Singapore Math: Dimensions Math (2016-2017)

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Report for 6th Grade

Alignment Summary

The instructional materials for Dimensions Math Grade 6 do not meet expectations for alignment to the CCSSM. In Gateway 1, the instructional materials do not meet the expectations for focus as they assess above-grade-level standards but do devote at least 65% of instructional time to the major work of the grade. For coherence, the instructional materials are partially coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade. In Gateway 2, the instructional materials meet the expectations for rigor and balance, but they do not meet the expectations for practice-content connections. Since the materials do not meet the expectations for alignment to the CCSSM, they were not reviewed for usability in Gateway 3.

6th Grade

Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Alignment (Gateway 1 & 2)

Partially Meets Expectations

Gateway 3

Usability

Usability (Gateway 3)

Not Rated

Overview of Gateway 1

Focus & Coherence

The instructional materials reviewed for Dimensions Math Grade 6 partially meet expectations for focus and coherence in Gateway 1. For focus, the instructional materials do not meet the expectations for assessing grade-level standards, but the amount of time devoted to the major work of the grade is at least 65 percent. For coherence, the instructional materials are partially coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade.

Gateway 1

v1.0

Gateway 1

Focus & Coherence

Criterion 1.1: Focus

Criterion 1.2: Coherence

Criterion 1.3: Coherence

Partially Meets Expectations

Criterion 1.1: Focus

Materials do not assess topics before the grade level in which the topic should be introduced.

The instructional materials reviewed for Dimensions Math Grade 6 do not meet expectations for not assessing topics before the grade level in which the topic should be introduced. The instructional materials include assessment items that align to standards above this grade level.

Indicator 1A

The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for Dimensions Math Grade 6 do not meet expectations for assessing grade-level content. The FAQ page on the website for Singapore Math states, “There are currently no tests, but the workbook could be used as a test bank.” In Dimensions Math workbooks 6A and 6B, above grade-level items are present and could not be modified or omitted without a significant impact on the underlying structure of the instructional materials. For example:

Students solve multi-step percent problems involving discount, tax, and percent increase/decrease (7.RP.3). For example, in Lesson 7.2 pages 150-151, problem 6 states, “A pair of pants is discounted 25 percent. They now cost $104.25. What was the cost before the discount?”
In Lesson 10.3, pages 58-63, students write equations in the form px + q = r (7.EE.4a). For example, problem 6 states, “The internet service at the airport costs $11 to sign on and an additional $2.50 per half hour. Let h represent the amount of time Frances used the internet, and t represent the total cost in dollars. Write an equation that represents this scenario.”

Criterion 1.1: Focus

Indicator 1A

Criterion 1.2: Coherence

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials reviewed for Dimensions Math Grade 6 meet expectations for devoting the large majority of class time to the major work of the grade. The instructional materials spend at least 65% of instructional time on the major work of the grade.

Indicator 1B

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Dimensions Math Grade 6 meet expectations for spending a majority of instructional time on major work of the grade.

The approximate number of chapters devoted to major work of the grade (including assessments and supporting work connected to the major work) is 11 out of 13, which is approximately 86 percent.
The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 28.5 out of 34, which is approximately 84 percent.
The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 107 out of 134, which is approximately 80 percent.

A lesson-level analysis (which includes lessons and sublessons) is most representative of the instructional materials because it addresses the amount of class time students are engaged in major work throughout the school year. As a result, approximately 84 percent of the instructional materials focus on major work of the grade.

Criterion 1.2: Coherence

Indicator 1B

Criterion 1.3: Coherence

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

The instructional materials reviewed for Dimensions Math Grade 6 partially meet expectations for being coherent and consistent with the Standards. The instructional materials contain supporting work that enhances focus and coherence simultaneously by engaging students in the major work of the grade and foster coherence through connections at a single grade. The instructional materials include an amount of content that is partially viable for one year, do not attend to the full intent of some standards, and do not give all students extensive work with grade-level problems.

Indicator 1C

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

The instructional materials reviewed for Dimensions Math Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Supporting standards/clusters are connected to the major standards/clusters of the grade with two exceptions. No connections are explicitly stated. Examples of supporting work that engage students in the major work of the grade include:

In Lesson 6.1, students calculate average weight, average height, and average distance (supporting standard 6.SP.5c), and these are connected to unit rates (major standard 6.RP.2).
In Lesson 10.1, students graph points to draw a polygon on the coordinate plane (supporting standard 6.G.3), and this is connected to graphing points in all four quadrants (major standard 6.NS.8).
In Lesson 1.4, students use division of multi-digit numbers (supporting standard 6.NS.2) when writing equivalent expressions and solving equations (major clusters 6.EE.A,B).
In Chapter 3, students evaluate expressions (major cluster 6.EE.A) that include multi-digit decimals (supporting standard 6.NS.3).
In Chapters 11 and 12, students evaluate expressions arising from area and volume formulas (supporting standards 6.G.1,2), and this connects to writing and solving equations for unknown lengths (major standards 6.EE.2,7).
In Lesson 13.1B, students evaluate expressions (major standard 6.EE.1) to find the mean of a data set (supporting standard 6.SP.5c).
In Lesson 13.2, students calculate percentages (major standard 6.RP.3) from analyzing histograms (supporting cluster 6.SP.B).
In Lesson 6.2, students solve problems involving unit rates (major standard 6.RP.3) by dividing multi-digit numbers (supporting standard 6.NS.2).
In Lesson 12.1, students use the ratio of length to width to height of a right rectangular prism (major standard 6.RP.3) to find the volume of the prism (supporting standard 6.G.2).

Examples of missed connections between supporting and major work include:

These is no connection between finding factors (6.NS.4) and generating equivalent expressions (6.EE.3).
In Lesson 11.1, problem 13 involves division of fractions (major standard 6.NS.1) within the context of area, surface area, and volume (supporting cluster 6.G.A). There are no other opportunities to connect 6.G.A and 6.NS.1.

Indicator 1D

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials for Dimensions Math Grade 6 partially meet expectations that the amount of content designated for one grade level is viable for one year.

As designed, the instructional materials can be completed in 134 days. The total days were computed in the following manner:

Each lesson was counted as one day of instruction.
A “lesson” with subsections (i.e., 1a, 1b, 1c) counted as three lessons or three days.
A practice day was added for each chapter.

The total days were computed based on a pacing chart provided in the teacher guide. The suggested time frame for the materials and/or the expectations for teachers and students are not viable. Some significant modifications would be necessary for materials to be viable for one school year.

Indicator 1E

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials for Dimensions Math Grade 6 partially meet expectations for being consistent with the progressions in the standards. In general, materials follow the progression of grade-level standards, though they don’t always meet the full intent of the standards. In addition, lessons utilize standards from prior grade levels, though these are not always explicitly identified in the materials.

Examples where standards from prior grades are utilized but not identified include:

In Lesson 1.1, students write numeric expressions for statements (5.OA.2). This material is not identified as content from a prior grade.
In Lesson 2.1, the materials reference Multiplication of a Proper Fraction by a Whole Number as learning from a previous grade, but the materials do not identify Multiplication of a Proper Fraction by a Fraction as previous learning. Instead, the materials treat this topic and Division of a Whole Number by a Fraction and Division of a Fraction by a Whole Number (Lesson 2.2) as grade-level topics, though they are prior learning (5.NF.4).
In Lesson 1.4, students examine division as sharing and division as grouping (3.OA.2 & 3.OA.6), but the material do not reference this as prior learning.
Lesson 3.4 is not identified as work from a prior grade (5.MD.1).

The materials typically develop according to the grade-by-grade progressions in the standards, but one missed opportunity is in the unit on The Number System (Chapters 1-3). The standards include students representing numbers on a number line, but students are not given that opportunity. The model most commonly used in this unit is the bar model. There is some emphasis on a number line with the introduction of integers to help students compare values.

The “Notes on Teaching” in Teaching Notes and Solutions provide some direction for teachers to explicitly relate the content to prior learning:

In Lesson 2.1, the lesson states, “In Grade 5, we interpreted a fraction as a division of the numerator by the denominator…let’s recap the multiplication of a fraction by a whole number.”
In Lesson 3.1B, adding and subtracting decimals are connected to prior knowledge: “Since both whole numbers and decimals are written using the base-ten number system of numbers, we can use the same method for adding and subtracting whole numbers to add and subtract decimals.”
In Lesson 11.1, the materials state, “In earlier grades, we learned the area of a rectangle can be found by multiplying the side lengths.”

Examples where the student workbooks reference content learned in earlier grades include:

In Workbook 6B, page 109 states, “In earlier grades, we learned that the area of a rectangle can be found by multiplying the side lengths. That is, area of a rectangle = length x width.”
In Workbook 6B, page 154 states, “In earlier grades, you have already come across shapes like rectangular prisms and cubes. To review, a rectangular prism is a three-dimensional solid shape.”

The instructional materials do not attend to the full intent of some standards. Examples include:

In Chapter 1, students find the GCF of two numbers but do not have an opportunity to “use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor,” as stated in 6.NS.4.
In Lesson 3.4, students convert measurements to different units, but ratio reasoning is not used for these conversions (6.RP.3d).
In Chapter 5, rate language is not used to develop the concept of ratios (6.RP.2).
In Chapters 5 and 6, ratios are not represented with tables (6.RP.3a).
Unit rate is defined in Chapter 6 on page 172, but there is no opportunity for students to “understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0” (6.RP.2).
In Chapter 8, students write algebraic expressions for given statements but do not have an opportunity to “identify parts of an expression using mathematical terms.” (6.EE.2b)
The materials do not provide an opportunity for students to “view one or more parts of an expression as a single entity.” (6.EE.2b)
In Chapter 10, students use coordinates and absolute value to find distances between points on a coordinate plane (6.NS.8) but do not apply this understanding to real-world problems.

The materials do not give all students extensive work with grade-level problems for some standards. Examples include:

In Chapter 2, students divide whole numbers, and in Chapter 3, students divide decimals by decimals using the standard algorithm. The materials do not provide opportunities for extensive work dividing multi-digit numbers using the standard algorithm (6.NS.3).
On pages 175 and 189, students examine the net of a triangular prism (6.G.4), but they do not “represent three-dimensional figures using nets made up of rectangles and triangles, and use nets to find the surface area of these figures,” as stated in the standard.

Indicator 1F

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials for Dimensions Math Grade 6 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the standards.

The materials include learning objectives that are visibly shaped by CCSSM cluster headings, and there are correlations between Dimensions Math Grade 6 learning objectives and CCSSM cluster headings. Examples include:

In Chapter 5, learning objectives are shaped by 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.”
Examples of learning objectives shaped by 6.RP.A are: “Compare quantities using ratios and use ratio language to describe a ratio relationship between two quantities; simplify ratios to obtain equivalent ratios; and relate ratios and fraction and apply ratio relationships to solve real-world problems.” In Chapter 4, some learning objectives are: “Recognize the use of positive and negative numbers in the real-world context; compare and order positive and negative numbers and plot them on a number diagram; and interpret the absolute values of a positive and negative quantity in real-world situations.” These objectives are shaped by 6.NS.C, “Apply and extend previous understandings of numbers to the system of rational numbers.”
In Chapter 2, some learning objectives are: “multiply fractions by whole numbers; multiply fractions by fractions; divide whole numbers by fractions and fractions by whole numbers; divide fractions by fractions; and solve word problems involving multiplication and division of fractions.” These objectives are shaped by 6.NS.A, “Apply and extend previous understandings of multiplication and division to divide fractions by fractions.”

The materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Examples include:

In Chapters 11 and 12, students use formulas to calculate area, volume, and surface area (6.G.A) involving measures that are given in both decimal and fraction forms (6.NS.B).
In Chapter 13, students investigate appropriate use of measures of center in different contexts (6.SP.A) and make comparisons among the three measures (6.SP.B).
In Lesson 8.1, students evaluate expressions (6.EE.A) using multiplication and division of fractions (6.NS.A).
In Lesson 9.1, students solve equations (6.EE.B) with fractional coefficients (6.NS.A).
In Lesson 1.3, students model the distributive property (6.NS.B) to solve area problems (6.G.A).
In Lesson 10.3A, students write and solve equations (6.EE.B) to describe relationships between dependent and independent variables (6.EE.C).

Students do not compare rates of two or more quantities using graphs of quantities, missing a connection between 6.NS.C and 6.RP.A.

Criterion 1.3: Coherence

Overview of Gateway 2

Rigor & Mathematical Practices

The instructional materials for Dimensions Math Grade 6 partially meet expectations for rigor and the mathematical practices. The instructional materials meet the expectations for rigor and balance by giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency and spending sufficient time working with engaging applications of the mathematics. The instructional materials do not meet the expectations for practice-content connections because they do not identify the mathematical practices, use them to enrich the content, or carefully attend to the full meaning of each practice standard.

Gateway 2

v1.0

Gateway 2

Rigor & Mathematical Practices

Criterion 2.1: Rigor

Criterion 2.2: Math Practices

Partially Meets Expectations

Criterion 2.1: Rigor

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

The instructional materials for Dimensions Math Grade 6 meet expectations for rigor and balance. The instructional materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency, spend sufficient time working with engaging applications of the mathematics, and do not always treat the three aspects of rigor together or always treat them separately.

Indicator 2A

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

The instructional materials for Dimensions Math Grade 6 partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Conceptual understanding is developed by connecting models, verbal explanations, and symbolic representations of concepts, with an emphasis on the use of bar models. The following examples illustrate how the materials develop those standards addressing conceptual understanding.

Standard 6.RP.A: Understand ratio concepts and use ratio reasoning to solve problems.

In Lesson 5.1, page 138 Class activity, students use green and red cubes to explore equivalent ratios. Students are guided through the use of the manipulatives; the activity moves into the procedure of multiplying or dividing a number by the same amount.
In Lesson 5.2, pages 145-152, students use bar models to represent and solve real-world ratio problems in examples 11 - 17.
In Lesson 6.1, page 166 Class Activity, students use cubes to understand average. In page 168 example 3, students use a bar model to solve an average problem.
In Lesson 6.2, page 175, students use division with bar models to find unit rates.
Lessons 7.1 and 7.2 use bar models and 100s grids to develop the meaning of percent. In Class Activity 1, students model fractions, decimals, and percents on 100s grid. In Class Activity 2, students use bar models to represent percentages.

However, the materials do not include work with ratio tables (6.RP.3a).

Standard 6.EE.3: Apply the properties of operations to generate equivalent expressions.

In Lesson 8.1 page 2, a table connects expressions, verbal descriptions, and tiles as a model to help students understand an expression. In the examples that follow, students are not asked to create models to represent the expressions.
In Lesson 8.2, page 20 Class Activity 2, students play a game discussing why two terms are or are not like terms. The first two examples use a diagram to show why two terms are like terms.
In Lesson 9.1B, page 33 Class Activity 1, students are given a visual model of a scale to demonstrate adding and subtracting the same quantity to both sides of an equation. Examples 4-14 on pages 35-42 show bar models as a visual method of solving the equations.

6.EE.5 Understand solving an equation or inequality.

In Lesson 9.2, page 50 BrainWorks #13, students write an inequality to represent the number of days Rachel needs to clear orders for 30 cakes and graph the solution. They then determine how the inequality and graph will change if Rachel must clear the cake orders in eight days or less.

6.NS.A: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

In Chapter 2, pages 56-62, examples of bar models are provided to model problems involving both measurement and partitive division, extending previous understandings of multiplication and division to divide fractions by fractions. Students see how dividing a fraction by a fraction is equivalent to multiplying the fraction by the reciprocal.
Exercise 3.2, page 85 Multiplication of Decimals: “Without doing any calculations, determine which of the following multiplication expressions will give a product that is larger than ___? Explain your answers.”
6A Workbook, page 88 Division of Decimals: “Without doing any calculations, determine which of the following division expressions will have a quotient that is less than the dividend. Explain your answer.”

Indicator 2B

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials for Dimensions Math Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. The instructional materials develop procedural skills and fluencies and provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level.

Examples of the materials developing procedural skills and fluencies throughout the grade level include:

Chapter 3 addresses adding, subtracting, multiplying, and dividing multi-digit decimals using the standard algorithm for each operation (6.NS.3). Each lesson utilizes the standard algorithm and contains fluency practice in the lesson through Basic Practice and Further Practice in the Exercises. For example, in Lesson 3.1, #3 includes addition and subtraction problems written in a vertical format, #4 and #7 provide questions written in a horizontal format, and #9-12 students use the standard algorithm to solve problems.
Chapter 8 addresses writing and evaluating numerical and algebraic expressions (6.EE.1,2). In Lesson 8.1, there are several examples that demonstrate the procedural skills for writing and evaluating numerical and algebraic expressions, and there are several opportunities for students to practice these skills.

Examples of students demonstrating procedural skills and fluencies independently include:

In Chapter 6, students divide multi-digit numbers to calculate rates and solve problems involving percentages (6.NS.2). Students also find the average of sets of numbers containing decimal values (6.NS.3) (pages 167-170) and compute with decimals as they calculate rates (pages 173-177).
In Chapter 9, students solve equations of the form px = q involving multi-digit numbers and decimals (6.NS.2,3; 6.EE.7).
In Chapters 11 and 12, students divide multi-digit numbers and perform operations with decimals as they find area, volume, and missing side lengths of two- and three-dimensional figures (6.NS.2,3; 6.EE.2c).

Indicator 2C

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

The instructional materials for Dimensions Math Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single- and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

Opportunities for students to engage in application and demonstrate the use of mathematics flexibility primarily occur in Math@Work and Brainworks included with each lesson. Additionally, the Problem Solving Corner included with some chapters engages students in application problems. Some of the Extend Your Learning Curve problems included in the chapter reviews are intended as non-routine problems (page 1 of Teacher Guide 6B and page 27 of Book B).

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level.

Basic Practice tasks tend to be single-step, routine application problems. For example, in Lesson 10.3 Basic Practice #2, students identify the independent variable and the dependent variable in given scenarios (6.EE.9).
Further Practice tasks tend to include language and/or types of numbers that increase the level of complexity, or the task may require more than one step. For example, in Lesson 2.2 Further Practice #3, students evaluate expressions involving multiple operations using the order of operations (6.EE.1).
In BrainWorks tasks, students often make decisions and explain the decisions that are made. Many of these problems are non-routine because they aren’t similar to an example presented earlier in the materials, and they can usually be solved in a variety of ways. For example, in Lesson 7.2 BrainWorks #15, students choose between two options regarding pocket money and explain their choice (6.RP.3c).

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts.

In Lesson 2.2, students determine how many fractional lengths of ribbons could be cut from a whole- number length (Book A, page 52 example #9) and how to divide paint into jars in fractional amounts (Book A, page 62 example #16) (6.NS.1).
In Lesson 6.2, Class Activity 2 on page 171, students apply unit rates (6.RP.3) throughout the lesson and in the Problem Solving Corner on pages 183-187.
In Lesson 12.2, Math@Work, students find the surface area of rectangular and triangular prisms in the contexts of determining how much paint is needed to cover a cube and how much fabric is needed to create a tent, respectively (6.G.4).

Indicator 2D

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials for Dimensions Math Grade 6 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 the program materials. Examples include:

Conceptual understanding is generally developed through the Class Activities, examples, and corresponding Try It! tasks. For example, students use pictures and data tables to determine various ratios in Try It! tasks in Chapter 5 on pages 132-136.
Procedural skill and fluency is treated independently in the Basic Practice and Further Practice problems of every chapter. For example, in Chapter 2 page 63, students evaluate given expressions by dividing a fraction by a fraction. In Chapter 3 page 94, students divide decimals by either another decimal or a whole number.
Application is developed independently in Further Practice and Math@Work that appear at the end of each section. For example, Lesson 6.1 Problem #11 states, “Three students are going to watch a movie. Each has a whole number of dollars, and no student has more than $20. If the average amount is $16, what is the smallest possible amount one of them could have?”
Chapter 11 Area of Plane Figures contains the three aspects of rigor treated in a single lesson but taught on separate days. Lesson 11.1 starts with Class Activity 1 on page 112, where students are guided through an activity to develop conceptual understanding of the area of parallelograms and “derive the formula for the area of a parallelogram.” This is followed by examples and Try It! problems where students find the area of parallelograms, developing procedural fluency. Finally, students “consolidate and extend the material covered thus far,” by applying the mathematics to problems like the BrainWorks problem on page 122, finding the area of the yellow parallelogram in the Republic of Congo flag. This lesson is expected to take three days.

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

In Lesson 8.1B, Class Activity 1 page 7, students represent the number of toothpicks needed for a given amount of squares. After completing a table, they write an algebraic expression that would give the number of toothpicks for n squares (conceptual understanding). They use the algebraic expression to find the number of toothpicks needed to make four different types of squares (application and procedural skill).
In Lesson 8.1C, students create a table of values and use bar models to represent real-world problems algebraically, which integrates application, conceptual understanding, and procedural skills used together.
In Lesson 3.2, students integrate procedural skill with application by determining the total cost of 12.8 pounds of wire that cost $23.16 per pound.

Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials for Dimensions Math Grade 6 do not meet expectations for practice-content connections. The instructional materials prompt students to construct viable arguments and analyze the arguments of others, and they partially assist teachers in engaging students to construct viable arguments and analyze the arguments of others and explicitly attend to the specialized language of mathematics.

Indicator 2E

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for Dimensions Math Grade 6 do not meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Mathematical Practices are not identified in the materials. In the Syllabus, page 39, Mathematical Processes (Reasoning, Communication and Connections, Application and Modeling, and Thinking Skills and Heuristics) are identified, but these are not referenced in the remainder of the materials. Teachers are not provided with guidance or directions for how to carry out lessons to ensure students are developing the mathematical processes.

Examples where the Mathematical Practices are not identified and do not enrich the mathematics content include:

For MP4, students use physical models in problems. For example, in Chapter 6 Section 6.1, Class Activity 1, students use blue and yellow blocks to model averages for player A and player B. However, students do not represent the situation mathematically with an equation or a method that would help them generalize information to draw conclusions.
For MP5, students are directed which tools to use in problems, and students do not discuss which tools to select or use strategically. The instructional materials show different methods for solving problems, but students do not choose which method to use or which method would be most appropriate for problems.
For MP7, the materials do not identify looking for and making use of structure. For example, in Chapter 1 BrainWorks Exercise #8, Daniel makes use of structure to write $3^5$ as an equivalent expression for $3^2\times3^3$ . However, the materials do not identify the use of structure or provide guidance for teachers as to how MP7 could be used.

Indicator 2F

Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for Dimensions Math Grade 6 do not meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. The materials do not attend to the full meaning of three MPs.

For MP4 Model with mathematics, students solve real-world, contextual problems, but students do not model with mathematics in those problems. Examples of the materials not attending to the full meaning of MP4 include:

In Student Workbook 6A, Lesson 1.4, students “draw a model and equation to match” for two real-world problems. On page 32 problem 3, students are prompted to “draw a model and solve” for a real-world problem. In these problems, there are no opportunities for students to revise initial assumptions or models once calculations have been made.
In Lesson 2.1C, examples 14-16 show students how to use bar models to solve real-world problems and write the solution mathematically from the models. Problems like this are also encountered in Lessons 5.2 and 7.2. In these problems, students do not make assumptions, define quantities, or choose what model to use, and there are no opportunities for students to revise initial assumptions or models once calculations have been made.
In Lesson 8.1C, students complete an example by using a given table to model the relationship in age between two children, creating an expression from the table, and using the expression to determine the age of one child. In Try It! on page 14, students complete a similar problem on their own. In this problem, students do not make assumptions, define quantities, or choose which model to use.

For MP5 Use appropriate tools strategically, the materials rarely demonstrate the use of tools to solve problems, other than a tape diagram. The instructional materials do not introduce and engage students in the use of various tools, including technology. Examples of the materials not attending to the full meaning of MP5 include:

In Lesson 2.1, students are shown how to use a bar model as a mathematical tool for solving a problem involving multiplication of fractions. Other tools are not introduced or used, and students do not choose which tool to use.
In Lesson 10.2, students are shown a number line to help define absolute value in the introduction. Further examples in the lesson show distance on a coordinate plane through the use of absolute value, but students do not choose which tools to use to find distances.
In Lesson 12.2B, students are shown a net of a triangular prism, but students do not use nets as a tool for finding surface area in subsequent examples in the lesson.
Students do not use ratio tables or number line diagrams in problems with ratios, rate, or percentages. They are shown tape diagrams as a tool for working with ratios, but students do not choose the tool.

For MP8 Look for and express regularity in repeated reasoning, there are limited opportunities for students to examine repeated calculations and look for general methods and/or shortcuts. Examples where students do not engage in MP8 include:

In Lesson 2.2, the materials demonstrate how to divide a whole number by a fraction, draw a model to represent “how many 1/3’s and 2/3‘s are in 6,” and complete a table by using the reciprocal of the divisors to write equivalent multiplication expressions. Students are asked to: “(a) Look at the patterns in the divisors and the quotients. What happens to the quotient as the divisor gets smaller? (b) What do you notice about the quotients of the division expressions and the products of the equivalent multiplication expressions?” However, students are given a summary of the activity showing the generalization, “Dividing a whole number by a fraction is the same as multiplying by its reciprocal,” as an algebraic expression.
In Lesson 3.2, the materials demonstrate what happens to a rational number written in decimal form as it is multiplied by powers of 10. Students relate their results to place value. Students do not look for and express regularity in repeated reasoning. In example 5 on page 81, students determine which decimal factors produce products of a given size, but the remainder of the lesson includes teacher-led explanations of the repeated reasoning rather than students engaging in the mathematical practice.
On page 94, students divide rational numbers written in decimal form, but none of the divisions result in repeating decimals, which means students do not engage in MP8. In BrainWorks question 16 on page 94, students compare two different students’ reasoning about the value of repeating decimals, but they do not look for and express regularity in repeated reasoning themselves.

Indicator 2G

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

Indicator 2G.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Dimensions Math Grade 6 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The student materials include questions for discussion in the margins where students explain their thinking, and the teacher materials indicate students should discuss their explanations with a partner or in a group. The materials contain some problems where students are specifically asked to justify a claim with mathematics, build a logical progression of statements to explore the truth of a conjecture, or analyze situations by breaking them into cases. Examples include:

In Chapter 1 page 4: “Which is greater: Two to the third power or three to the second power? Explain.”
In Chapter 1 page 15: “Are 72 and 96 common multiples of 4 and 6? Explain.”
In Lesson 5.1, problem 15 states, “A bag contains some red balls and blue balls. The ratio of the number of red balls to the number of blue balls is 4:7. If the total number of balls in the bag is not more than 40, what are the possible numbers of blue balls in the bag? Explain how you find the answers using equivalent ratios.”
In Chapter 7 page 198, students explain whether they would want 10 percent of $20 or 20 percent of $10.
In Chapter 13 page 199: “Can we arrange the data in descending order instead of ascending order? Why?”

The student materials include problems in each chapter for students to critique someone else’s work or explanation. The Workbooks contain additional similar tasks. Examples include:

In Chapter 3 page 101 Write in Your Journal, students determine if 6.5 x 10 = 6.50 is a student’s correct application of the rule, “Add a zero when you multiply by 10,” and explain their reasoning.
In Workbook 6A, Lesson 1.1B page 7, students explain Leo’s error in evaluating $6^3$ as 18.
Chapter 7 page 209 states: “A shop owner sold 10 cell phones and made a total gain of 20 percent. What was her profit for each cell phone? A student solved this question as follows: 10 phones → 20 percent, 1 phone → 20%/10 = 2 percent. Her profit for each cell phone was 2 percent. Is this solution correct? Explain.”

Indicator 2G.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for Dimensions Math Grade 6 partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The instructional materials provide little assistance to teachers in engaging students in constructing viable arguments and analyzing the arguments of others, and the assistance that is provided is general in nature.

In the Syllabus page 13, the Mathematics Framework describes five components that make up Problem Solving: Attitudes, Metacognition, Processes, Concepts, and Skills. Under Processes, the syllabus describes how “in the context of mathematics, reasoning, communication, and connections take on special meanings.” On page 16, there is a section that describes ways “to help teachers focus on these components in their teaching practice,” and on page 18, the Syllabus states, “To support the development of collaborative and communication skills, students must be given opportunities to work together on a problem and present their ideas using appropriate mathematical language and methods.”

On page 39, the Syllabus specifies that Reasoning, Communication, and Connections include:

Using appropriate representations, mathematical language (including notations, symbols and conventions), and technology to present and communicate mathematical ideas;
Reasoning inductively and deductively, including: explaining or justifying/verifying a mathematical solution/statement; drawing logical conclusions; making inferences; and writing mathematical arguments; and
Making connections within mathematics and between mathematics and the real world.

The syllabus does not give specific direction to teachers about creating these opportunities for students, and this information is not found in any of the other materials besides the Syllabus.

Within the remainder of the instructional materials, there are no prompts, suggested questions, or frameworks for teachers suggesting ways to engage students in constructing viable arguments and/or analyzing the arguments of others. There is no guidance for teachers as to what constitutes a viable mathematical argument, such as the use of definitions, properties, counterexamples, cases, or if-then statements, and there is no guidance for analyzing the arguments of others, such as repeating or restating to check for understanding, asking clarifying questions, or building on a previous idea.

The teacher materials contain some directions to engage students in discussions, but there is no guidance for teachers on constructing viable arguments or analyzing the arguments of others. Examples of directions for teachers to engage students in discussions include:

On page 6, “Have students talk about DISCUSS boxes with a partner or group.”
On page 17, “Have them solve the problem and share and discuss their solutions.”
On pages 28 and 54, “Have students work together with a partner or in groups . . . then compare solutions with their partner or group. If they are confused, they can discuss together.”
On page 60, “Have students study examples 18-20 and do Try It! 18-20 on their own, then compare their solutions with partners or in a group.”

Indicator 2G.iii

Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for Dimensions Math Grade 6 partially meet expectations for explicitly attending to the specialized language of mathematics.

In general, the materials accurately use numbers, symbols, graphs, tables, and mathematical vocabulary. However, the materials do include some vocabulary and definitions that are not consistent with the CCSSM. Examples include:

In Lesson 1.1, “Remark” on page 2, the following information is given: “There will be a convention that when a difference between two numbers is asked for, it will be the larger minus the smaller unless otherwise specified. For division, the quotient of two numbers will be the larger divided by the smaller unless otherwise specified.” This may reinforce a common misconception that division is always the larger number divided by the smaller number.
“Simplest form” is used in Chapter 2 with fractions and in Chapter 5 with ratios, but it is not used in the CCSSM.
In Student Workbook 6A problem 6 page 15, “Cora and Alyssa solved the expression independently and got different solutions: $6 + 3\times 6 \div2 + 4$ . Cora says that the solution is 19. Using the Order of Operations convention, which girl is correct? What was the other girl thinking?” The term convention is used once in the Remark section on page 3, but a formal definition or explanation of convention is not provided.
Simplify is used throughout Chapter 8, but it is not used in CCSSM.

Criterion 2.2: Math Practices

Overview of Gateway 3

Usability

This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two

Gateway 3

v1.0

Gateway 3

Usability

Criterion 3.1: Use & Design

Criterion 3.2: Teacher Planning

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

Criterion 3.5: Technology

Not Rated

Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

Indicator 3A

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

Indicator 3B

Design of assignments is not haphazard: exercises are given in intentional sequences.

Indicator 3C

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

Indicator 3D

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

Indicator 3E

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

Criterion 3.1: Use & Design

Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

Indicator 3F

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

Indicator 3G

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

Indicator 3H

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Indicator 3I

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Indicator 3J

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

Indicator 3K

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Indicator 3L

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

Criterion 3.2: Teacher Planning

Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

Indicator 3M

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

Indicator 3N

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Indicator 3O

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Indicator 3P

Materials offer ongoing formative and summative assessments:

Indicator 3P.i

Assessments clearly denote which standards are being emphasized.

Indicator 3P.ii

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Indicator 3Q

Materials encourage students to monitor their own progress.

Criterion 3.3: Assessment

Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

Indicator 3R

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Indicator 3S

Materials provide teachers with strategies for meeting the needs of a range of learners.

Indicator 3T

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Indicator 3U

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

Indicator 3V

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Indicator 3W

Materials provide a balanced portrayal of various demographic and personal characteristics.

Indicator 3X

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

Indicator 3Y

Materials encourage teachers to draw upon home language and culture to facilitate learning.

Criterion 3.4: Differentiation

Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

Indicator 3AA

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

Indicator 3AB

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

Indicator 3AC

Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

Indicator 3AD

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

Indicator 3Z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Criterion 3.5: Technology

Downloadable Resources

2016-2017