor do students consider units involved in a problem and attend to the meaning of quantities.

• In Section 7.5, Real-World Problems: Algebraic Expressions, Independent Practice, Problem 6, page 40, students solve real-world problems involving algebraic expressions. The problem states, “Uriah has (2x - 1) one-dollar bills and (4x + 2) five-dollar bills. Van has 3x dollars more than Uriah. a) Find the total amount of money that Uriah has in terms of x. b) Find the number of pens that Uriah can buy if each pen costs 50 cents. c) If x = 21, find how much money Van will have now if Uriah gives her half the number of five dollar bills that she has.” Teacher guidance states, “Assesses students’ ability to solve real-world problems involving algebraic expressions.” Teacher guidance gives a generic reference to have students review strategies learned in the chapter which is repeated throughout the materials. Students are not asked to explain or discuss what the numbers or symbols in the expression represent.

• In Chapter 11, Surface Area and Volume of Solids, Math Journal, page 251, students reason abstractly and quantitatively about volume. The materials state, “Devin thinks that the volume of Cube A is twice the volume of Cube B since the edge length of Cube A is twice the edge length of Cube B. Explain why Devin is incorrect.” A visual representation of Cube A, with side length A, and Cube B, with side length B, is provided. Teacher guidance states, “Review with students the various strategies learned in this chapter. Encourage students to build a representation using unit cubes. Some students may benefit from the use of concrete models to guide their thinking. Challenge students to express any mathematical patterns or relationships they can identify using mathematical expressions.” Teacher guidance gives a generic reference to have students review strategies learned in the chapter which is repeated throughout the materials.

Materials identify focus Mathematical Habits for MP2 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:

• Chapter 5, Rates and Speed, Chapter Wrap-Up, is noted as addressing MP2 on pages 294-302 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to reason abstractly and quantitatively are provided.

• Section 8.1, Solving Algebraic Equations, is noted as addressing MP2 on pages 57-66 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to reason abstractly and quantitatively are provided.

### Indicator 2f

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.

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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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 have the opportunity to construct viable arguments in connection to grade-level content leading to the intentional development of MP3, identified as mathematical habits in the materials. However, the teacher guidance is often repetitive, not specific, and often distracts from the intentional development of MP3. Examples include:

• In Lesson 2.2: Number Lines and Negative Numbers, Independent Practice, Problem 38, page 68, students construct viable arguments to understand negative numbers. The problem states, “Your friend says that the statement 0 < -15 is correct. Explain why this statement is correct.” Teacher guidance states, “Assess students’ understanding of negative numbers and ability to compare numbers using > or <.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument.

• In Section 13.6, Real-World Problems: Measures of Central Tendency and Variability, Independent Practice, Problem 2, page 371, students construct viable arguments about which measure of central tendency to use to solve real-world problems. The problem states, “The table shows the results of a survey carried out on 80 families. a) Find the mean, median, and mode. b) Which measure of central tendency best describes the data set? Explain your answer.” A table showing the number of children and number of families is provided. Teacher guidance states, “Assess students’ ability to decide which measure of central tendency to use to solve real- world problems.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument.

Students have the opportunity to critique the reasoning of others in connection to grade-level content leading to the intentional development of MP3, identified as mathematical habits in the materials. Examples include:

• In Section 3.1, Dividing Fractions, Math Talk, Page 99, students critique the reasoning of others when dividing fractions. The materials state, “Connor’s solution to $$\frac{6}{7} ÷ \frac{3}{8}$$ is as shown. Do you agree with Connor’s solution? Why? Explain how you would divide a fraction by a fraction.” Teacher Guidance states, “Fraction models help students to visualize dividing a fraction by a fraction, but ultimately they should apply the rule of using the reciprocal of the divisor, instead of the dividend. In pairs, have students discuss if they agree with Connor’s solution. Invite students to share their strategies to divide a fraction by a fraction without using fraction circles. Be sure students know the mathematics behind dividing by a fraction and cross-simplification.”

• In Section 4.1, Comparing Two Quantities, Try, Problem 2, page 170, students critique the reasoning of others when writing ratios to compare two quantities with the same unit. The problem states, “Ms. Clark buys 4 bags of apples and 7 bags of oranges. Each bag has an equal number of fruits. The ratio of the number of apples to the number of oranges is ____:____. The number of apples is ____ the number of oranges.” Math Talk note states, “Zoe says that for every 7 oranges, there are 4 apples. Is she correct? Why?” Teacher guidance states, “Is Zoe correct? Why? Reinforce students’ understanding of ratios by using a visual to illustrate the situation.”

Math Journal Activities provide opportunities for students to engage in the intentional development of MP3. Examples include:

• In Chapter 6, Percent, Math Journal, page 341, students construct viable arguments about writing percents. The problem states, “Sara and Kyle checked some books out of the library. 20% of the books Sara checked out were fiction books, and 40% of the books Kyle checked out were fiction books. Your friend thinks that Kyle checked out more fiction books than Sara. Explain the error in your friend’s thinking. Use an example to support your reasoning.” Teacher guidance states, “Review with students the key concepts learned in this chapter. Encourage students to write down an example to support their reasoning. You may want to pose these questions to students who are struggling to reason out their answers: Can your friend be correct? Can he or she be wrong too? Can you think of more examples to support your reasoning?”

• In Chapter 7, Algebraic Expressions, Math Journal, page 41, students construct viable arguments as they simplify expressions. The materials state, “William simplified the two expressions shown. 10w - 5w + 2w = 10w - 7w = 3w 12 + 24x = 12(2x). Are William’s answers correct? If not, explain why they are incorrect?” Teacher guidance states, “Review with students the various strategies learned in this chapter. Encourage students to use multiple strategies (visuals and/or substitution) to determine the correctness of the situation. Walk among the students to provide support as needed. You may want to post these questions to students who are struggling to reason out their answers: ‘How can we determine if William is correct? What can we say to William to help him identify his error?’ You may also want to provide visuals or encourage students to use models to support their reasoning.”

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:

• Section 5.4, Average Speed, is noted as addressing MP3 on pages 271-176 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to construct arguments or critique the reasoning of others is provided in the lesson.

• Section 6.2, Fractions, Decimals, and Percents, is noted as addressing MP3 on pages 315-322 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to construct arguments or critique the reasoning of others is provided in the lesson.

### Indicator 2g

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.

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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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.

Students have limited opportunities to model with mathematics in connection to grade-level content, identified as mathematical habits in the materials. Additionally, MP4 is referred to as “use mathematical models” in the student and teacher materials. Students are told which models to use and teacher guidance is often repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP4. Examples include:

• In Chapter 2, Put On Your Thinking Cap!, Problem 1, page 75, students identify integer opposites on a number line. The problem states, “You can interpret a negative sign in front of a number as meaning ‘the opposite of’. So, -3 means the opposite of 3. a) What number is -(-3) the opposite of? b) What number is -(-3) equal to? Draw a number line to explain your answers.” Teacher guidance states, “Requires students to identify opposites of negative numbers and provides a preview for considering integer operations.” Students are told what type of model to use (number line) rather than independently modeling mathematics, and the mathematics is not connected to everyday life.

• In Section 3.1, Dividing Fractions, Activity, Problem 1, page 94, students work with a partner to divide a proper fraction by a unit fraction. The problem states, “1) Fold a rectangle strip of paper into quarters. Use your model to find how many quarters there are in $$\frac{1}{2}$$. Then find, $$\frac{1}{2}$$ ÷ $$\frac{1}{4}$$. 2) Ask your partner to find $$\frac{1}{2}$$ × 4. 3) Compare your answers in 1 and 2. What do you notice?” 4) Trade places. Repeat 1 to 3 to find how many eighths there are in $$\frac{1}{2}$$.” Teacher guidance states, “Let’s work in pairs to divide proper fractions by a unit fraction using paper strips. Provide each student with a copy of Paper Strips. Ask students to use a paper strip to show the number of quarters in one-half. How many quarters are in 1 whole? How does the answer help us find the number of quarters in $$\frac{1}{2}$$. Students are told what type of model to use (paper strips) rather than independently modeling mathematics, and the activity is heavily scaffolded/guided.

• In Chapter 7, Algebraic Expressions, Put On Your Thinking Cap! Problem 2, page 41, students write algebraic expressions using bar models. The problem states, “Amanda is twice as old as Xavier. Cole is three times as old as Xavier. If Cole is y years old, write an expression of their total ages 5 years ago. Draw a bar model to explain your answer.” Teacher guidance states, “Requires students to persevere in solving problems. Go through the problem using the four-step problem-solving method. Discuss with students how they can represent the situation with a bar model. You may want to ask them to consider the total number of equal parts.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials, and students are instructed on what type of model to create (bar model).

Materials identify focus Mathematical Habits for MP4 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:

• Section 2.1, The Number Line, is noted as addressing MP4 on pages 45-59 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to model mathematics are provided.

• Chapter 3, Fractions and Decimals, Chapter Wrap-Up and Performance Task are noted as addressing MP4 on pages 152-164 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to model mathematics are provided.

Students do not have the opportunity to use appropriate tools strategically. Materials identify focus Mathematical Habits for MP5 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are never intentionally addressed in the activities and problems. Examples include:

• Section 1.1, Prime Factorization, is noted as addressing MP5 on pages 159-172 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.

• Section 2.2, Negative Numbers, is noted as addressing MP5 on pages 57-68 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.

• Section 10.5, Area of Triangles, is noted as addressing MP5 on pages 159-172 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.

• Section 11.1, Prisms and Pyramids, is noted as addressing MP5 on pages 215-222 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.

• Section 11.3, Volume of Rectangular Prisms, is noted as addressing MP5 on pages 237-242 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.

### Indicator 2h

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.

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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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.

Students have some opportunities to attend to precision and the specialized language of mathematics in connection to grade-level content, identified as mathematical habits in the materials. However, little to no student guidance is provided. Examples include:

• In Chapter 1, Whole Numbers, Prime Numbers, and Prime Factorization, Math Journal, Problem 2, page 31, students have the opportunity to attend to precision as they find factors of numbers. The problem states, “Write two statements to relate any two of the numbers. 70, 4,900, 343,000.” Teacher guidance states, “Monitor student’s methods. For those who are choosing calculation as their approach, they may struggle to recognize the relationship between the given numbers. For students getting lost in the computation, it might be helpful to prompt them to think of 7 as 7 tens, 4,900 as 49 hundreds, and 343,000 as 343 thousands when generating known factors and numerical relationships.”

• In Section 3.1, Dividing Fractions, For Language Development, TE page 92, attends to the specialized language of mathematics as teachers are guided to, “Discuss the meaning of inverse operations: the root of inverse is invert, which means to turn something upside down or reverse positions. So inverse operations reverse each other: Multiplication by a number reverses division by that number. Then write $$\frac{2}{3}$$ and tell students that these fractions are reciprocals. Point out that the reciprocal of a fraction is the fraction with the numerator and denominator inverted.”

Students have some opportunities to attend to precision and the specialized language of mathematics in connection to grade-level content, identified as mathematical habits in the materials. However, there is no student guidance and teacher guidance is repetitive and not specific, preventing intentional development of the full intent of MP6. Examples include:

• In Section 1.3, Squares and Cubes, Independent Practice, Problem 11b, page 27, students have the opportunity to attend to precision as they evaluate expressions. The problem states, “Find the value of each expression. Describe any pattern(s) you see. $$3^2$$ - $$2^2$$.” Teacher guidance states, “requires students to evaluate numerical expressions containing exponents and describe patterns using mathematical language.” Neither teacher guidance nor student directions prompt students to attend to the precision of mathematics.

• In Section 4.1, Comparing Two Quantities, Independent Practice, Problem 16, page 182, students have the opportunity to use the specialized language of mathematics to describe ratios. The problem states, “Describe a situation that each ratio could represent 5:16.” Teacher guidance states, “Assess students’ ability to write two ratios to compare quantities.” Neither teacher guidance nor student directions prompt students to use specific mathematical terms to explain their thinking or communicate their ideas.

• In Chapter 8, Equations and Inequalities, Math Journal, Problem 1, page 97, students have the opportunity to use the specialized language of mathematics to describe a real-world situation and translate a given scenario symbolically. The problem states, “Construct a real-world problem involving an equation. Write the equation to represent this situation and solve it.” Teacher guidance states, “This journal provides opportunities for students to use precise mathematical language when describing a real-world situation and to translate a given scenario symbolically. Review with students the various strategies learned in this chapter. Encourage students to work independently to construct real-world problems involving equations and inequalities. Ask students to trade problems with one another and challenge them to solve the problems. Walk among the students to provide support as needed.” Neither teacher guidance nor student directions prompt students to use specific mathematical terms to explain their thinking or communicate their ideas.

There are some instances when the materials attend to the specialized language of mathematics; however, these lessons were not identified as aligned to MP6. Examples include:

• Section 8.4, Solution of Simple Inequalities, For Language Development, TE page 84, states, “Make sure that students understand the meaning of inequality. Point out that the prefix in- means ‘not.’ While an equation is a statement that two quantities are equal, an inequality is a statement that two quantities are not equal.”

• Section 12.2, Dot Plots, For Language Development, TE page 277, states, “Help students to understand the meaning of symmetrical and skewed. Explain that when a dot plot is symmetrical, the arrangement of dots is balanced. A vertical line drawn at the mid-point of the range has about the same number of dots on each side of the line. The two sides look like mirror images of each other. Tell students that skewed is the opposite of symmetrical. A skewed dot plot has a tall stack of dots at one end of the range and a ‘tail’ on the other.”

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

• Section 3.3, Adding and Subtracting Decimals Fluently, is noted as addressing MP6 on pages 121-126  in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to attend to precision and attend to the specialized language of mathematics are identified in the lesson.

• Section 7.1, Using Letters to Represent Numbers, is noted as addressing MP6 on pages 5-14  in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to attend to precision and attend to the specialized language of mathematics are identified in the lesson.

### Indicator 2i

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.

0/2
+
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Indicator Rating Details

The materials reviewed for Math in Focus: Singapore Math Course 1 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.

Students have minimal opportunities to look for and make use of structure in connection to grade-level content, identified as mathematical habits in the materials. There is no student guidance, teacher guidance is often repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP7. Examples include:

• In Chapter 7, Algebraic Expressions, Put On Your Thinking Cap!, Problem 1, page 41, students find perimeter. The problem states, “Find the perimeter of the figure in terms of x, given that all the angles in the figure are right angles. If x = 5.5, evaluate the expression.” Teacher guidance states, “requires students to make use of structure to solve problems. Prompts: What information can we gather from the problem? What can we do to help us solve the problem? Allow students to carry out the plan and observe if they can solve the problem. Invite volunteers to show their work on the board.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials and does not require students to look for or use structure in solving.

• In Chapter 9, The Coordinate Plane, Put On Your Thinking Cap!, Problem 2, page 143, students find missing coordinates of a parallelogram. The problem states, “ABCD is a parallelogram. The coordinates of A are (-5, -4), the coordinates of B are (2, -3), and the coordinates of D are (-3, 1). Give the coordinates of point C.” Teacher guidance states, “Requires students to use their knowledge of coordinate planes and polygons to provide clues that result in the construction of a quadrilateral. Encourage the use of precise mathematical language in the development of their clues. Explain to students that they may choose any point on the graph paper as the origin. The locations of most points are described in relationship to the origin. Support students who are having a difficult time in getting started by having them review the language used in 1.” Materials misidentify MP7 in this problem, students do not have the opportunity to look for patterns or structures to make generalizations and solve problems, or look for and explain the structure within mathematical representations.

• In Section 10.1, Area of Triangles, Math Sharing, page 168, students compare areas of various triangles. The materials state, “Look at the four triangles. What do you notice about the areas of different triangles with equal bases and heights? Discuss.” Teacher guidance states, “Using a method of their choice, challenge students to compare the bases, heights and areas of the given triangles.” Additionally, the problem lends itself to MP5.

Materials identify focus Mathematical Habits for MP7 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:

• Section 5.2, Real-World Problems: Rates and Unit Rates, is noted as addressing MP7 on pages 253-260 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make use of structure are provided in the lesson.

• Section 7.4, Expanding and Factoring Algebraic Expressions, is noted as addressing MP7 on pages 30-34 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make use of structure are provided in the lesson.

Students have one opportunity to look for and express regularity in repeated reasoning in connection to grade-level content, identified as mathematical habits in the materials. There is no student guidance and teacher guidance is repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP8. According to the Standards for Mathematical Practice Guide, students have one opportunity to engage in MP8:

• In Section 13.1, Mean, Independent Practice, Problem 16, page 324, students find missing numbers for the mean. The problem states, “Find five different numbers whose mean is 12. Explain your reasoning.” Teacher guidance states, “Assess students’ ability to find the total and a missing number from the mean.”

## Usability

### Criterion 3a - 3h

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

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.

N/A

### Indicator 3b

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.

N/A

### Indicator 3c

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

N/A

### Indicator 3d

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.

N/A

### Indicator 3e

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

N/A

### Indicator 3f

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

N/A

### Indicator 3g

This is not an assessed indicator in Mathematics.

N/A

### Indicator 3h

This is not an assessed indicator in Mathematics.

N/A

### Criterion 3i - 3l

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

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

N/A

### Indicator 3j

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.

N/A

### Indicator 3k

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

N/A

### Indicator 3l

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

N/A

### Criterion 3m - 3v

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

### Indicator 3m

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

N/A

### Indicator 3n

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

N/A

### Indicator 3o

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.

N/A

### Indicator 3p

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

N/A

### Indicator 3q

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.

N/A

### Indicator 3r

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

N/A

### Indicator 3s

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

N/A

### Indicator 3t

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

N/A

### Indicator 3u

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

N/A

### Indicator 3v

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

N/A

### Criterion 3w - 3z

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

### Indicator 3w

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.

N/A

### Indicator 3x

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

N/A

### Indicator 3y

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

N/A

### Indicator 3z

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

N/A
abc123

Report Published Date: 2021/10/25

Report Edition: 2020

Title ISBN Edition Publisher Year
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

## Math K-8 Review Tool

The K-8 review criteria identifies the indicators for high-quality instructional materials. The review criteria supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our review criteria evaluates materials based on:

• Focus and Coherence

• Rigor and Mathematical Practices

• Instructional Supports and Usability

The K-8 Evidence Guides complement the review criteria by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.

Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways.

Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?

Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom.

In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.