6th Grade - Gateway 2

Materials partially meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Frequently opportunities are missed. Opportunities for students to work with standards that specifically call for conceptual understanding occur by use of pictures, manipulatives and strategies but frequently fall short by not providing higher order thinking questions to truly determine students' understandings.

Cluster 6.RP.A calls for understanding ratio concepts and using ratio reasoning to solve problems.

• There are 13 Focus lessons on 6.RP.1 and eight focus lessons on 6.RP.2. Many of the lessons are doing "dual duty" as many lessons are marked for both standards. The directed and explicit structure of the lessons reduces students' opportunities to struggle with the understanding of the mathematics. There is one Open Response lesson on ratios in the year.

Standard 6.EE.5 focuses on understanding solving an equation or inequality as a process of answering a question.

• In the following lessons, problems that are part of the practice sections are incorrectly aligned to 6.EE.5, which reduces opportunities to develop conceptual understanding: 1.11, 2.11, 2.14, 3.4, 5.6, 6.3, 7.4 (Math Box Problem 1) and 8.4. The misaligned problems in these lessons have students evaluating numerical expressions as opposed to demonstrating an understanding of solving an equation or an inequality as a process of answering a question.

Some attention to Conceptual Understanding is found in the Professional Development boxes throughout the Teacher Edition.

• On page 80 of the Teacher Edition, the Professional Development box explains that fractions can serve as area models or as number line strips and provides an example.
• On page 286 of the Teacher Edition, the Professional Development box explains that working with grids can help "students reason about percents conceptually before they use an algorithm to convert fractions to decimal equivalents and percents."

The instructional materials reviewed for Grade 6 partially meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials lack activities to build fluency computing with multi-digit numbers, 6.NS.2 and 6.NS.3. Standards 6.NS.2 and 6.NS.3 have a total of 215 exposures in the instructional materials. Exposures could include problems in the Math Boxes, problems in the Math Journal, direct instruction during the Focus lesson, problems during the online or hands-on game, and/or homework problems.

Standard 6.NS.2 has 61 exposures within the curriculum and is listed as the focus of three lessons.

• There is only one focus lesson that explicitly teaches students the standard algorithm for division. Lesson 3.5 is the only lesson where there is focused instruction on the standard algorithm.
• In Lesson 1.3 "Finding Equal Shares with the Mean," 6.NS.2 is not the focus. Standard 6.NS.2 is only included because division is used to find the mean, and the standard algorithm is not specifically addressed.
• In Lesson 1.5 about comparing measures of center, 6.NS.2 is not the focus. Standard 6.NS.2 is only included because division is used to find the mean, and the standard algorithm is not specifically addressed.
• There are 44 exposures for practice aligned to 6.NS.2, but only about half of those opportunities occur after the standard algorithm is discussed in Lesson 3.5.

Standard 6.NS.3 has 154 exposures within the curriculum and is listed as the focus of nine lessons.

• Lesson 3.3 focuses on the standard algorithm for addition and subtraction of multi-digit decimals.
• Lesson 3.4 focuses on the standard algorithm for multiplication of multi-digit decimals.
• Lesson 3.6 focuses on the standard algorithm for division of multi-digit decimals.
• Lesson 3.7 includes an application problem that involves all operations with multi-digit decimals.
• There are 93 exposures for practice aligned to 6.NS.3, but only a few more than half of those opportunities occur after the standard algorithms are discussed in Lessons 3.3, 3.4, 3.6 and 3.7.

Math Boxes are used during each lesson. These problems, typically 5-6 problems, do not connect to each other but are pulled from several different clusters and/or domains and are designed for student practice and maintenance of previous skills. Most lessons in the materials have a "Mental Math and Fluency" section which allows students to practice fluencies required in grade 5. However, often lessons develop a specific procedure and reinforce that procedure. The teacher often guides students thinking with direct instruction and procedural guided questioning.

The materials partially meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

Most problems are presented in the same way throughout the entire curriculum. There is little variety of problems or types of problems. Problems are presented as short, one correct answer problems. Some of the problems are tied together through concepts and ideas, but many times lessons are completely disjointed from one anther.

Standard 6.NS.1 has 77 exposures within the curriculum and is listed as the focus of four days of Focus lessons.

• The Focus portions of Lessons 2.5, 2.6, 2.7 and 2.8 are aligned to 6.NS.1.
• About half of these exposures, including lessons 2.13, 3.1, 3.3, 3.5, 4.4 and 4.13, only involve computing quotients of fractions free from interpreting or being part of a word problem. The practice listed for lesson 3.4 does not actually align to 6.NS.1 as problem 4 in the Math Boxes only has students writing the reciprocal of a number. The practice listed for lessons 2.14, 3.2 and 3.8 are good examples of properly addressing the interpret and compute aspects of 6.NS.1.
• Only two lessons, 2.6 and 2.7, truly focus on the standard.
• Lessons 2.5 and 2.8 are also listed as focus lessons for this standard. However, in lesson 2.5, there are no quotients of fractions interpreted or computed, and in lesson 2.8, students are actually comparing when the divisor is a whole number to multiplying by the reciprocal of that divisor. The divisors in lesson 2.8 are whole numbers.

Standard 6.EE.9 has 50 exposures in the curriculum and is listed as the focus of 10 days of Focus lessons.

• There are 10 focus lessons, 4.3, 7.3, 7.4, 7.6, 7.8 to 7.11, 8.7 and 8.8, that are listed as aligning to 6.EE.9, but only seven of the lessons actually align to the standard.
• Lesson 4.3 does not have students use any variables, and lessons 7.3 and 7.4 have students write the expression in electronic spreadsheets which detracts from using two variables in one equation.

Standard 6.EE.7 has 58 exposures in the curriculum and is listed as the focus of 11 days of Focus lessons.

• The Focus portions of Lessons 2.6, 3.3, 3.6, 5.9, 6.3, 6.4, 6.5, 6.8, 7.8, 7.9 and 8.6 are aligned to 6.EE.7.
• Many of the exposures/lessons overlap with the previous standard.
• In lessons 2.6, 3.3, 3.6 and 5.9, students do not have the opportunity to write equations with variables and proceed to solve those equations. In these lessons, students write numerical expressions and evaluate those numerical expressions.
• In lessons 6.3, 6.8 and 7.8, almost all of the equations that are written are of the form px + q = r, which means the lessons more closely align to 7.EE.4.A instead of 6.EE.7. Also, in lesson 6.8, students do not get to solve some of the equations that are written, and in lesson 7.8, the equations that are written are used to complete tables of data and create graphs, not solving real-world or mathematical problems.
• In lessons 6.4 and 6.5, students compare bar models and pan balances to develop a conceptual understanding of solving an equation, but these lessons do not provide students with opportunities to solve real-world or mathematical problems by writing equations and solving them.
• In lesson 7.9, students are expected to perform computations with quantities from a real-world context, but there is no direct connection to students writing equations to model the context of the problems before solving them.
• In lesson 8.6, students are expected to find the mean of a set of data by thinking about the mean as a balancing point, but, as in lesson 7.9, there is no direct connection to writing equations and using those equations to find the mean.

The Grade 6 Everyday Mathematics instructional materials partially meet the expectations for balance. Overall, the three aspects of rigor are treated separately within the materials, and the lack of lessons on conceptual understanding and application do not allow for a balance of the three aspects.

Despite efforts to include conceptual understanding and application, problems are all too often presented in a formulaic way. Questions give away the answers or prompt specific thought patterns. The order of questions often lead students to a specific procedure. Contexts are frequently routine and problems are posed in a way in which students can solve them by relying on the procedural skill. All aspects of rigor are almost always treated separately within the curriculum including within and during lessons and practice.

The instructional materials reviewed for Grade 6 partially meet the expectations for identifying the MPs and using them to enrich the Mathematics content.

The MPs are identified in the Grade 6 materials for each unit and the focus part of each lesson.

• For Unit 3, page 229 discusses how MP1 and MP2 unfold within the unit and lessons.
• For Unit 5, page 461 identifies which MPs are in the focus parts of the lessons within the unit.
• For Unit 7, page 655 explains the development of MP5 and MP6 in this unit.
• Within the lessons are spots where the MPs are identified.

However, within the lessons limited teacher guidance on how to help students with the MPs is given. Because there is limited guidance on implementation, it is difficult to determine how meaningful connections are made. Additionally, it is difficult to determine if the MPs have meaningful connections since the materials break them into small parts and never address the MPs as a whole. The broken apart MPs can be seen on pages EM8-EM11.

The materials partially meet the expectation for prompting students to construct viable arguments and analyze the evidence of others. MP3 is not explicitly called out in the student material. Although the materials at times prompt students to construct viable arguments, the materials miss opportunities for students to analyze the arguments of others, and the materials rarely have students do both occur together.

There are some questions that do ask students to explain their thinking on assessments and in the materials. Sometimes there are questions asking them to look at other's work and tell whether the student is correct or incorrect and explain. Little direction is provided to make sure students are showing their critical thinking, process or procedure, or explaining their results. Many questions that prompt students to critique the reasoning of others tell the student if the reasoning was originally correct and incorrect. It should be noted though that student materials never explicitly call out entire MPs at once; MP3 is broken into GMP 3.1 and GMP 3.2 in the materials.

The open response lessons could be opportunities for students to construct arguments for or against a mathematical question. However, besides just working in groups, there is little prompting from the teacher for students to discuss the answers of other groups or students

The following are some examples of where the materials indicate that students are being asked to engage in MP3 (Unit 1 and 6 claim MP3 to be a focus):

• For Unit 1, about half of the 14 lessons have opportunities for students to construct viable arguments, but some of those opportunities, such as in lessons 1-4 and 1-5, are only for students that volunteer and are chosen by the teacher.
• For Unit 1, there are only two lessons where students are expected to analyze the arguments of others. Both of these lessons also have students construct viable arguments. Neither of them prompt students to use critiques of their arguments to improve their arguments.
• Math Boxes, page 24, question 5 asks students to explain their reasoning for selecting an answer.
• Math Boxes, page 48, question 5 asks students to explain how they solved problem 2.
• Math Boxes, page 79, question 5 asks students to explain how the found the balance point in problem 3.
• For Unit 6, less than half of the 11 lessons identify opportunities for students to construct viable arguments. Lesson 6-6 has an opportunity for students to construct an argument that includes multiple parts. However, there is no opportunity for students to have their arguments critiqued so that the arguments might be improved.
• For Unit 6, there are only two lessons where students are expected to analyze the arguments of others. Neither of these lessons also identify opportunities for students to construct viable arguments.

The materials partially meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. The Grade 6 materials sometimes give teachers questions to ask students to have them form arguments or analyze the arguments of others, but typically the materials do not give both at the same time.

In the teacher's guide and lessons, the teachers have very specific, almost scripted, directions for students. Most, if not all, of the Math Master worksheets are presented in a step-by-step directive that does not allow for students to evaluate, justify, or explain their thinking. Usually only one right answer is available to the posed problem, and there is not a lot of teacher guidance on how to lead the discussion given besides a question to ask. There are many missed opportunities to guide students in analyzing the arguments of others. Students spend time explaining their thinking but not always justifying their reasoning and creating an argument.

The following are examples of lessons aligned to MP3 that have missed opportunities to assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others:

• Lesson 1-2 states, "Have students share how they matched a dot plot to each statistical question." The missed opportunity here is for teachers to guide students in a rich discussion about how they used the dot plot.
• Lesson 1-4 states, "Have the students pose arguments for why the two sides are not balanced," but teachers are not given guidance to help students pose the arguments.
• Lesson 1-9 is the open response lesson. It tells the teacher to ask “which histogram better supports your view” and "which features of the graph helped you to make your argument,” but does not provide guidance to the teacher to guide students in a rich discussion.
• Lesson 2-5 cites MP3; however, the questions posed have right and wrong answers and do not have students engaging in constructing viable arguments or analyzing the arguments of others. There is no direct guidance to help the teacher engage students in MP3.
• Lesson 2-8 has students explain whose strategy is correct. Again, there is not instruction or guidance for the teacher to help the students explore the explanations of others.

The instructional materials reviewed for Grade 6 partially meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics; however, often the correct vocabulary is not used.

• Each unit includes a list of important vocabulary in the unit organizer which can be found at the beginning of each unit.
• Vocabulary terms are bolded in the teacher guide as they are introduced and defined but are not bolded or stressed again in discussions where students might use the term in discussions or writing.
• Each regular lesson includes an online tool, "Differentiating Lesson Activities." This tool includes a component, "Meeting Language Demands," that includes vocabulary, general and specialized, as well as strategies for supporting beginning, intermediate, and advanced ELLs. An example of this, from Lesson 7-4, includes "For Beginning ELLs, use visuals, restatements, role play, and read-clouds to help students understand task directions and written statements."
• Everyday Math comes with a reference book that uses words, graphics and symbols to support students in developing language.
• Some units have a heavy load of required mathematical vocabulary. In unit 5, there are 28 vocabulary words needed for students in Grade 6 to understand the unit. Some of these words include compose, cubic units, decompose, net, scale drawing, surface area and others. In contrast, Unit 8 only has 12 vocabulary words for the unit which is a much more manageable number for students in Grade 6.
• Correct vocabulary is often not used. For example, name-collection box instead of equivalent equations or equivalent expressions, nested parenthesis instead of brackets, number model instead of expression, responding variable instead of dependent, and manipulated variable instead of independent variable.