8th Grade - Gateway 2

The instructional materials for Ready Grade 8 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The materials include problems and questions that develop conceptual understanding throughout the grade level. The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade through the use of visual models, real-world connections, mathematical discourse prompts, concept extensions, and hands-on activities.

• In Unit 3 Lesson 11 Represent Proportional Relationships, students graph and compare proportional relationships. Students examine the connection between tables, graphs, and equations and are guided through a series of questions to help them interpret the slopes of the graphs. They apply their understanding to a new problem and make a connection between price per pound represented in an equation to the same rate represented in a graph. Students examine multiple representations of proportional relationships and make connections between the various representations and the underlying unit rates.
• In Unit 2 Lesson 7 Compare Functions, students create a table to be used with developing an expression and create a graph with the same data to examine multiple representations of functions.
• In Unit 4 Lesson 25 Distance in the Coordinate Plane, students create a model for finding the distance between two points. Students draw a right triangle, find the lengths of the legs, and develop an understanding of how the Pythagorean Theorem is used to find the distance between two points on the coordinate plane.

The instructional materials for Ready Grade 8 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill.

The instructional materials attend to procedural skills in the following ways: lessons that address specific standards, activities, and problem sets. Many of these opportunities are provided in the Additional Fluency Practice book for 8th grade. In the Practice and Problem Solving book the development of procedural skills happen daily. In addition, the resources include math games for centers and unit practice are included in the units. Some examples of lessons where procedural skills are used and practiced include:

• In Unit 4 Lessons 26 and 27, standard 8.G.9 is explicitly taught, and students practice using formulas in lesson problems.
• In Unit 3 Lessons 13 and 14, standard 8.EE.7 is explicitly taught, and procedural skills are practiced in problems throughout the lessons as students substitute numbers for variables including rational numbers to solve equations and/or find equivalent solutions.
• In Unit 1 Lesson 5 Operations and Scientific Notation, students are given guided instruction on the procedure for adding, subtracting, and multiplying numbers in scientific notation. Students then practice solving problems independently that involve addition, subtraction, and multiplication of numbers in scientific notation. Question 6 reads, “Find the perimeter of the rectangle in scientific notation.” A rectangle is shown with one length and width measurement given in scientific notation.
• In Unit 3 Lesson 13 Solve Linear Equations with Rational Coefficients, students are given guided instruction on the procedure for solving an equation with rational coefficients. This guided instruction includes five steps to solving the equation. Students then practice the procedure for solving the equations in six questions.
• Question 2, “Suppose that you first want to eliminate the fraction in the example equation. What would your first step be? Is -2 still the solution when you start by eliminating the fraction first? Explain.”
• Question 4, “Describe the first step you would use to solve the equation 20 = 7y + 2 - y. Is that the only possible first step?”

The instructional materials reviewed for Ready Mathematics Grade 8 meet the expectation 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 opportunities for students to apply mathematical knowledge and/or skills in a real-world context.

• In Unit 2 Lesson 10 Graphs of Functional Relationships (8.F.5), students interpret slopes of graphs that reflect Mr. Dilla driving through the neighborhood. There are several different representations of his driving to work. During Independent Practice students are to “Think about a trip that you have taken, or a route you travel often. Sketch a graph of the trip and write a description.” Students choose how to represent the independent and dependent variables, for example, graphing distance versus time or speed versus time, and there is opportunity for discussion of different approaches.
• Lesson 27 Problem and Practice Solving Problem 5 states: “A toy company makes sphere-shaped and cone-shaped toys. Two of the toys are shown at the right. Lawrence says that because the height and radius of the cone are equal to the radius of the sphere, the two toys have the same volume. Explain why Lawrence is incorrect.” Students apply their understanding of volume to the cone and sphere in order to analyze Lawrence’s thinking.
• The Unit Assessment Form B for Unit 5 Problem 3 allows students to apply their understanding of representation of survey data in multiple ways and analyze the results: “Dottie surveys students at her school to find out if there is an association between grade level and a favorite hobby. Of the 7th graders responding to the survey, she found that 18 prefer reading, 19 sports, and 15 crafts. Of the 8th graders responding to the survey, she found that 11 prefer reading, 25 sports, 9 drawing, and 11 crafts.” Students complete 3 parts, in Part A students make a two-way table to display the results of the survey. In Part B, students make a relative frequency table, and in Part C students answer: “Do the data support the idea that the older students get, the more likely they are to favor sports? Explain.”

The instructional materials reviewed for Ready Mathematics Grade 8 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately.

The Teacher’s Resource Book Program Overview Built for Rigor and Engagement outlines how the materials balance conceptual understanding, procedural skills and fluency, and application. The consistent structure of the lessons presents opportunities to “Understand,” where they develop conceptual understanding, followed by the sections Think It Through, Connect, and Apply, where they explore different procedural skills and application of the mathematics of the lesson. In addition, the Practice and Problem Solving book includes problems that present further opportunities to engage with the three aspects of rigor.

Students engage in the three aspects of rigor independent of each other. For example:

• Unit 1 End Unit Assessment Form A Problem 7 requires students to apply their understanding of volume of a cube to find a side length: “Matteo is making a wooden cube to hold his little sister’s jewelry. The volume of the cube is 343 $$in^3$$. What is the length of one side of the jewelry box?”
• In Unit 4 Lesson 23 Understand the Pythagorean Theorem, students develop conceptual understanding of the geometric proof of the Pythagorean Theorem with a hands-on activity.
• Unit 3 Lessons 15, 16, and 17 show how aspects are treated separately. Unit 3 Lesson 15 develops conceptual understanding of systems of equations through activities which students solve a system graphically and determine the number of solutions to the equation. Lesson 16 begins to develop procedural skill by having students practice solving systems of equations algebraically. Lesson 17 has students apply those skills to multi-step real-world problems.

Balance is displayed in each unit with multiple lessons where two or three aspects of rigor are interwoven.

• Unit 2 Lesson 9 Analyze Linear Functions develops conceptual understanding as well as application of analyzing linear functions. The lesson contains application questions asking students to construct functions and interpret slope and y-intercept. Students write a function for a real-world situation and identify the rate of change and initial value. Conceptual understanding is developed by having students graph a situation, model it as a table and equation, and then use a visual model of a double number line to aid with interpretation of the slope and y-intercept.
• In Unit 4 Lesson 21 Understand Angle Relationships, students explore relationships of angles cut by a transversal to develop conceptual understanding. Students develop procedural skill through practice problems included in the Practice and Problem Solving section.

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

The Teacher Resource Book identifies the MPs for each lesson in the Table of Contents and in the CCSS Focus section as part of the Lesson Overview. SMP TIPs are found in the Teacher Resource Book throughout the lessons, and these tips highlight the integration of particular MPs within the lessons. Some examples of where the MPs are identified and used to enrich the mathematics content include:

• In Unit 4 Lesson 27 Solve Problems with Cylinders, Cones and Spheres, the SMP Tip for MP 7 states, “Students make use of the structure of the expressions for volume to think about the ratio of the volume of the cone to the volume of the cylinder. Remind students that sometimes they can figure out what they want by analyzing an expression without actually doing the computation.”
• In Unit 5 Lesson 9 Analyze Linear Functions, the SMP Tip for MP 8 states, “Students have calculated the rate of change by finding the change in y-values and dividing by the change in x-values. They made the connection between rate of change and the slope formula. Through repeated reasoning they are able to make sense of change in y divided by change in x to find slope as shown in the modeled instruction. Ask them to use the table and graph on the page to describe as many ways as they can to find the slope. (SMP 8)”
• In End of Unit Performance Tasks students reflect on mathematical practices. For example, in the Unit 4 Performance Task, students draw a model based on a situation. Students explain how the model assisted the students to decide lengths of the sides for each triangle. (MP 4)

The Mathematical Practices Handbook describes each of the MPs for students and provides students with questions to elicit thinking and discuss with a partner.

The instructional materials reviewed for Ready Grade 8 meet the expectation for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials offer students multiple opportunities to construct viable arguments and/or analyze the arguments of others.

Examples where students are prompted to construct viable arguments and/or analyze the arguments of others include:

• Lesson 1 Quiz prompts: “Is Josh correct? Explain why or why not.”
• In Unit 2 Lesson 10, students critique the reasoning of others when they analyze how a student got the wrong answer. They pair/share with the question, “Does Amir’s answer make sense? Discuss.”
• In the Unit 3 Performance Task, students give advice to a friend (construct an argument) about which online gaming club to join. They are given a checklist and a question, “Did you justify your advice to Sharleen?

The instructional materials reviewed for Ready Grade 8 meet the expectation for assisting teachers to engage students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. The materials provide teachers with SMP TIPs to help facilitate students to construct arguments and/or analyze the arguments of others.

Examples where teachers are supported to help students construct viable arguments and analyze the arguments of others include:

• In Unit 2 Lesson 8, teachers put students in pairs to present arguments and discuss whether they agree with their partner’s logic.
• In Unit 3 Lesson 11, teachers ask questions in the Mathematical Discourse section which encourage critiquing the reasoning of others. For example, “Suppose another classmate said the unit rate is 5 because it takes 5 minutes to run 1 mile. Is the classmate correct? Explain.”
• In Unit 5 Lesson 31 - Find Out More, students give two examples of categorical variables and explain why it would not be possible to display the variables in a scatter plot. The SMP Tip states, “Discussion of student examples of categorical data will allow students to listen to and critique the reasoning of others. By listening to the logical reasoning of others, students should decide if the reasoning makes sense, and, if necessary, ask useful questions to clarify. (SMP 3)”

The instructional materials reviewed for Ready Grade 8 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.

• Lesson Vocabulary is identified at at the beginning of every lesson and used correctly to refer to mathematical terms and topics. For example, Unit 4 Lesson 25 includes a definition of the Pythagorean Theorem in the teacher materials as well as the student materials. Pythagorean Theorem is defined as, “In any right triangle, the sum of the squares of the lengths of the legs, a and b, is equal to the square of the length of the hypotenuse, c.”
• Each lesson has an English Language Learners section in the Teacher Resource Book that contains some ways to support vocabulary development for all students. Some examples are:
• In Unit 1 Lesson 2, the teacher reviews the meaning of inverse operations and calls on volunteers to explain the inverse of squaring a number.
• In Unit 2 Lesson 8, teachers are given the following Mathematical Discourse prompts: “How are the two equations on the page similar? How are they different? How are the graphs of the two equations similar? How are they different?” The guidance for teachers attends to the mathematical vocabulary students use in discussion.
• Teachers pose questions to students and attend to precision using appropriate terminology. For example, in Unit 4 Lesson 24, teachers pose the following question, “Suppose you have a triangle that is not a right triangle. Can you use the Pythagorean Theorem to find the length of one of the sides? Explain.”
• In the Teacher Instruction Book, mathematical vocabulary is defined in the Find Out More section.
• Teachers are prompted in the Teacher Resource Book to have students use precise mathematical language. For example, in Unit 2 Lesson 7 Find Out More, students describe the initial value and rate of change for given snowfall amounts shown on a graph.
• In the Student Practice and Problem Solving Book, mathematical terms are defined, along with a picture example in small square sections labeled “Vocabulary.”