6th Grade - Gateway 1

The instructional materials reviewed for Open Up Resources 6-8 Math, Grade 6 meet expectations that they assess grade-level content. The assessments are aligned to grade-level standards.

For example:

• Unit 1 End-Unit Assessment Problem 4 assesses 6.EE.1. Students find the area of a square when given a side length and then the side length of a square when provided an area: “A square has a side length 9 cm. What is its area? A square has an area of 9 cm². What is its side length?” Providing this context for students connects the grade-level expectation of evaluating whole number exponents to their previous understandings of area of squares.

• The Unit 4 End-Unit Assessment assesses dividing fractions, 6.NS.1, which states that students should compute and solve real-world problems that involve division of fractions by a fraction, by using visual models and equations. The seven questions on this End-Unit Assessment assess all aspects of 6.NS.1. Problems 1 and 7 are set in a real-world context, Problems 2 and 3 connect to multiplication of fractions, Problem 4 assesses knowledge of the standard algorithm for the division of fractions, and Problems 5 and 6 use visual representations.

Assessments are located in the teacher materials in each of the first eight units. Unit 9 Putting It All Together is an optional culminating unit and has no assessments. Assessments are limited to seven problems, but these are often broken into multiple prompts, assessing numerous standards. There are also four Mid-Unit Assessments for a total of 12 assessments.

The instructional materials reviewed for Open Up Resources 6-8 Math, 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, is five out of eight, which is approximately 62.5%.
• The number of lessons devoted to major work of the grade, including assessments and supporting work, is 88 out of 133 total non-optional lessons, or approximately 66%.
• The number of days devoted to major work, including assessments and supporting work, is 102 out of 153 days, which is approximately 67%.

A lesson-level analysis is most representative of the instructional materials because this calculation includes all lessons with connections to major work with no additional days factored in. As a result, approximately 66% of the instructional materials focus on major work of the grade. An analysis of days devoted to major work includes 20 days for review and assessment, but the materials do not dedicate items to be used for the review.

The instructional materials reviewed for Open Up Resources 6-8 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. Multiple lessons in the Grade 6 curriculum incorporate supporting standards in ways that support and/or maintain the focus on major work standards. Connections are strongest as they relate to Grade 6 work with solving problems related to geometric measurement and the major work related to writing and evaluating expressions, including formulas to solve real-world problems.

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

• Unit 1 Lessons 5, 6, 9, 10, and 18 connect standards 6.EE.2 and 6.G.A as students substitute numerical values for variables in order to solve for the area or surface area of an object. Within these lessons, 6.G.A is the focus, and 6.EE.2 naturally emerges as students generate and use the developed formula and substitute the appropriate numerical values for calculation. In Lesson 5, students first explore and then create formulas for base-height definitions and relationships as they relate to area. They continue to find base and height and calculate area for a sequence of parallelograms (6.EE.2a). The final task in Lesson 5 includes two parallelograms in which students find the base and height and then evaluate the formula they created in task 2 to find the area (6.EE.2c).

• Unit 3 Lesson 17 is a culminating lesson connecting 6.RP.A back to the Unit 1 focus of 6.G.A. Students work collaboratively on a culminating task involving finding the area of a room and the cost of the paint based on size of the unit and related discounts.

• In Unit 4, Lessons 14 and 15 connect the unit focus 6.NS.1 to supporting standard 6.G.2. After work on understanding fraction division, students apply the concept to a variety of area/volume problems and a culminating task.

• Unit 6 Lesson 4 connects 6.NS.3 to 6.EE.B as students represent situations with equations and practice solving. This connection happens throughout the lesson as decimal values are incorporated into many equations that can be solved mentally.

Instructional materials for Open Up Resources 6-8 Math, Grade 6 meet expectations that the amount of content designated for one grade level is viable for one year.

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. As designed, the instructional materials can be completed in 177 days.

• The provided scope and sequence found in the Course Guide, Grade 6 includes materials for 153 instructional days. There are 133 non-optional lessons, twelve summative assessments, and eight review days.

• 128 of the non-optional lessons are designed to address grade-level standards, and five lessons serve to connect prior knowledge of previous grade-level standards to the lessons in the unit.

• Six optional lessons are also present throughout the first eight units. Unit 9 Putting it All Together is optional and includes an additional six lessons requiring up to 18 additional days depending on the number of lessons completed. There are a total of 177 instructional days if all optional lessons are completed.

• Each unit is comprised of 15 to 19 lessons. Within each unit, lessons contain a Warm-Up, two or three Activities, Lesson Synthesis, and a Cool-Down. Guidance regarding the number of minutes needed to complete each component of the lesson is provided in the teacher materials.

The instructional materials for Open Up Resources 6-8 Math, Grade 6 meet expectations for the materials being consistent with the progressions in the standards.

The instructional materials clearly identify content from prior and future grade levels and use it to support the progressions of the grade-level standards. The instructional materials also relate grade-level concepts explicitly to prior knowledge from earlier grades.

The materials are intentionally designed to address the standards the way they are laid out in the progressions, and the Unit Overview clearly explains how the standards and progressions are connected. Units begin with lessons connected to the standards from prior grades that are relevant to the current topic. Standards from the grade level and prior grades, and standards that will be addressed later in the year are identified in the sections as “addressing,” “building on,” and “building towards,” respectively. For example:

• Unit 1 Lesson 4 Warm-Up is identified as “building on” 4.G.2 and 5.G.B. The lesson activities are labeled as “addressing” 6.G.1. The lesson affords students a variety of opportunities to compose or decompose quadrilaterals using right triangles (4.G.2 and 5.G.B) leading to “defining attributes of parallelograms.” (6.G.1)

• Unit 5 Lesson 8 a “culminating lesson on multiplication” addresses 6.NS.3 as students employ the standard algorithm for multiplication after “building on” 5.NBT.7 by using diagrams to show partial products. 6.EE.A is identified as a standard this lesson is “building towards” as students will apply these skills later in Unit 6 when working with algebraic expressions.

The Warm-Ups in lessons frequently work with prior-grade standards in ways that support learning of grade-level problems and make connections to progressions from previous grades. For example:

• Unit 2 Lesson 7 Warm-Up makes explicit connections between Grade 4 and Grade 5 fraction and decimal equivalence work on the number line to skills related to equivalent ratio work in Grade 6.

• Unit 7 Lessons 2, 3, and 6 include Warm-Ups that make explicit connections between prior-grade work with using the number line and making comparisons with fractions as indicated in the Number Operations-Fractions progression.

The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems.

In the Course Guide under Course Information and Scope and Sequence, there is a chart which reflects the mathematics in the materials. All grade-level standards are represented across the 9 units. Tasks are aligned to grade-level work and are connected to prior-grade knowledge. For example:

• Work with ratios begins in Unit 2. Lessons emphasize ratio language and using concrete models. Lessons lead to the use of diagrams. Lesson 6 makes explicit connections to previous work with number lines as an introduction to a continuous model with double-number line diagrams. Students build on the work of prior grades to develop a tool for looking at equivalent ratios and then exploring unit rates. Lesson 11 then includes problem contexts that reach the limitations of using double-number lines to introduce the use of ratio tables.

• In Unit 5 Arithmetic in Base Ten, students compute sums, differences, products, and quotients of multi-digit whole numbers and decimals using algorithms. The first lesson focuses on calculating with money, the Warm-Up in the second lesson addresses place value, and the subsequent lessons have students calculate decimals in various problem-based activities providing opportunities to build fluency. A rationale connected to the progression documents is given in the materials, “In previous grades, students learned how to add, subtract, multiply, and divide whole numbers and decimals to the hundredths place. In this unit, they will extend this knowledge to include to all positive decimals.”

A typical lesson has a Warm-Up, one or more Activities, and a Cool-Down. Additionally, every lesson provides practice problems that can be used as independent or group work. Some lessons also provide an “Are you ready for more?” problem. These problems are an opportunity for students to explore grade-level mathematics in more depth and often make connections between the topic in the lesson and other concepts at grade level. They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own.

Overall, the materials give students extensive work with rigorous, grade-level problems.

The instructional materials for Open Up Resources 6-8 Math, Grade 6 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings, including:

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

• The Unit 2 Overview states, “Students learn to understand and use the terms ‘ratio, rate, equivalent ratios, per, at this rate, constant speed, and constant rate,’ and to recognize when two ratios are or are not equivalent. They represent ratios as expressions and represent equivalent ratios with double number line diagrams, tape diagrams, and tables. They use these terms and representations in reasoning about situations involving color mixtures, recipes, unit pricing, and constant speed.” The lessons include goals for understanding important ratio vocabulary, recognizing equivalent ratios, and using a variety of representations to explore and understand the concepts. For example: “I can explain the meaning of equivalent ratios using a color mixture as an example.”

• In the Unit 3 Course Guide, a connection is made to understanding developed in Unit 2, how learning about unit rate is formalized, as well as how understanding of percents and percentages is related to unit rate. Again, there is a link between the understanding of ratio concepts and using them to solve problems.

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

• Unit 4 Lessons 1 through 4 include tasks that revisit prior-grade work with division. Lesson 1 begins the unit with a foundation of how the size of the divisor affects the size of the quotient, Lesson 2 attends to the different meanings of division, and in Lesson 3 there are interpreting division situations which are aligned to the cluster heading of applying and extending previous learnings of multiplication of division to divide fractions by fractions with understanding.

6.EE.A Apply and extend previous understandings of numbers to the system of rational numbers.

• In Unit 6 Lesson 7 Overview, students connect using number lines and contextual situations to “understand” the terms “positive number” and “negative number,” “understand and use absolute value notation,” and “understand” the concept of “infinitely many solutions.” Extending previous number understandings to rational number concepts is present throughout the unit, especially as it relates to previous understanding of number on continuous models like the number line and coordinate plane.

6.G.A Solve real-world and mathematical problems involving area, surface area, and volume.

• The Unit 1 Course Guide description states explicitly that mathematical problems are used for problem exploration because “tasks set in real-world contexts that involve areas of polygons are often contrived and hinder rather than help understanding.” Lessons 1 through 11 reflect an explicit alignment to the cluster heading regarding area, and Lessons 12 through 18 connect with surface area. Lesson 19 closes the unit with tasks which include real-world contexts and mathematical modeling using concepts developed over the unit.

Materials consistently include problems and activities that 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. Multiple examples of tasks connecting standards within and across clusters and domains are present. These connections build deeper understanding of grade-level concepts and the natural connections which exist in mathematics.

• Unit 1 Lesson 5 Activities connect standards 6.G.1 and 6.EE.2 when formulas are derived for finding the area of special quadrilaterals and triangles by evaluating expressions. Lessons 9 and 10 continue to develop these two domains simultaneously as students write, read, and evaluate expressions from formulas.

• Unit 4 Lesson 4 standards 6.G.1, 6.G.2, and 6.NS.1 are connected when finding and using fractional dimensions of triangles and prisms by multiplying and dividing rational values to calculate area or volume or to find specific dimensions of the given shape.

• Unit 6 Lessons 16 and 17 address both 6.EE.9 and 6.RP.3b. Students extend prior learning with ratio understanding and equivalent ratios in a paint-mixing context, write equations that show a relationship between two quantities, and explore dependent and independent variable relationships. Students create tables of values, graph them, and explore the patterns they see.

• In Unit 8 Lesson 9, students determine the mean for a numerical data set and understand the interpretation of the mean as a "leveling out" of the data or an indication of "fair share" as well as understand that the mean is a measure of center that summarizes the data using a single number, thus connecting clusters 6.SP.A and 6.SP.B.

• The Unit 8 Lesson 12 Warm-Up builds fluency with dividing by decimal values (6.NS.3) in order to calculate mean and MAD (6.SP.5c) more efficiently in the two Activities that follow.