6th Grade - Gateway 1

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) 15cm, B) 24 in^2, C) 31 mm^3, D) 9ft^2, E) 12 in, F) 28 cm^3, G) 42 m^2” (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)

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.

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?”

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.

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.

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.”