8th Grade - Gateway 2

The instructional materials for Common Core Coach Suite Grade 8 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials develop conceptual understanding using concrete visual models during class instruction and present students with opportunities to explain their thinking through Discussion Questions and prompts. For example:

• Common Core Coach Lesson 10 Solving Systems of Two Linear Equations Graphically addresses the understanding that the solutions to a system of two linear equations in two variables correspond to the point of intersection of their graphs, because points of intersection satisfy both equations simultaneously (8.EE.8a). Discussion Questions for Examples A and B prompt students to demonstrate conceptual understanding by explaining how they would test a conjecture and how they know whether or not the given system of equations has infinitely many solutions. Three of the 19 practice problems prompt the student to explain. Problems 17 and 18 state: “Without graphing, determine whether each system of equations will have no solution, one solution, or infinitely many solutions. Explain your answer.”
• Common Core Coach Lesson 13 Introducing Function addresses the understanding that a function is a rule that assigns to each input exactly one output (8.F.1). Students see examples of functions represented in verbal descriptions, input-output tables, graphs, and mappings. Students are prompted to “Explain how you can tell from a graph whether a relation is a function.” Problem 11 states: “Yoshi draws a vertical line on the graph of a relation. His vertical line intersects only 1 point of the graph. He determines that the relation is also a function. Assess Yoshi’s conclusion and explain why it is or is not necessarily correct.” Students engage in conceptual understanding to explain the solutions to these prompts.

However, students are given few opportunities to independently demonstrate conceptual understanding. During Independent practice, students solve problems similar to the examples from class instruction, with slight differences in the context and/or numbers. Students rarely create visual representations on their own. There are Practice questions with labels such as “Write Math,” “Describe,” or “Compare” where students explain mathematical concepts. The questions elicit students' ability to restate the mathematics ideas addressed during class instruction by the teacher. The materials address conceptual understanding standards in a proceduralized way and do not enhance the student's ability to form a conceptual understanding of major work within the grade. For example:

• In Common Core Coach Lesson 8 Relating Slope and Y-intercept to Linear Equations, the materials use similar triangles to explain why the slope is the same between points on a non-vertical line in the coordinate plane and the derivation of the equations y=mx and y=mx+b (8.EE.6). Students do not have an opportunity to develop a conceptual understanding of the relationship between similar triangles, slope, and equations in slope intercept form. In the Understand section students are given several examples:
  • In Example 1, students are given two similar triangles on a line in the coordinate plane. Example 2 leads students through a derivation of the equation y = mx. In the final example, students are given a procedure for finding the equation of a line. “1. Find ordered pairs from the graph; 2. Determine the slope, m, of the line; 3. Determine if y = 2x is the equation of the line. Compare the graph of y = 2x with the given graph; 4. The slope-intercept form for a linear equation is y = mx+b, where m is the slope of the line and b is the y-intercept. Adding 4 units to each point of y = 2x results in the equation y = 2x + 4, where the slope is 2 and the y-intercept is 4.” All Practice problems are related to using the procedural steps outlined in the example and working with the equation of a line.
• In Common Core Coach Lesson 19 Understanding Congruence of Two Dimensional Figures (8.G.A) Understand and Connect sections, students are given the definition of congruence as well as examples of different ways to show that the figures are congruent. Students are prompted to “Identify the type of rigid motion that could be used to show in one step that each pair of triangles is congruent” and have six practice problems to identify congruent figures. However, students do not explain why the figures are congruent, the underlying concept for the lesson.

The instructional materials for Common Core Coach Suite Grade 8 partially meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

Many lessons in the suite provide students with opportunities to use computation skills. Common Core Coach lessons conclude with two pages of practice problems, Common Core Support Coach lessons conclude with three practice problems, and Common Core Performance Coach lessons conclude with independent practice problems. Additional pages for practicing procedural skills are found in Appendix A of the Common Core Coach Teacher’s Guide. For example:

• Common Core Coach Lessons 3, 4, 5, and 6 address cluster 8.EE.A (Work with radicals and integer exponents). In Lesson 4 Cube Roots Practice Problems 13 through 24, students develop procedural skills as students solve problems in the form of $$x^2 = y$$. Problem 22 states, “Solve the equation for x. $$x^2 = 2.25$$”

Throughout the Common Core Suite, students are given examples with step-by-step procedures at the beginning of each lesson. The majority of practice problems involve working with these procedures; however, students do not have opportunities to independently demonstrate the full intent of some standards that address procedural skills. For example, in Cluster 8.EE.C, students analyze and solve linear equations and pairs of simultaneous linear equations:

• In Common Core Coach Lessons 10 and 11, students have opportunities to solve systems of equations both graphically and algebraically. However, within these lessons there are limited opportunities to develop procedural skills with solving systems of linear equations resulting in infinitely many or no solutions.
• Common Core Support Coach Lesson 6 Linear Equations with Rational Coefficients includes 10 problems where students solve linear equations with one variable. Problem 7 states, “Solve for z. If there are infinitely many solutions, write infinitely many solutions. If there is no solution, write no solution. 0.5(10z + 20) - 5 = 0.1(50z + 50)”

In Grade 8, there are few lessons that specifically address procedural skills, so there are few opportunities for students to independently demonstrate procedural skills throughout the year. There are lessons in all three components of the suite that develop procedural skills, but students do not independently demonstrate those skills. Many procedural skills needed to solve problems are scaffolded so that students fill in spaces and do not demonstrate the skills on their own.

The instructional materials for Common Core Coach Suite Grade 8 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of the grade. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied.

In the Common Core Coach Teacher’s Edition, the Table of Contents denotes lessons that apply skills to real-world problems. Common Core Support Coach does not label specific lessons as application. In Common Core Performance Coach, there is one Performance Task at the end of each domain that applies concepts and skills to real-world problems. Non-routine problems are addressed in the Performance Tasks, and there are five Performance Tasks throughout the year.

Students have some opportunities to engage in routine application problems; however, they are often given a solution strategy. There are few opportunities for students to engage with non-routine problems, and those opportunities present in either Common Core Support Coach or Common Core Performance Coach are not assigned to all students.

Examples where students engage with routine application problems:

• In Common Core Coach Lesson 12 Problem Solving: Using Systems of Equations, examples are presented for a four-step method for problem solving. Practice problems 3-5 are routine word problems that follow the given four-step method.
• Digital Assessment, Domain Assessment: Expressions and Equations: Problems 24 and 29 prompt students to identify a system of equations for a word problem (24) and solve a system of equations using elimination (29). Both problems are similar to the examples provided in the lesson.
• In Common Core Performance Coach Lesson 15: Using Functions to Model Relationships, the Practice problems are routine applications of problems with questions asking for the initial value and rate of change. In Practice Problem 8, students are given a graph that shows the height of a balloon during the first 30 seconds it is released. Practice Problem 8 states, “The graph shows the height of a balloon during the first 30 seconds after it is released. Part A: Find the initial value and the rate of change of the function represented by the graph. The initial value is ___. The rate of change is ___. Part B: Write an equation in slope-intercept form that shows the relationship between the number of seconds, x, and the height in feet, y. Interpret the meaning of the initial value and the rate of change in terms of the situation.” This context is similar to the lesson examples.

Examples where students engage in non-routine application problems:

• Common Core Performance Coach Lesson 11 Solving Systems of Two Linear Equations Algebraically Practice Problems 1, 5 and 6 provide students with an opportunity to solve non-routine mathematical problems involving systems. For example, Practice Problem 1 gives a point (2, -4) and prompts students to find the values of a and b in the system: ax + 2y = -2, ax + by = 10. However, not all students will be assigned these problems.
• Common Core Support Coach Lesson 7 Linear Equations in Two Variables provides examples for solving systems of equations through graphing and algebraically. Students solve seven word problems using systems of equations. Three of these problems are non-routine and do not identify the variables or equations for students.

The instructional materials for Common Core Coach Suite Grade 8 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the program; however, they are mostly treated separately, with an emphasis on procedural skill and fluency over the other aspects of rigor.

Common Core Coach designates lessons that are specifically identified as procedural skill, concept, or problem solving (application) lessons. However, the majority of the materials present the mathematics procedurally.

The Common Core Coach Teacher’s Manual states,“1. Concept Lessons begin with an underlying concept that connects directly to the skill or skills taught in that lesson. 2. Skill Lessons start directly with a skill and work through many variants of its application. All skills are developed through Examples. 3. Problem-Solving Lessons apply skills to real-world problem situations. Students will use a four-step problem-solving process to approach mathematical problems.”

Throughout the Common Core Coach suite, students engage with mathematics through scaffolding and problem-solving strategies that proceduralize lessons addressing conceptual understanding and application. For example, in Common Core Support Coach Lesson 16 Using Functions to Model Relationships, students identify the slope (rate of change) and y-intercept (initial value) in twelve routine practice problems based on the examples in the lesson.

The instructional materials reviewed for Common Core Coach Suite Grade 8 partially meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

The Standards for Mathematical Practice (MPs) are identified in the Teacher Editions of all three components of the suite. The MPs are identified in the “teacher notes” and are mostly found during the discussion portion of the lessons. Common Core Coach Teacher’s Manual page 10 states, “The Standards for Mathematical Practice are connected to content throughout Common Core Coach. The Standards for Mathematical Practices are aligned to interactive questions, practice questions, and assessment items. Correlations to Standards for Mathematical Practices are indicated by tags.” The following examples illustrate how the MPs are identified across the Common Core Suite:

• Common Core Coach Lesson 20 Rigid Motion on the Coordinate Plane identifies MP1 and MP5: “TRY MP1 MP5 Contrast the rule with the rule in the example. Have students predict the translation of the triangle from the rule.” This is associated with the following prompt found in the student materials: “Now use the following rule to translate $$\triangleABC$$ to form $$\triangleA"B"C"$$:$$(x, y) → (x - 5, y + 6)$$.”
• Common Core Support Coach Teacher Edition includes a “Spotlight on Mathematical Practices” section in each lesson, providing the teacher with more detail on where the MPs are woven into the lesson and “notes that support teachers at point-of-use to develop strong mathematical behaviors.”
• Common Core Performance Coach Teacher’s Manual, page xx, states, “The following Standards for Mathematical Practice are leveraged throughout Performance Coach.” The MPs are listed, defined, and describe examples of use within a mathematical context.

Although MPs are identified throughout the suite, they do not serve to enrich the mathematical content. Since they are identified primarily in discussion questions, the materials lack guidance for teachers on how the highlighted MPs connect to the mathematics in which students are engaged. Thus, the treatment of the MPs is fragmented across the suite and do not provide opportunities for students to make connections and interact with the MPs in a meaningful way. For example:

• In Common Core Coach, each lesson identifies the MP’s within the “Understand” section with a heading “Discuss” or “Try,” along with some example problems titled “Check, Discuss, or Model.” However, there is no guidance on how to use the MPs to help enrich the mathematics within the lesson.
• In the Common Core Support Coach Teacher’s Manual “Introduce Concepts and Vocabulary” and the “English Language Learners” sections, connections are made to MP6. While teachers are given suggestions on how to assist students with developing vocabulary (e.g., Lesson 1 vocabulary decimal states: “Have students identify the root word, terminate, in the term and determine a definition for the word.”), teachers are not provided with further direction within the lesson on how to carry out this lesson to ensure that students are developing MP6.
• In Common Core Performance Coach Lesson 12 Introducing Functions, the Teacher’s Manual provides aligned discussion questions to MPs:
  • MP5 - "Explain how a vertical line test can be used to decide if a graph shows a function or only a relation. Why won’t a horizontal line test work?”
  • MP2 - "How can you change a relation into a function? a function into a relation?”
  • MP6 - "How are relations and functions alike? How are they different?”

While each practice is represented in this suite, there are a few instances where the MPs do not enrich the content. For example:

• In Common Core Support Coach Lesson 3, the content is not enriched by MP4 when the teacher is prompted to, “Have students discuss other examples of very large numbers.” Students are given one example, “The mass of the sun is 1,988,920,000,000,000,000,000,000,000,000 kilograms.” This is not the intent of MP4, as students do not model with mathematics.
• In Common Core Support Coach Lesson 18, the content is not enriched by MP5 when the teacher is prompted to, “Have partners discuss briefly before group discussion. As needed, have students use graph paper to model the scale on a coordinate grid. How would changing the scale of the x-axis impact the look of the scatter plot?” This is not the intent of MP5, as students are given a tool and do not have the opportunity to use tools strategically.

The instructional materials reviewed for Common Core Coach Suite Grade 8 partially meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. There is little teacher guidance on how to lead discussions beyond the provided discussion questions, and there are missed opportunities to guide students in analyzing the arguments of others.

In Common Core Coach, items marked as addressing MP3 are often related to the teaching of the content with little or no assistance to teachers to engage students in both constructing viable arguments and analyzing the reasoning of others. Most often the materials prompt a discussion about “the topic” to assist students, but there are limited questions or prompts for teachers to support students’ development of arguments. Often suggestions for teachers regarding MP3 focus on students checking their work. Teachers are not provided with strategies for students to analyze the work of others in any of the lessons. For example:

• In Lesson 11, in the paragraph labeled “Discuss,” the teacher is prompted, “Have students explain in their own words what elimination means. Discuss how to use the meaning of the word to understand the name of the method.” This is shown as aligned to MP3. Questions that prompt students to understand mathematical terminology and methods do not assist teachers in helping students to construct viable arguments or analyze the arguments of others.
• In Lesson 13 Introducing Functions "Understand-Connect," the teacher is prompted, “Have students explain in their own words what a function is and how to recognize whether a relation is a function. Encourage them to provide a method for recognizing whether a relation is a function, both graphically and from a list of ordered pairs.” This is shown as aligned to MP3. Questions that prompt students to explain in their own words the definition of a function do not assist teachers in helping students to construct viable arguments or to analyze the arguments of others.

Common Core Support Coach provides limited assistance to teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. Most often when MP3 is identified, teachers are directed to “Have partners discuss briefly before group discussion.” Some lessons contain a section titled “Spotlight on Mathematics” that offers additional support for teachers in developing critical thinking by offering probing questions to use with students. In addition, teachers are frequently provided a prompt and sentence starter to assist students. However, these probing questions and prompts do not allow for students to construct arguments or critique the reasoning of others.

• Lesson 1 “Spotlight on Mathematical Practices” states, “Help students explain their reasoning by asking probing questions: How can you use a place-value chart to compare these numbers?” The materials do not assist teachers in helping students construct their own argument or analyze the arguments of others as students are not asked a question that requires an argument. Instead it encourages students to explain how to use a place-value chart to help them compare numbers.
• In Lesson 15 “ Support Discussion," the following discussion question is given and aligned to MP3: “Have partners discuss briefly before group discussion. Suggest that they consider the types of angles from the Plug In as well as the ones here.” This discussion question does not assist the teacher in helping students to construct viable arguments or analyze the arguments of others.

In Common Core Performance Coach, there are no directions to assist teachers in engaging students in constructing arguments or analyzing the arguments of others. Although discussion questions and journal prompts are provided, there are no prompts for teachers, or example student answers to guide the teacher. MP3 is addressed within the discussion questions at the beginning of lessons and within the journal prompt that accompany most lessons. Additional support for the teacher related to MP3 is not present within the lessons.

• In Lesson 2, a Discussion question is presented: “How could you check two approximations of a square root to see which one is more accurate?” This question potentially assists students with constructing their own arguments related to estimating a square root. However, there is little guidance for the teacher as to how and when to present the question, and no guidance to the teacher on having the students analyze the arguments of others.
• In Lesson 8, a Journal Prompt labeled as MP3 states, “Why do you think that the ratios of the triangles’ corresponding side lengths and the slope of the line are equal? Explain.” There is no guidance to the teacher on having students construct arguments or analyze the reasoning of others to provide an answer.