6th Grade - Gateway 2

The instructional materials reviewed for Grade 6 partially meet the expectations for giving attention to procedural skill and fluency. Overall, the materials give opportunities to practice with procedural skills when those skills are first introduced, but the materials do not give opportunities with different procedural skills throughout the year so that fluency is completely addressed and developed.

• In lesson 6 of chapter 1, students encounter 50 percent of problems that require them to divide by a multi-digit number.
• The test prep and mixed review sections in each lesson provide an opportunity for repeated practice, but the items do not have students divide by multi-digit numbers throughout the year (page 177, problem 23; page 187, problem 54).
• In lesson 6 of chapter 2, students are given problems that allow for more practice with dividing rational numbers and have opportunistic strategies that can be used (i.e., 16j=80 or 2.5g=17.5) and also generic cases where students must use the algorithm (5.6k=19.152).
• Students simplify expressions in chapter 3, but they have little opportunity to become fluent with generating a variety of equivalent expressions instead of only simplified equivalent expressions (6.EE.A).

Students multiply and divide decimals (6.NS.B.3) in chapter 1, but they have limited opportunities to do so throughout the text.

The instructional materials reviewed for Grade 6 partially meet the expectations for giving attention to applications. Overall, the materials are designed so that teachers and students spend limited time working with engaging applications of the mathematics, without losing focus on the major work of each grade, and there are very few opportunities for students to engage in non-routine problems.

• There are some non-routine, real-world problems given in the homework as challenge questions, but there is typically only one question provided per lesson.
• There is a problem-solving handbook in the beginning of the book. Problem-solving strategies are modeled, and then, five or six problems are provided. The problems require students to solve each situation by entering the situation from a different angle of the standard and are connected to the major work of the grade.
• Each chapter has a “Guided Problem Solving” activity toward the end of the chapter, and these have real-world scenarios or contexts.
• The “What You Might Think” questions have generic questions that scaffold student thinking, but the “What You Might Write” sections are too suggestive to encourage alternate solution paths (page 77).

In chapter 6, approximately one out of five exercises, including the lessons and chapter test, include applications.

The instructional materials reviewed for Grade 6 partially meet the expectations for balance. Overall, the three aspects of rigor are not always treated together nor are they always treated separately within the materials, but there is not a balance of the three aspects of rigor within the grade.

• The materials have an emphasis on procedural skill and fluency by providing numerous skill-driven problems. The materials do not create a balance between the three aspects of rigor as they don’t fully promote the development of conceptual understanding or provide enough non-routine problems.

Examples of models to develop conceptual understanding are provided on a limited basis, and some real-world problems are provided in the homework sections.

The instructional materials reviewed for Grade 6 partially meet the expectations for identifying and using the MP to enrich the mathematics content throughout the grade. Overall, the MP are identified in different places in the materials, but the places where they are identified do not provide enough explanation as to how the practices enrich the mathematics content. 

• The MP are identified on pages T26-T31.
• Each practice is defined and explained how it is used throughout the materials, such as: MP1 found in the Guided Problem Solving exercise within the homework problems; MP3 and Error Analysis; and MP5 found in the Activity Labs and Choose a Method exercises.
• The pages are provided as to where to find the use of the MP; however, when viewing the pages, there is no additional guidance provided on the use of the MP.
• There is reference to MP1 in the student edition found on pages xxxii – xlix, but the practice itself is not identified or referenced.
• MP2 is noted in the teacher edition to be located on pages xx – xxii in the student edition, but the pages are an index of “Connect Your Learning” activities.

The MP are found in the student edition on page viii, but there is no explanation of their meaning or importance.

The instructional materials reviewed for Grade 6 partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials provide opportunities for students to construct viable arguments and analyze the arguments of others, but the instructions for those opportunities do not always prompt students to give explanations for their answers. 

• There are opportunities for students to construct viable arguments and analyze the arguments of others in the content standards through check your understanding and homework exercises titled “Error Analysis” and “Writing in Math.”
• On page 77, students are asked how much fuel would a plane have when landing, but there is no expectation of explaining the reasoning.
• Page 110 provides a reasoning problem, but it only asks to apply an algorithm to use the distributive property.
• Under the section for MP3, page 11 is listed as having examples of the practice. Problems 30 and 31 are labeled Reasoning and ask the student to place parentheses to make the statement true, but students do not have to provide any information about the thinking behind their choice.
• In the Writing in Math, problem number 35, students are told to explain the steps to find the value of an expression, but they do not have to build an argument about why subtraction comes first in this case.
• In More Than One Way, students are supposed to “analyze and critique the solution plans of two students.” On pages 133 – 134, two different methods are explained step by step.  Students are asked to choose a method to model the quotient of 3¾ divided by 1/8.  They are not asked any follow up questions in order to evaluate the methods.

Some sections have an “error analysis” exercise so that students can find and correct an error; however, they are told that there is an error to correct.

The instructional materials reviewed for Grade 6 partially meet the expectations for explicitly attending to the specialized language of Mathematics. Overall, the materials typically address Mathematical language appropriately, but there are few opportunities for students to demonstrate using the language correctly.

• Materials provide vocabulary with definitions throughout the lessons when key terms are being introduced. Key terms are highlighted when used within the lessons (Key Concepts).
• Vocabulary is built as definitions within the examples of each lesson. Key terms and concepts are identified by highlighted text and often followed by a diagram that models the word or an example of the process.
• Since little explanation is required in most of the lessons, students do not have as many opportunities to practice using the language.
• New vocabulary is called out at the beginning of each section, and references to new vocabulary are highlighted within each section.

A key example of lack of attention to the specialized language of Mathematics is evident when dividing by fractions, particularly on page 137 when students are told to remember division of fractions by thinking “invert and multiply.”