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Report Overview
Summary of Alignment & Usability: Math Expressions, Grade 6 | Math
Math 6-8
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations for alignment to the CCSSM. The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and the materials partially connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). Since the instructional materials partially meet expectations for alignment, they were not reviewed for usability in Gateway 3.
6th Grade
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Report for 6th Grade
Alignment Summary
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations for alignment to the CCSSM. The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials partially meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and the materials partially connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). Since the instructional materials partially meet expectations for alignment, they were not reviewed for usability in Gateway 3.
6th Grade
Alignment (Gateway 1 & 2)
Usability (Gateway 3)
Overview of Gateway 1
Focus & Coherence
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focusing on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.
Gateway 1
v1.0
Criterion 1.1: Focus
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for not assessing topics before the grade level in which the topic should be introduced. The materials assess grade-level content and, if applicable, content from earlier grades.
Indicator 1A
The instructional materials reviewed for Math Expressions Grade 6 meet expectations that they assess grade-level content.
The assessments are aligned to grade-level standards and do not assess content from future grades. The Grade 6 Assessment Guide includes a Beginning of Year Test, Middle of Year Test, End of Year Test, and tests for each Unit. Each Unit Test includes multiple choice, multiple-select, short answer, constructed response, and a separate performance task assessment. The materials include a form A and form B assessment for each unit.
Digitally available assessments are PARCC and Smarter Balanced aligned practice tests. Each digital platform includes a variety of practice tests. Digital assessments assess grade-level content.
Examples of on-grade level assessment items include:
Unit 2, Form B, Item 8, “The coordinates of four points are A(1,8), B(9,6), C(9,3), and D(1,5). Part A: Plot and label the points. Using a ruler, connect the points to form polygon ABCD. Part B: Find the area of the polygon. Use the correct unit of measure. Show your work. Part C: Using a ruler, connect points A and C on the coordinate grid above. What is the area of each of the triangles formed by this line? Explain your answer.” (6.G.1 and 6.G.3)
Unit 4, Form A, Item 1, “Which of the following measures could represent the surface area of a solid? Choose all that apply. A) , B) , C) , D) , E) , F) , G) ” (6.SP.5b and 6.G.4)
Unit 8, Form A, Item 14, “Calculate the range and interquartile range for the data displayed in the dot plot. Show your work.” (6.SP.5c and 6.SP.3)
Grade 6, Middle of Year Test, Item 1, “For every 5 hamburgers sold at a restaurant, 18 hot dogs were sold. What is the ratio of hot dogs to hamburgers sold?” (6.RP.1)
Grade 6, End of Year Test, Item 32, “Circle any possible solutions to the equation. x + 6 = 4x.” (6.EE.5)
Grade 6, PARCC Test Prep: Standard 6.NS.C.7b Practice Test, Item 4, “Rex’s house is located at point (2, −5) on a coordinate plane. The location of Terrell’s house is the reflection of the coordinates of Rex’s house across the x-axis. In what quadrant is Terrell’s house?” (6.NS.7b)
Criterion 1.2: Coherence
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for students and teachers using the materials as designed devoting the large majority of class time to the major work of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade.
Indicator 1B
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for spending a majority of instructional time on major work of the grade.
The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6 out of 9, which is approximately 67%.
The number of Big Ideas, CCSSM clusters, devoted to major work of the grade (including assessments and supporting work connected to the major work) is 17 out of 22 , which is approximately 65%.
The number of lessons devoted to major work (including assessments and supporting work connected to the major work) is approximately 94 out of 125, which is approximately 75%.
A lesson level analysis is most representative of the instructional materials as the lessons include major work, supporting work, and the assessments embedded within each unit. As a result, approximately 75% of the instructional materials focus on major work of the grade.
Criterion 1.3: Coherence
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The instructional materials are also consistent with the progressions in the standards and foster coherence through connections at a single grade.
Indicator 1C
instructional materials reviewed for Math Expressions Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.
Examples of connections between supporting work and major work include the following:
Unit 4 connects major cluster 6.EE.A with supporting cluster 6.G.A throughout the entire unit. For example, students use their understanding of writing and evaluating expressions in which letters stand for numbers to find the surface area of various shapes.
Unit 5, Lesson 5 connects supporting cluster 6.G.A to major work cluster 6.EE.A. Students use algebraic expressions to find the surface area of prisms. For example, Student Activity Book, Question 4, “Write an expression for the surface area of the prism, using only the numbers 3, 4, and 5. Show your expression to another student, and explain how the terms of the expression match the parts of the net.”
Unit 6 connects 6.G.A to the major work cluster 6.EE.A. In Lesson 1, Student Activity Book page 306, students solve problems to find volume and surface area by using variable formulas to arrive at the correct solution. Problem 23 uses the dimensions of an aquarium, “How many cubic feet of water will it take to fill this aquarium? What is the surface area of the water that is open to the air?”
Indicator 1D
The instructional materials for Math Expressions Grade 6 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 152 days. The Pacing Guide can be found on page I18 in the Teacher Edition. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications.
The program is designed with nine units and 107 lessons. Most lessons require one day.
The Pacing Guide notes 27 lessons that could take two days, but this is not noted in the Day at a Glance for each lesson.
All Units designate two days for Unit Assessments.
Teachers start each lesson with a 5-minute Quick Practice and each lesson is comprised of several activities with estimated time ranging from a total of 55-65 minutes per lesson. Math Activity Centers are tailored for all levels of achievement across readiness and learning styles. They can be completed within the lesson or after, however, the time required for the activity is unstated.
Indicator 1E
The instructional materials for Math Expressions Grade 6 meet expectations for the materials being consistent with the progressions in the Standards. Content from prior and future grades is identified and connected to grade-level work, and students are given extensive work with grade-level problems.
The materials clearly identify content from prior and future grades and connect concepts to grade level work. Each unit includes a Unit Overview providing a Learning Progression. The Learning Progression states connections between the standards of the prior grade, current grade, and future grade. Additionally, each unit contains a Math Background Section. This section contains in depth information for the teacher articulating the learning progressions and the progression of the content between lessons. For example:
Unit 3, the Learning Progression chart makes connections between Grade 5, Grade 6, and Grade 7 within The Number System domain as it relates to operations with whole numbers, fractions, and decimals. “In Grade 5, students used equivalent fractions as a strategy to add and subtract fractions and applied and extended previous understandings of multiplication and division to multiply and divide with fractions. In Grade 6, students will apply and extend previous understandings of equivalent fractions to add, subtract, multiply and divide and interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions. In Grade 7, students will apply properties of operations as strategies to add, subtract, multiply, and divide rational numbers, represent addition and subtraction on a horizontal or vertical number line, understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, and convert a rational number to a decimal using long division.”
Unit 7, the Math Background, quotes from the Learning Progressions for Ratios and Proportional Relationships. “Ratio and Rate Language - As students work with tables of quantities in equivalent ratios (also called ratio tables), they should practice using and understanding ratio and rate language. It is important for students to focus on the meaning of the terms “for every,” “for each,” “for each 1,” and “per” because these equivalent ways of stating ratios and rates are at the heart of understanding the structure in these tables, providing a foundation for learning about proportional relationships in Grade 7.”
The instructional materials attend to the full intent of grade-level standards by giving all students extensive work with grade-level problems. Within each lesson, students practice grade level problems within Quick Practice, Student Activity Book pages, Homework, and Remembering activities. During modeled and guided instruction, students are given opportunities to engage in the grade level work by doing various examples with teacher and peer support. The independent practice in the Student Activity Book aligns with the lesson and provides students the opportunity to work with grade level problems to extend concepts and skills. For example:
Unit 3, Lesson 13 (6.NS.A and 6.NS.B), students review multiplication and division with fractions to build reasoning skills, finding an unknown factor in a fraction equation by relating multiplication and division, and completing a quick practice routine to build fluency with fraction operations. Problem 1, “On Gino’s farm, there is a rectangular wooded area in a cornfield. This wooded section has an area of 8/15 square mile and is 2/3 mile long. How wide is the wooded section?”
Unit 5, Lesson 1, students discuss their thoughts about algebra leading into a discussion about variables and expressions. The Teacher Edition for this lesson states, “Explain that variables and expressions are important parts of the language of algebra. Discuss the definitions of variable and expression on Student Activity Book page 211 and work through Exercises 1 and 2 as a class. Emphasize that an expression does not have an equal sign.” Students complete Exercise 3 individually, then several students share their expressions. The accompanying Homework and Remembering provide students with additional practice simplifying numerical expressions and evaluating algebraic expressions.
Students are introduced to Grade 7 proportional reasoning in Unit 1, Lessons 10 through Lesson 14. This content is mathematically reasonable; however, the materials do not include guidance for teachers to connect with grade-level work, for example:
Unit 7, Lesson 4, Understanding Cross Multiplication, “Students will learn to use the cross-multiplication method for solving proportions.”
Lesson 12, Non-Proportion Problems, Differentiate Proportion from Non-Proportion Problems, Problem 10, includes guidance for the teacher, “Problem 10 is a proportion problem because in order for the paint to be the same color, the same recipe or the same ratio of blue to yellow paint will need to be used to make the new smaller batch…” Problem 10, “Peachy Paint Company used 20 cans of blue and 15 cans of yellow paint to make Grasshopper Green paint. They have 8 more cans of blue paint. How many cans of yellow paint do they need to make more Grasshopper Green paint?”
Lesson 13, Days at a Glance, What will students learn? states, “Students will learn to use basic ratios to solve proportions and use the greatest common factor to find a basic ratio.” In the Student Activity Book for this lesson, Identify and Solve Basic Ratio Problems, students are directed to “Solve each proportion problem. Circle the number of the problem that is not a proportion problem and tell why it is not.” Students are given 6 problems, only one is not a proportion.
Each lesson contains Math Center Activities, as well as Homework and Remembering (spiral reviews) pages which provide additional practice with grade-level problems. For example:
Unit 1, Lesson 10, Homework, students use factor puzzles to solve proportion problems.
Unit 4, Lesson 4, Remembering, students solve a rate problem, use a coordinate grid to graph ordered pairs, determine the perimeter and area of a triangle, and find the surface area of given 3-dimensional shapes.
Indicator 1F
The instructional materials for Math Expressions Grade 6 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.
Each unit is structured by specific domains and big ideas. Learning objectives within the lessons are clearly shaped by CCSSM cluster headings. For example:
Unit 3, “Operations with Whole Numbers, Fractions, and Decimals” is shaped by cluster 6.NS.B, “Compute fluently with multi-digit numbers and find common factors and multiples.”
Unit 4, Big idea 1, “Nets and Surface Area of Prisms” is shaped by cluster 6.G.A, “Solve real-world and mathematical problems involving area, surface area, and volume.” Learning objectives in this section include, “Students explore nets for rectangular prisms and calculate the surface area of a prism, students use nets to model and find surface area of non rectangular prisms, and students will solve real world problems involving surface area of prisms.”
Unit 9, Lesson 2, the learning objective states, “Students determine distance and locate and plot integers on a number line.” This is shaped by 6.NS.C, “Apply and extend previous understandings of numbers to the system of rational numbers.”
Materials include problems and activities connecting two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. For example:
Unit 5, Lesson 15, connects cluster 6.EE.A with cluster 6.EE.B, when students determine whether a given value is a solution of an equation or inequality. Student Activity Book, Problem 1, Students evaluate 7x - 1 for x = 11 and are asked if x = 11 is a solution of 7x - 1 = 83, 7x - 1 < 83, 7x - 1 > 83.
Unit 6, Lesson 1, cluster 6.G connects to domain 6.EE. This lesson connects concepts in 6.G and 6.EE as students discuss cubic units (inch, foot, yard) and use formulas to calculate surface area and volume of a rectangular prisms.
Unit 7, Lesson 13, domain 6.RP connects to domain 6.EE, when students engage in solving problems which convert measurements within the same system of measurement. Students apply their understanding of arithmetic to algebraic expressions to support solving proportions and unit rates. Student Activity Book, Problem 3 states, “A can holds 344 mL of seltzer. How many liters is this? Find your answer in two ways: by writing and solving a proportion and by using a unit rate.”
Overview of Gateway 2
Rigor & Mathematical Practices
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations for Gateway 2, rigor and balance and practice-content connections. The instructional materials meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and they partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Gateway 2
v1.0
Criterion 2.1: Rigor
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations, by giving appropriate attention to developing students’ conceptual understanding and procedural skill and fluency. The instructional materials also do not always treat the aspects of rigor separately or together.
Indicator 2A
The instructional materials for Math Expressions Grade 6 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.
The materials identify Five Core Structures: Helping Community, Building Concepts, Math Talk, Quick Practice, and Student Leaders as the five crucial components that are the organizational structures of the program. “Building Concepts in the classroom experiences in which students use objects, drawings, conceptual language, and real-world situations - all of which help students build mathematical ideas that make sense to them.”
The instructional materials present opportunities for students to develop conceptual understanding. For example:
In Unit 1, Lesson 8, students discuss relationships between drawings and ratio tables. Questions include, “How is the ratio table related to the multiplication table? How is the ratio table related to two rate tables? How are the constant increases shown in the drawing and in the ratio table?"
Unit 3, Lesson 1, “Place value and operations with decimals and fractions are complex. There are many aspects of the symbols, words, and meanings of the operations that must be related and understood. A major recurring teaching task is leading the attention of your students to multiple aspects of a situation, problem, or numeric representation. This might be while you are explaining something, or you may need to help students do this when they are explaining.”
In Unit 5, Lesson 12, students explore ways to represent constant speed. They use equations, tables, and graphs, and use these relationships to identify constant change.
The instructional materials include opportunities in the Student Activity Book for students to independently demonstrate conceptual understanding. For example:
Unit 2, Lesson 3, “Erica’s Solution, Here is how Erica found the area of a parallelogram with no vertices over the base.” Erica’s solution includes finding the area of a rectangle, subtracting the area of two right triangles, to find the area of the parallelogram. Problem 1, “Can you compute the area of the parallelogram using the formula A=bh? Explain.”
Unit 6, Lesson 3, Problem 25, “A box in the shape of a rectangular prism has a volume of and a height of 2 1/2 inches. What are possible whole number dimensions for the length and width of the base?” Students use their understanding of volume to find the missing dimensions for the base.
Unit 7, Lesson 3, Check Understanding, “Draw one drawing to represent the fraction 4/5 and one drawing to represent the ratio 4/5.”
Indicator 2B
The instructional materials for Math Expressions Grade 6 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.
The instructional materials provide regular opportunities for students to attend to the standards. For example, 6.NS.2, fluently divide multi-digit numbers using the standard algorithm; and 6.NS.3, fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
The instructional materials develop procedural skill and fluency throughout the grade-level. Each lesson includes a “Quick Practice” described as “routines [that] focus on vitally important skills and concepts that can be practiced in a whole-class activity with immediate feedback”. Quick Practice can be found at the beginning of every unit on the pages beginning with the letters QP. Student materials and instructions are also found in the Teacher Resource Book on pages beginning with Q. Examples include:
Unit 2, Teacher Resource Book, Find Perimeter and Area, “The Student Leader chooses one of the slips of paper and reads the name of a figure, for example: Triangle. A volunteer sketches the figure on the board, chooses a base, draws a height, and describes how to find the area and perimeter of the figure.” Students practice using the formulas for area and perimeter.
Unit 7, Teacher Resource Book, Find Unit Rates, “Student Leader 1 asks the class for the ratio given in the table on the board, for example: the ratio of apple juice to cherry juice is 8 to 3. Student Leader 2 writes ‘1’ in the second column of the second row in the table, and says: “Say the unit rate using cups.” The class responds, “There are 8/3 cups of apple juice per 1 cup of cherry juice.”
The instructional materials provide opportunities for student to independently demonstrate procedural skill and fluency throughout the grade-level. These include: Path to Fluency Practice, and Fluency Checks. For example:
Unit 3, Lesson 5, Path to Fluency problems in exercises 20 – 57 provide computational practice in multiplying and dividing decimals. (6.NS.3)
Fluency Checks are provided throughout the materials. For example, in Unit 4, Lesson 3, Fluency Check includes 15 four-digit by two-digit division problems.
In addition, Homework and Remembering activity pages found at the end of each lesson provide additional practice to build procedural skill and fluency.
Indicator 2C
The instructional materials for Math Expressions 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.
Students engage with application problems in many lessons for the standards that address application in solving real-word problems. In Unit 4, Lesson 3, Student Activity Book, students solve contextual problems using visual 3-D models and given dimensions to find surface area. “Daniel needs to know how many square feet of metal it takes to build this warehouse including the roof. The warehouse will have a concrete floor. How many square feet of metal does he need?”
Each lesson includes an Anytime Problem listed in the lesson at a glance, and Anytime Problems include both routine and non-routine application problems. For example, Unit 2, Lesson 4, Anytime Problem, “In Ms. Park’s class, there are 16 students with glasses or brown hair (or both). If 7 students have glasses and 14 students have brown hair, how many students have both glasses and brown hair?”
The instructional materials present opportunities for students to engage routine applications of grade-level mathematics. Examples include:
Unit 1, Lesson 9, Student Activity Book, students create ratio tables to represent various situations. For example, “Noreen makes 2 drawings on each page of her sketchbook. Tim makes 5 drawings on each page of his sketchbook.” “John can plant 7 tomato vines in the time it takes Joanna to plant 4 tomato vines.”
Unit 1, Lesson 10, Student Activity Book, students solve routine proportion problems. “Noreen did 72 push-ups while Tim did 32 push-ups. When Tim had done 12 push-ups, how many had Noreen done?”
Unit 7, Lesson 7, Student Activity Book, Question 5, students use representations to solve rate problems. “At a factory, an assembly line processes 100 cans every 3 minutes. How long will it take the assembly line to produce 250 cans?” This question requires applying mathematical representations to answer. It also requires students to know which representations could be used in order to solve problems.
Remembering pages at the end of each lesson are designed for Spiral Review anytime after the lesson occurs. One feature of the Remembering problems are those titled Stretch Your Thinking, which often present opportunities for students to engage with non-routine problems. For example:
Unit 3, Lesson 9, Remembering, Stretch Your Thinking, Exercise 19, “Lucy has 3 3/4 feet of ribbon. She needs 1 1/2 feet for one project. She needs 1 5/6 piece for another project. Will Lucy have more or less than 1/2 foot of ribbon left after she completes both projects? Explain.”
Unit 5, Lesson 14, Remembering, Exercise 4, “Jenna is having a sidewalk sale. She pays $12 for a permit. She collects $1.50 for every item she sells.” Stretch Your Thinking, Exercise 5, “Look at the situation in Exercise 4. Suppose at a second sale Jenna pays $15 for a permit and sells each item for $1.75. If she sells 30 items at each sale, at which sale does she make more money? Explain.”
Unit 8, Lesson 5, Remembering, Stretch Your Thinking, Exercise 8, “The average number of people at a concert over 3 nights is 125. On the first night, there were 151 people at the concert. The same number of people attended on the second and third nights. How many people attended on each of the second and third nights? Explain.”
Indicator 2D
The instructional materials for Math Expressions Grade 6 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.
All three aspects of rigor are represented in the materials. For example:
Each lesson has a 5-minute Quick Practice providing practice with skills that should be mastered throughout the year.
There are Performance Tasks throughout the series, where students use conceptual understanding to perform a mathematical task. For example, Unit 3 Performance Task, “If the students had 3 pounds of frozen chopped spinach, what is the maximum number of servings of oeufs a la florentine that they could make? Explain and show your work.”
Fluency Checks are included throughout the series, where students practice procedural skills and fluency. For example, Unit 3, Lesson 5, Multiplication and Division, Homework and Spiral Review, Fluency Check 1, students solve multiplication with decimals similar to Problem 8, “0.87 x 29.”
Application problems are embedded into practice in the Student Activity Book. For example, Unit 5, Lesson 10, Homework and Remembering, Problem 1, "Benny is writing a report. For every 7 paragraphs, he uses 4 pieces of art. How many pieces of art will Benny use is his reports is 56 paragraphs long?"
Examples where student engage in multiple aspects of rigor:
Unit 7, Lesson 2, Ratio as Quotient, introduces students to using unit rates to describe any ratio. Students use two recipes and division to find the unit rate for each recipe. Problem 1, “Find the amounts of cherry juice in each drink for 1 cup of orange juice. Remember that when you divide both quantities in a ratio table by the same number, you get an equivalent ratio.” Students engage in application and procedural skill and fluency, however, the reminder emphasizes procedural skill.
Student Activity Book, Unit 2, Lesson 5, Solve Real World Problems, students solve application problems which involve conceptual understanding of area and perimeter. Problem 25, “An attic playroom is in the shape of a rhombus with a base of 12 ft and a height of 6 ft. How much carpeting is needed? How much wood molding is needed to go around the room?” Students are instructed to solve the problems and draw pictures to help them.
Student Activity Book, Unit 9, Lesson 7, students solve: “Victor’s checking account has a balance of $10 and is charged a $2 service fee at the end of each month. Question 1: “Suppose Victor never uses the account. Complete the table below to show the balance in the account each month for 6 months. Then use the data to plot points on the coordinate plane to show the decreasing balance over time.”
Criterion 2.2: Math Practices
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified and partially attend to the full meaning of each practice standard. The instructional materials partially use accurate mathematical terminology and also partially support teachers and students in students constructing viable arguments and analyzing the arguments of others.
Indicator 2E
The instructional materials reviewed for Math Expressions Grade 6 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.
Mathematical Practice Standards are clearly identified in a variety of places throughout the materials. For example:
The Mathematical Practices are identified in both volumes of the Teacher’s Edition. Within the introduction, on page I13 in the section titled The Problem Solving Process, the publisher groups the Mathematical Practices into four categories according to how students will use the practices in the problem solving process. Mathematical Practices are also identified within each lesson.
Each time a Mathematical Practice is referenced, it is listed in red with a brief description of the practice.
At the beginning of each Unit is a section devoted to the Mathematical Practices titled Using the Common Core Standards for Mathematical Practices which includes guidance for the teacher. Within this section, each Mathematical Practice is defined in detail. In addition, an example from the Unit is provided for each practice. For example, in Unit 7, Lesson 10, Activity 2, Percents of Numbers, identifies “MP1 Make Sense of Problems. Be sure students understand the problem. Discuss what information we know and what we need to find. We know the adult dose. We know what percent the child dose is to the adult dose. We need to find the child dose.”
Examples of Mathematical Practices that are identified, and enrich the mathematical content include:
Unit 1, Lesson 7, MP6 - Attend to Precision|Verify Solutions. “Once students have identified the unit rate from the graph, they can use that rate to complete the table. Or, they can use points on the graph to complete the table. You can suggest that they use one method first, and then use the other method to verify their solution.”
Unit 2, Lesson 2, identifies MP7 - Look for Structure. Teachers are guided to “Ask students to think about many different right triangles to which this formula applies; huge right triangles, tiny right triangles, and medium right triangles. Will the formula A= 1/2 bh work for all these right triangles? Explain.”
Unit 4, Lesson 5, Problem 12, MP2 - Reason Abstractly and Quantitatively | Connect Symbols and Words. “Students should connect that finding 'how many square inches' means finding the surface area as they solve Problems 7-12. Students also need to use the description of the object and the picture to decide how much of the surface area needs to be found. Finally, students need to use the correct operations to calculate the number of square inches needed.”
Indicator 2F
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations for carefully attending to the full meaning of each practice standard.
The materials do not attend to the full meaning of Mathematical Practice 5.
Mathematical Practice 5: The instructional materials do not meet the full meaning of MP5 as tools are chosen for students, and there are few opportunities for students to choose tools strategically. For example:
Unit 1, Lesson 3 identifies MP5. Students use a provided multiplication table and identify the two highlighted columns to determine a rate table. Students are using a given tool to complete the task.
Unit 2, Lesson 3 identifies MP5. Students copy rhombi from the board onto their MathBoards. Students are not choosing a math tool in this situation.
Unit 5, Lesson 2 identifies MP5 on Student Activity Book on page 215, questions 21-23. A grid is provided and students are told to use a straightedge to draw and then answer questions. Students are not selecting a tool to use. They are using the provided grid and a suggested tool.
Indicator 2G
Indicator 2G.i
The instructional materials reviewed for Math Expressions Grade 6 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.
Math Expressions includes a Focus on Mathematical Practices lesson as the last lesson within each unit. Activity 3 of each of these lessons prompts students to determine whether a mathematical statement is true or false or to establish an arguable position surrounding a mathematical statement. These activities provide students opportunities to construct an argument and critique the reasoning of others. Student volunteers ask questions of other students to verify or correct their reasoning. Examples of Focus on Mathematical Practices lessons include, but are not limited to:
Unit 2, Lesson 10, students develop an argument for the following statement: “When the perimeter of a rectangle increases, the area always increases.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.
Unit 4, Lesson 6, students develop an argument for the following statement: “When the dimensions of a rectangular prism are doubled, the surface area doubles.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.
Unit 7, Lesson 7, students develop arguments for the following statements: “All integers are rational numbers. All rational numbers are integers.” Volunteers share their positions and explanations with the class. The class asks the volunteers questions and verifies or corrects reasoning errors.
Puzzled Penguin problems are found throughout the materials and provide students an opportunity to correct errors in the penguin’s work. These tasks focus on error analysis, and many of the errors presented are procedural. Examples of Puzzled Penguin problems include:
Unit 3, Lesson 10, Puzzled Penguin problem, students solve: “Dear Math Students, I am planting a flower garden. The space I have to plant in is 3 1/2 feet wide and 4 1/2 feet long. I told my friend that the area of my flower garden is 12 1/4 square feet. She says I’ve made a mistake! Can you help me find my mistake and the correct area? Your friend, Puzzled Penguin.”
Unit 5, Lesson 3, Puzzled Penguin problem, “Dear Math Students, Here’s how I analyzed m * (4 + m). Did I do it right? If not, help me understand what I did wrong. Step 1: I circled the part in parentheses. Step 2: There are no powers so I didn’t need to do anything. Step 3: I circled the multiplication. Step 4: I looked for addition and subtraction and circled the terms. Your friend, Puzzled Penguin.” Each step is illustrated to the side of the narrative. Student analysis needs to identify that the error is in treating the expression (m + 4) as m, and 4.
Unit 7, Lesson 6, Puzzled Penguin problem, students analyze a ratio table illustrating 2 purple units for every 5 orange units. “I made my own sand mixture. I mixed 2 parts purple and 5 parts orange. Then I wrote this multiplicative comparison. The amount of purple sand is 2/5 times the amount of the total mixture. My friend says I made a mistake. Did I? If I did, can you tell me what mistake I made and help me correct it?” In this example, students need to demonstrate understanding of part to part and part to whole comparisons.
Unit 8, Lesson 8, Puzzled Penguin problem, students find the error that was made on drawn box plots. “Dear Math Students, I was given this set of data. 21, 21, 22, 23, 24, 25, 26, 26, 26, 29, 30. Here is the box plot I made to represent the data. Can you help me understand what I did wrong?” The box plot does not include a line to identify the median of the data.
In addition, Remembering pages at the end of each lesson often present opportunities for students to construct arguments and/or critique the reasoning of others. For example:
Unit 4, lesson 3, Remembering, Stretch Your Thinking, Exercise 7, “Jerome has a block of craft foam shaped like a square prism. He cuts the foam block in half. Jerome thinks that because the surface area of the original black was , the surface area of each piece is . Is he correct? Explain.”
Unit 5, Lesson 8, Remembering, Stretch Your Thinking, Exercise 10, students construct arguments as they explain “The diagram shows 4⋅(m⋅3). Change the diagram so this it shows (4⋅m)⋅3. How do the diagrams show that 4⋅(m⋅3) = (4⋅m)⋅3 = 12m?
Unit 8, Lesson 6, Remembering, Stretch Your Thinking, Exercise 4, students construct arguments to explain “The balance point of a set of 4 numbers is 14. How can you move one point to make the balance point 16?”
Indicator 2G.ii
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Overall, the teacher materials provide students multiple opportunities to construct viable arguments, however there are missed opportunities to support teachers in engaging students in analyzing the arguments of others throughout the materials.
Throughout the Teacher Edition, MP3 is identified with explanations and guidance for teachers, either in reference to specific parts of the lesson, or in specific activities such as Math Talks. However, this guidance often supports teachers to engage students in explaining their methods, instead of constructing arguments or critiquing reasoning. For example:
Unit 1, Lesson 9, identifies MP3 Construct a Viable Argument | Compare Representations. “Next discuss how the three ratio tables are alike and how they are different.” The teacher prompt states: “Let’s look across Ratio Tables 1, 2, 3. How are they alike and how are they different?” Teachers are given no guidance to engage students in how the unit rates is represented and used to compare different quantities: drawings on a page in notebooks, marching band formation, and planting tomatoes. Students do not need to use mathematics to construct an argument or analyze the reasoning of others.
Unit 7, Lesson 2, Unit Rate Strategy, MP3 Construct a Viable Arguments | Compare Methods. “Students discuss the three variations of the unit rate strategy shown in Exercise 12: Gen’s use of the ratio table, Claire’s Factor Puzzle, and Joey’s method of “going through 1”. “The Math Talk in Action below shows a sample discussion.” Math Talk includes this teacher prompt: “What do you notice about the three methods?” There is no further guidance for teachers to engage students in comparing methods to construct arguments or analyze the arguments of others.
Unit 7, Lesson 7, Math Talk, Problem 3, “Pokey the snail travels 25 centimeters every 2 minutes. How far will Pokey go in 15 minutes?” Three examples of student work are provided for the teacher: the use of a ratio table, cross-multiplication, and reasoning. No guidance is provided to teachers to facilitate mathematical arguments on the solution strategies. In student work samples 1 and 2, students represent the quantities in different ways. Teacher guidance states: “187 1/2 cm and 187.5 both represent the same length. We usually use decimals with metric units because metric units are based on 10 just like decimals. However, both of your answers are correct.” There is no support to elicit the underlying mathematics of equivalence in fractions and decimals.
Examples of materials assisting teachers in engaging students in constructing viable arguments:
Unit 1, Lesson 8, Student Activity Book, students work through ratio/rate table exercises together. In the Teacher Edition, Math Talk, guides teachers to make sure that students notice and discuss specific points about unit rates and ratio tables. Students construct arguments when they answer Question 1, “How are the tables alike? How are they different?”
Unit 2 Lesson 10, teachers share the statement: “When the perimeter of a rectangle increases, the area always increases.” Students decide if this statement is true or false and develop an argument that supports their position.
Unit 7, Lesson 4, Puzzled Penguin, students analyze how he cross-multiplied and then write a response to, “My work can’t be right because my answer is only 2 minutes! What did I do wrong?” Teachers are guided to have volunteers share their responses. The responses should demonstrate an understanding of how to use cross-multiplication to solve proportions.
There are instances where MP3 is identified in A Day at a Glance for a lesson, but there is no guidance for teachers on how to engage students to construct arguments or analyze the arguments of others.
Indicator 2G.iii
The instructional materials reviewed for Math Expressions Grade 6 partially meet expectations that materials use accurate mathematical terminology.
New vocabulary is introduced at the beginning of a Lesson or Activity.
The Teacher’s Edition provides instruction for teachers on how to develop the vocabulary, with guidance for teachers to discuss and use of the vocabulary.
The student materials include Unit Vocabulary Cards that students can cut out and use in school or at home to review vocabulary terms.
The Student Activity resource contains activities that students can do with the vocabulary cards; however, the teacher materials do not provide guidance as to when students should engage in these activities to support learning the vocabulary.
There is an eGlossary providing audio, graphics, and animations in both English and Spanish of the vocabulary needed in the lessons.
Study POP! is an interactive digital charades app that includes Math Expressions vocabulary to help students practice and develop mathematical vocabulary. Study POP! is listed at the beginning of many lessons, but is not referenced during the lesson.
Examples of how vocabulary is incorporated within lessons include:
Unit 3, Lesson 7, lists equivalent fractions, simplifying, and unsimplifying as vocabulary at the beginning of the lesson. During Activity 2, the teacher leads a whole group Math Talk about the effects of rewriting fractions.
Unit 7, Lesson 6, identifies multiplicative comparison as vocabulary at the beginning of the lesson.
In addition, there are instances where teachers are told to look for precise use of words, facts, and symbols. For example:
Unit 2, Lesson 10, “MP6 - Attend to Precision: The sentences must include precise mathematical words, facts, and symbols. The drawings should include side lengths and calculations for perimeter and area.” Students decide if the statement, “When the perimeter of a rectangle increases, the area always increases,” is true or false and develop an argument to support their position.
Unit 4, Lesson 6, “MP6 - Attend to Precision: The sentences must include precise mathematical words, facts, and symbols. The drawings should include side lengths and calculations for the area of the faces.” Students decide if the statement, “When the dimensions of a rectangular prism are doubled, the surface area doubles,” is true or false and develop an argument to support their position.
There are instances where mathematical vocabulary is not accurate. For example:
Unit 1, Lesson 9, Basic Ratios and Equivalent Ratios. The materials state: “A basic ratio has the least possible whole numbers. 4:7 is a basic ratio because no whole number (except 1) divides evenly into 4 and 7."
Unit 7, Lesson 4, the term “cross-multiply” is introduced and used as a method for solving equations that involve proportional relationships. The use of “cross-multiply” obscures the precise, mathematical process that is occurring and does not attend to the specialized language of mathematics, such as properties of equality and identity, involved in the process.