7th Grade - Gateway 2

The instructional materials for Grade 7 partially meet the expectations to develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Overall, the instructional materials present inquiry labs and higher order thinking questions as a way to develop conceptual understanding, however, the conceptual development is generally a quick activity which hurriedly ends with a set of rules to follow, and there is little time for students to discover their own knowledge.

• Conceptual understanding is called for in 7.NS.A, this is primarily covered in chapters 3 and 4.
  • Chapter 3 covers operations with integers 7.NS.1. The lessons on adding, subtracting, and multiplying integers have an associated inquiry lab that helps students visualize the operations using counters. The lessons with counters are not incorporated into the lessons themselves and other then a few practice problems in the inquiry lab students don’t use the counters to complete problems.
  • The lessons in chapter 3 use number lines in the examples as a way to explain operations with integers, but do not expect students to use the number lines for practice. In order to be in full alignment with the intent of the standards, students should have multiple opportunities to develop their ability to demonstrate understanding using number lines.
  • Chapter 4 covers operations with all rational numbers (7.NS.2 and 7.NS.3). Two inquiry labs give some conceptual understanding of rational numbers. The inquiry lab prior to lesson 1 uses number lines to explain ordering negative fractions. The inquiry lab prior to lesson 3 uses a number line to show adding and subtracting positive and negative fractions. Both labs offer a nice introduction to conceptual understanding, however, that understanding is not reinforced in the corresponding lessons and practice problems.
  • The lessons on multiplying and dividing rational numbers start by showing the algorithm and do not develop a conceptual understanding.

• Conceptual understanding is called for in 7.EE.A. This is primarily covered in Chapter 5.
  • In chapter 5 students use the properties of operations to generate equivalent expressions. This chapter gives students some practice with algebra tiles to help students get a conceptual understanding of equivalent expressions. For example, the materials develop a nice conceptual understanding of factoring linear expressions. At the start, an inquiry lab shows students how to use algebra tiles to get a visual of factoring. In the lesson, students are shown two ways to factor. Method 1 encourages the use of a model, and method 2 shows students how to use the greatest common factor.
  • At the end of each lesson there are Higher Order Thinking Problems. These give students the opportunity to develop conceptual understanding. Students are asked to explain their understanding by justifying conclusions, finding errors or looking at a problem's structure.

The materials reviewed for Grade 7 partially meet the expectations of attending to fluency and procedural work within the lessons. Overall, there are multiple opportunities for students to develop procedural skill and fluency, however, one of the standards that calls for fluency 7.EE.B.4 is not thoroughly attended to.

• Procedural skills and fluency are called for in 7.NS.A which is primarily covered in Chapters 3 and 4. The chapters are comprised of lessons that give many questions that help students develop fluency.
  • In the Teachers Edition, on the side bar, there are questioning strategies that give students the chance to articulate procedures. For example, on pages 204 - 205 the teacher is encouraged to refer to example problems on adding integers and asks students to notice when the signs are the same or different and then state what happens to the sum in those cases
  • The Higher Order Thinking (H.O.T) Questions give students a chance to articulate their understanding of procedural skills. For example, Question 14 on page 208 asks students to name the property illustrated by the following x + (-x) = 0 and x + (-y) = -y + x. The H.O.T questions also give students the ability to apply procedural skills to unfamiliar types of problems. For example, Question 15 on page 238 asks: The product of two integers is -21. The difference between the integers is -10. The sum of the two integers is 4. What are the two integers?
  • Students are given opportunities to continually engage in fluency through-out the year. After students have practiced operations with integers in Chapter 3, they continue to use operations with integers in chapter 5 when they are working with expressions.

• Procedural skills and fluency is called for in 7.EE.A.1 which is primarily covered in Chapter 5. This chapter also includes ample fluency based practice problems in the guided and independent practice sections. As well as H.O.T questions and teacher directed questioning strategies to help students articulate and apply procedural skills.

• Procedural skills and fluency is called for in 7.EE.B.4. According to the glossary of Common Core Standards at the beginning of the Teachers Edition this topic is covered in Chapter 6. Chapter 6 focuses on solving equations and inequalities. There are no lessons that specifically have students use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Though there are places in the lesson that touch on this, there is a lack of problems that would allow students to develop fluency in constructing simple equations.

The instructional materials for Grade 7 partially meet the expectation so that teachers and students spend sufficient time working with engaging applications of mathematics, without losing focus on the major work of the grade. Overall, the materials have multiple opportunities for application, but many of the application problems are one-step, routine word problems in which students are directed on the procedure to follow in order to solve the problem.

• The materials incorporate the following application type of lessons throughout the chapters.
  • The Power Up Performance Tasks at the end of each chapter offer students multi-step abstract questions where they solve problems by using a variety of solution paths.
  • At the end of each unit there is a unit project. This project gives students the opportunity to research a topic and relate that information to the mathematics of the unit.
  • The materials have problem solving investigation through-out each chapter. It gives students step-by-step ways to use a problem solving strategy

• Application problems are called for in 7.RP.A. The majority of this standard is covered in Chapters 1 and 2.
  • The lessons begin with a real-world link, where students perform a task that introduces the lesson. For example, Chapter 1, Lesson 1 "Rates", students work with a partner to take each other’s pulse and record the results thus beginning the conversation about rates.
  • There are application problems included in the guided and independent practice sections, however, students are rarely given the opportunity to solve problems without being told how to solve them. The materials rarely give students the chance to pick their own process for solving a problem.

• Application problems are called for in 7.NS.A.3 the majority of this standard is covered in Chapter 4 Lessons 3 - 8.
  • The application problems covering this standard are incorporated into a series of lessons where each lesson teaches the various operations with rational numbers. For example, Lesson 5 "Add and Subtract Mixed Numbers" The majority of the lesson is on procedural skills and fluency with some word problems included at the end of the problem set. The word problems are generally routine and one-step in nature.

• Application Problems are called for in 7.EE.B.3 this standard is covered throughout chapters 2, 3 and 4.
  • Even though there are many lesson that lead students to covert between mathematical forms, there are very few opportunities for students to discuss why one form would be a better choice then another.

The Instructional materials reviewed for Grade 7 partially meet the expectation for identifying and using the MPs. Overall, the materials clearly identify the MPs and incorporate them into the lessons, however the MPs are often over identified.

• The MPs are incorporated into each lesson, so they are used to enrich the content and they are not taught as a separate lesson.
• There is a Mathematical Practice Handbook at the start of the textbook. This handbook explains each practice standard and gives example problems for each standard.
• There is a table of contents that specifically addresses the MPs and it lists the pages where you could find each of the practices. All of the MPs are represented.
• Each lesson identifies several mathematical practices. For example, Chapter 3 Lesson 3, claims to incorporate MP1 through MP7. The materials point to these practice standards in the student practice section of Lesson 3 and in the Ideas for Use in the side bar of the teachers edition.
• Items are often over identified. In the side bar of the teachers edition, teaching strategies are suggested. Often those strategies are identified as attending to multiple strategies. For example, Chapter 6 Lesson 6, "Alternate Strategy" in this activity, students review the symbols used for inequalities and what they mean. Students than come up with key words or phrases that can be used for each symbol. Then they are asked how they know when the symbols should be used for the problem in the Real-World Link. This activity claims to incorporate MP1, MP2 and MP3. However, there is no explanation or description as to how these practices are incorporated.

The materials reviewed for Grade 7 partially meet the expectation for appropriately prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards. Overall, every lesson's problem set, has one or more questions in which students have to explain their reasoning. However, students are only occasionally prompted within problem sets, and application problems to explain, describe, critique, and justify their work.

• In the practice problems nearly every lesson includes questions that are specifically labeled with the heading "Justify Conclusions" these questions ask students to explain how they got their answers.
• In a few lessons the questions are labeled in bold with the heading "Construct a Viable Argument" These questions often ask students to explain if something is true or not.
• In some lessons the questions are labeled in bold with the heading "Find the Error". In these classic error analysis problems students are presented with someone's solution and asked to simply identify the error. This does not attend to the full meaning of the standard, where students would need to refute claims made by others by offering counter examples and counterarguments. There were very few instances where students were asked to find a counter-example.
• Students had a few instances where they were asked to make conjectures. Often this was not really a conjecture as students were not asked to make a generalization, instead they were asked to solve a specific problem. For example, in chapter 7, lesson 2, question 11d, students are asked to generate a conjecture from one observation based on a small drawing of intersecting lines. An exemplary model of creating conjectures would ask students to gather a variety of evidence to look for patterns and then make conjectures.

The materials reviewed for Grade 7 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others. Overall, the materials direct teachers with many scaffolding questioning strategies asking higher level questions and offering some suggested activities that lead students to construct viable arguments and analyze the arguments of others. However, the materials lack suggestions or ideas that guide a teacher with setting up scenarios where students experiment with mathematics and based on those experiments construct and present ideas.

• In the side bar of the teachers edition the teacher is provided with many scaffolding questions. The Beyond Level questions do a great job of asking higher Depth Of Knowledge level questions and provide supportive structures to analyze student arguments.
• In the side bar of the teachers edition there are suggested activities for teachers to use with students. Very often these suggested activities have students compare or analyze critique, and analyze answers. For example, chapter 5, lesson 1, "Find the Fib" Students work on a team where one student creates 3 problems, two are solved correctly and one is incorrect. The other students find the one that is wrong and correct it.
• The Higher Order Thinking Problems in the students practice section of the materials incorporate some of the MPs that help students to construct viable arguments and analyze the arguments of others. Students are given opportunities to be persistent in their problem solving, to express their reasoning, and apply mathematics to real-world situations. However, further guidance on how to promote this and support students in the development of these skills is not given. This is coupled with the fact that many students are rarely given authentic opportunities to develop the true intent of any of the MPs mentioned above.

The materials reviewed for Grade 7 partially meet the expectation for attending to the specialized language of mathematics. Overall, the materials identify and define correct vocabulary but there are only sporadic places where vocabulary is integrated into the lessons.

• At the start of every chapter, there is a list of related vocabulary words that will be used in the chapter. Students are given a box that outlines key concepts and key words are highlighted in yellow and immediately defined.
• In the guided practice section students answer a "Building on the Essential Question" question, in which they have to understand the vocabulary to answer the question. For example, Question 3 on page 84 asks; How can you determine if a linear relationship is a direct variation from an equation? table? a graph?
• In each lesson that introduces new mathematical vocabulary there is a Vocabulary Start-Up, that frequently uses a graphic organizer to help students understand the new vocabulary. The materials offer related vocabulary at the start of the lessons, however, minimal reference is made back to them as the lesson progresses. In this way, students are not explicitly supported in coming back and revising/adding to their understanding of these terms. Assumption is made that mastery of vocab is immediate.
• At the end of the chapters there is a vocabulary check included in the chapter review.
• Students are given sporadic opportunities to express math vocabulary with the daily lessons. The materials lack consistent structures to make math terms meaningful and incorporate high levels of mathematical language. There are few places where students are given the opportunity to write or explain, in a way that the use of mathematical vocabulary is assessed. The vocabulary usually consists of key words highlighted for the introduction of the lesson with a given definition.