6th Grade - Gateway 1

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for assessing grade-level content. The assessments are aligned to grade-level standards. Examples include:

• In Unit 1, End-of-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-of-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 in this End-of-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. 
• In Unit 6, Mid-Unit Assessment, Problem 7 assesses 6.EE.6 by asking students to demonstrate their skills with working in context by writing and solving an equation of the form x + p = q. The problem states, “Mai poured 2.6 liters of water into a partially filled pitcher. The pitcher then contained 10.4 liters.” In part a, students select a bar model that represents the situation, and in part b, students write an equation that represents the situation. In part c, students solve the equation they wrote in part b, and in part d, students explain what the solution to the equation means in the situation.

Assessments are located on each Unit Page under the Assessments tab for each of the first eight units. Unit 9 is an optional unit and has no assessments. Assessments are limited to seven problems, but these are often broken into multiple prompts and assess numerous standards. There are also five Mid-Unit Assessments for a total of 13 summative assessments.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 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 5 out of 8, which is approximately 63 percent.
• The number of non-optional lessons devoted to major work of the grade, including assessments and supporting work, is 90 out of 134 total non-optional lessons, or approximately 67 percent.
• The number of days devoted to major work, including assessments and supporting work, is 104 out of 168 days, which is approximately 62 percent.

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 67 percent 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 indicate which items to use for the review.

The instructional materials reviewed for Kendall Hunt’s Illustrative Mathematics 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. Examples of the connections between supporting work and major work include the following:

• In 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 is connected as students generate and use the developed formula and substitute the appropriate numerical values for calculation. In Lesson 5, students explore and 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).
• In Unit 3, Lesson 17 is a culminating lesson connecting 6.RP.A to the Unit 1 focus of 6.G.A. Students work collaboratively on a culminating task involving finding the area of the walls in a room and the cost of the paint, which is purchased in 1-quart, 1-gallon, or 5-gallon containers with 20% off all quart-sized paint cans.
• In Unit 4, Lessons 14 and 15 connect 6.NS.1 with 6.G.2. After work on understanding fraction division, students apply the concept to a variety of area/volume problems and a culminating task.
• In 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.

The instructional materials for Kendall Hunt’s llustrative Mathematics 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 of the materials for teachers and students are viable for one school year as written and would not require significant modifications. As designed, the instructional materials can be completed in a school year.

• The provided scope and sequence found in the Grade 6 Course Guide includes materials for 168 instructional days. There are 134 non-optional lessons, 21 assessment days (13 summative), and 13 optional lessons.
• 129 of the non-optional lessons are designed to address grade-level standards, and 5 lessons connect standards from previous grades to the grade-level standards in the unit.
• 7 of the optional lessons are present throughout the first eight units, and Unit 9 is an optional unit which includes 6 lessons.
• Units 1-8 are comprised of 15 to 19 lessons, and each lesson is designed for 45-50 minutes. Within each unit, lessons contain a Warm-Up, two or three Activities, a 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 edition.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for 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 Grade 6 Narrative in the Course Guide describes 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:

• In Unit 1, Lesson 4, the 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)
• In Unit 5, Lesson 8 is a “culminating lesson on multiplication” that 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:

• The 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.
• In 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 Lessons and Standards, there is a table 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 includes problem contexts that reach the limitations of using double-number lines to introduce the use of ratio tables.
• In Unit 5, 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 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 all positive decimals.”

A typical lesson has a Warm-Up, one or more Activities, Synthesis, 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.

The instructional materials for Kendall Hunt’s Illustrative Mathematics 6-8 Math, Grade 6 meet expectations for fostering 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 overview, a connection is made to prior understanding developed in Unit 2, where learning about unit rate is formalized, as well as understanding how percents and percentages are 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.

• In 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 Lesson 3 includes interpreting division situations.

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

• In Unit 7, the Lesson 7 overview connects 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 overview 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 throughout the unit.

The 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.

• In Unit 6, Lessons 16 and 17 connect 6.EE.9 and 6.RP.3b. Students extend prior learning with ratio understanding and equivalent ratios in the context of mixing paint, write equations that show a relationship between two quantities, and explore dependent and independent variable relationships. Students create tables of values and graphs, 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.
• In Unit 8, Lesson 12, the Warm-Up uses dividing by decimal values (6.NS.B) to calculate mean and Mean Absolute Deviation (6.SP.B) more efficiently in the two Activities that follow.