8th Grade - Gateway 3

The instructional materials for EdGems Math Grade 8 meet the expectations for distinguishing between problems and exercises. Each Unit presents lessons with a consistent structure. The instructional sections, which vary by day, include: Warm-Up, Introduction to lesson using Lesson Presentation, Explore! Activity, and Focused Assignment or Online Practice. 

Within the Teacher/Lesson Guide, the student work is referred to as exercises, activities, and independent student practice. Examples include:

• In Lesson 9.2, Volume of Cones, Teacher/Lesson Guide, the lesson planning suggestion states, “This lesson may take 2-3 class periods (45-60 minutes per period). A suggested order for covering the content in this lesson in two days is below. Additional time may be given to use more Teacher Gem activities or independent student practice. Day 1: Explore! Activity: “Cones in a Cylinder”, Lesson Presentation, Exit Card (use to assign leveled practice), Leveled worksheet assignment Day 2: Focused Assignment or Online Practice, Teacher Gem Activity (8.G.9 Relay).” 

Within the Student Lessons throughout the materials, multiple examples are provided to show the steps to take when working with the content. The students then practice the content out of context before working with the content in story problems within context. Examples include:

• In Lesson 5.3, Solving Systems by Substitution, Student Lesson, students learn the content by solving problems without context and apply the content in real-world contexts. “In each pair, identify the equation that has an isolated variable. State which variable is isolated. 1. Equation 1 3x − 5y = 10; Equation 2 x = 4 − 4y.” Near the conclusion of the problems, students evaluate the reasoning of others. “Problem 23. Aiden solved a system of equations using substitution. He stated that his solution was x = 7. Francis said something seemed wrong about the solution because it was not a point on a graph but just an x-value. Do you think Aiden’s solution is complete? If not, what else does he need to do?”

The instructional materials for EdGems Math Grade 8 meet the expectations that the design of assignments is intentional and not haphazard. Overall, lessons are sequenced so students develop an understanding of mathematical concepts and skills. The structure of the lessons provides students with the opportunity to activate prior learning, build procedural skills, and engage with multiple activities that increase in complexity, utilizing concrete and abstract representations.

In each Teacher/Lesson Guide, there are Lesson Planning Suggestions. These suggestions sequence the content in an order to help students develop understanding. For example, in Lesson 5.2, Solving Systems by Graphing, Lesson Planning Suggestions builds from guided practice to more abstract and contextual work with the content, “This lesson may take two class periods (45-60 minutes per period). A suggested order for covering the content in this lesson is below.”

Day 1: 

• Explore! Activity - serves to motivate and set the stage for students to learn new material and persevere through a related mathematical task.
• Lesson Presentation - opportunities for students to learn and practice new mathematics 
• Leveled worksheet assignment - support students in developing independent mastery of the current lessons as well as reviewing material from previous lessons.

Day 2: 

• Exit Card-formative assessment opportunity after instruction
• Teacher Gem Activity-opportunities for students to learn and practice new mathematics
• Focused Assignment or Online Practice-support students in developing independent mastery of the current lessons as well as reviewing material from previous lessons.

The instructional materials for EdGems Math Grade 8 meet the expectations for having a variety in what students are asked to produce. In the practice pages, students develop concepts and skills by answering multiple questions on the content. An example is:

• In Lesson 1.1, Solving One- and Two-Step Equations, Proficient Practice, students solve equations through multiple examples, “Solve each equation. 1. y + 4/5 = 1 2/5, 2. 4x = 28, 3. p − 83 = 129”

Students connect content to real-world situations through the Student Gems, linked resources from Open Educational online sites. An example is:

• In Unit 5, Systems of Equations, Student Gem from PBS LearningMedia, students use the content of systems of equations to work out a problem that has a real-world connection. Summary (from the website): "Use mathematics to determine what is required to beat world champion Usain Bolt in a 200-meter race. This video focuses on systems of equations that are visualized by completing a table of values and looking for a point of intersection in a set of line graphs.”

With the Teacher Gems, students create arguments and justify their answers. An example is:

• In Lesson 8.2, Division Property of Exponents, Teacher Gem: Partner Math, students solve problems involving the Division Property of Exponents, they justify their answers to others to compare work, “4. Once the students have a new partner, they need to compare answers from the previous two tasks. If they agree, they sign their initials in the circle connecting the two task boxes. If students disagree, they work to determine who is correct prior to signing. Once students compare, the teacher should have posted what the next two tasks are and they work with this partner to complete the next two tasks.”

The instructional materials reviewed for EdGems Math Grade 8 meet expectations for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

• The series does not often incorporate the use of manipulatives, but when they are included, manipulatives are consistently aligned to the content in the standards. An example is located in , In Lesson 1.1, Solving One and Two-step Equations: “‘Algebra Tile Modeling’ is an activity that uses algebra tiles to help students create conceptual understanding of the equation solving process. Algebra tiles can be purchased or made from card stock. In this activity, students use the concept of zero pairs to remove positive or negative tiles, and students work in partner sets with one equation mat between them. One student can be responsible for manipulating the tiles on the board while the other records on the Explore! activity sheet. Student pairs share out Step 6 with the class to wrap up the activity.”
• Examples of manipulatives include: Algebra tiles, grid paper, rulers, patty paper/tracing paper, cylinders and cones, protractor, and x-y tables.

The instructional materials for EdGems Math Grade 8 meet the expectations for providing quality questions to help guide students’ mathematical development. There is a Communication Prompt in every lesson that provides questions to help guide students’ mathematical development. Examples include:

• In Lesson 2.3, Distance on the Coordinate Plane, Communication Prompt, “What process would you use to find the distance between (2, 7) and (4, 10)?”
• In Lesson 9.3, Volume of Spheres, Communication Prompt, “Why are volumes that use 3.14 for $$\pi$$  approximations? How could a solution be more exact?”

Questions are also located in the Mathematical Practice section, but these questions are not located in every lesson. Examples include:

• In Lesson 1.3, Solutions to Linear Equations, Mathematical Practices - A Closer Look, “MP1: After the Explore! activity, have students look at three equations written on the board (i.e., x = 2, 4 = 4 and 5 = 7). Ask students which statement is never true, always true or sometimes true. This will reinforce making sense of the types of solutions that arise from different situations.”
• In Lesson 4.2, Writing Linear Equations for Graphs, Mathematical Practices - A Closer Look, “MP2: Each time students find the slope or y-intercept of a graph for a real-world scenario, ask “What is the real-world meaning of ____?”

The instructional materials for EdGems Math Grade 8 meet the expectations for containing ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials.  Examples include: 

• In the Unit Overview, there is information provided to help teachers understand the materials in order to present the content. The Unit Overview provides a brief overview of the content contained within the unit. It also includes standards, learning progression, pacing, and assessments contained within the unit.
• The Student Gems contain links to outside technology-enhanced activities. These contain instructions for students, but there is no guidance for the teacher as to when to use these activities to enhance student learning.
• Explore! Summary and Suggestions provides details about the Explore! Activity and suggestions for how it can be implemented. For example, in Lesson 2.1, The Pythagorean Theorem, “A Rule for Right Triangles” is made up of two parts. In Part I, students draw right triangles and measure side lengths. Students then use these triangles to look for a relationship between the squares of the side lengths and the square of the length of the hypotenuse. Students work in partners or small groups and then share out as a full class what they observe about the relationship of the side lengths (as prompted in Steps 6-7).
• Mathematical Practices - A Closer Look, contains information related to each Mathematical Practice contained in the lesson and suggestions for teachers. An example in Lesson 3.3 Calculating Slope from Graphs states, “MP4: In this lesson, students learn a method to create a model (graph) of a table of values or two points. They then use this model to find the slope using a slope triangle. Students should be encouraged to fall back on this visual method of finding slope even after they learn to use the slope formula in Lesson 3.4.”
• There are Extra Examples found in each of the lessons for teachers to present if needed.
• Teaching Tips provide teachers with suggestions for teaching the content within the lesson. For example, in Lesson 5.5, Applications of Systems of Equations, Teaching Tip states, “This lesson may be difficult for many students. Each system is different than the last which means that students will not have the repetition that many are used to with many mathematics topics.”

The instructional materials for EdGems Math Grade 8 meet the expectations for explaining the role of the grade-level mathematics in the context of the overall mathematics curriculum.

The materials provide information that explains the progression of the content across multiple grades and within the series itself. Each Unit Overview includes Learning Progression which includes concepts and skills that students have experienced in the past and ones that they will experience in the future. For example in Unit 8, Angle Relationships, the Learning Progressions contain:

“In earlier grades, students have:

• Solved problems involving scale drawings of geometric figures. (7.G.1)
• Drawn geometric shapes with given conditions with a focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. (7.G.2)
• Used facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (7.G.5)

In future grades, students will:

• Prove theorems about lines and angles. (G-CO.9)
• Prove theorems about triangles. (G-CO.10)
• Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (G-SRT.3)”

The instructional materials for EdGems Math Grade 8 meet the expectations for providing strategies for teachers to identify and address common student errors and misconceptions. Each Unit contains a Common Misconceptions document that discusses the common misconceptions for each lesson. This document also provides mathematically sound strategies for the teacher to address student errors. An example is:

• In Lesson 2.1, The Pythagorean Theorem, Common Misconceptions, “When trying to describe the location of the hypotenuse on a right triangle, some students may find the word “opposite” confusing and may not be able to determine where the hypotenuse is located. Teachers may want to diagram this, using an arrow from the right angle to the longest side, and labeling it as the hypotenuse.”

The instructional materials for EdGems Math Grade 8  meet the expectations that assessments clearly denote which standards are being emphasized.

Standards for the unit are denoted at the unit level in the Unit Overview in the Standards and Recommended Pacing. The unit standards are also noted on the Pacing Guide for each lesson in the unit. The assessments, tiered assessments, and performance tasks indicate which standards are being assessed at the question level. Standards are provided on the Teacher Unit Page with a blue “i” icon for the assessments, tiered assessments, and performance tasks. Standards are noted using the blue “i” icon for Exit Cards on the Lesson pages. The standards are noted on the Unit Overview in the Assessment section for the Gem Challenges.

The instructional materials for EdGems Math Grade 8 meet the expectations for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners. Evidence includes:

• Every lesson in the Teacher/Lesson Guide has Teaching Tips which provide information on strategies to use when teaching the concept, manipulatives that might be useful, questions to focus on, and other tips depending on the lesson.
• At the beginning of each unit, Learning Progressions makes connections to both prior and future skills and standards to scaffold instruction.
• Each lesson provides a Warm-up to activate prior knowledge.
• The materials provide three different levels of practice for each lesson (Tiered, Proficient, and Challenge Practice) as well as tiered assessments; however, there are no strategies or instructions for teachers on how to sequence or scaffold these items.

The instructional materials for EdGems Math Grade 8 meet the expectations for providing teachers with strategies for meeting the needs of a range of learners.

Within the materials there are three levels of practice pages: Proficient (on level), Tiered (more focused development), and Challenge (extension). These pages allow for more development of the content knowledge regardless of the level of the students. The material presents different types of questions all related to the content from the lesson. An example is:

• In Lesson 1.2, Solving Multi-Step Equations, Proficient Practice for on-level students has questions focusing on the content, “Solve each equation and check your solution. 1. 3(2x + 4) = 60 2. 5x + 1 = 8x − 23”. The Challenge practice, focused on extension of the concept, includes solving the equations and taking it to the next step of graphing the solution on a number line: “Solve each inequality. Graph the solution on a number line. Research online for tips on solving and graphing linear inequalities. 2. 2f + 4 < 5f − 11”. The Tiered Practice is geared towards students that need more focused help with the content which is done by providing the steps to solving multi-step equations: “TO SOLVE MULTI-STEP EQUATIONS: 1. Simplify by distributing and combining like terms 2. Move all variables to one side of the equation 3. Solve the one- or two-step equation 1. 4(2x + 3) = 36”.
• Assessments and Tiered Assessments are located in each Unit.

The instructional materials for EdGems Math Grade 8 meet the expectations for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations. The materials have Performance Tasks embedded into each unit, and teacher guidance on how to help students solve these problems is limited. Evidence includes:

• Each Unit includes Performance Tasks, some of which include multiple entry-points. Within the Unit Overview, Assessment section, the Performance Tasks (Formative or Summative) for each unit are identified. For example, Unit 6, Triangle Puzzle, allows for students find congruent triangles and explain what makes them congruent.
• Each Lesson has an Explore! task which does include instructions on its use. For example: “In “Mirror, Mirror”, students create reflections using tracing paper (or patty paper) as well as on a grid. This activity allows for students to use common sense about reflections from everyday life and scaffold into using a coordinate plane. Step 5 asks students to discuss if the image and pre-image are congruent. If students were working in smaller groups, you may want to bring the full class together for this discussion.”

The instructional materials for EdGems Math Grade 8 meet the expectations for providing opportunities for advanced students to investigate mathematics content at greater depth. The materials provide some opportunities for advanced students to investigate the course-level mathematics at a greater depth. Each lesson includes Tiered Practice, Proficient Practice, and Challenge Practice sheets. The Challenge Practice sheets include more comlex problems than the Tiered and Proficient Practice sheets, and the number of problems is comparable. 

The instructional materials for EdGems Math Grade 8 meet the expectations for providing a balanced portrayal of various demographic and personal characteristics. Lessons contain a variety of demographic and personal characteristics. Evidence includes:

• Names and wording are chosen with diversity in mind. The materials include various names throughout the problems for example: Juan, Seth, Sherry, Mikayla, Julio, Jaylynn, Kobe, Ebizah, Ginger, Gracin, Kitts, Vadek. The names are used in ways that do not stereotype characters by gender, race, or ethnicity. 
• Images portrayed in many student lessons are of fair skinned children. For example, in Lessons: 1.4, 1.6, 2.2, 4.2, 4.5, 6.2, 7.3, the same images are used in both the English and Spanish versions of the Student Lessons. 
• One image of a person with a disability was observed. In Lesson 2.1, The Pythagorean Theorem, Problem 8, an image of a person in a wheelchair is used. “According to federal guidelines, wheelchair ramps must be built with a 1-foot vertical rise for every 12 feet of horizontal distance.  a. What is the length of a ramp that rises 1 foot? Round to the nearest hundredth.  b. If Max needs to build a ramp to reach a 2-foot porch, what horizontal distance will it cover?  c. How long will the hypotenuse of Max’s ramp be? Round to the nearest hundredth.”
• When multiple characters are involved in a scenario, they are often doing similar tasks or jobs in ways not expressing gender, race, or ethnic bias, and there is no pattern in one character using more/fewer sophisticated strategies.