8th Grade - Gateway 2

The instructional materials for EdGems Math Grade 8 meet expectations for developing 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 instructional materials include Teacher Gems and Student Gems which provide links to activities that build conceptual understanding. Explore! activities provide students the opportunity to develop conceptual understanding at the beginning of each new lesson. In addition, Exercises, Online Practice, and Gem Challenges include problems to allow students to independently demonstrate conceptual understanding. Evidence includes:

• Lesson 3.1, Understanding Functions, the Explore! activity develops understanding of functions by building on student knowledge of proportional relationships. (8.F.1) The activity begins with a description of the following situation, “To find the relative age of a dog, some people use the rule of thumb that every year in a dog’s life is equal to seven years in a human’s life. In Steps 1-3, students have the “opportunity to create an equation for a situation, create a table and then graph the function. This connects to work students have done with proportional relationships in Grade 7. Students are introduced to the term “function” at the end of Step 3 and use this understanding to look at two relationships that are not functions and explain why.” 
• Lesson 4.5, Introduction to Non-Linear Functions, Teacher Gems include the activity, “Always, Sometimes, Never” which provides students with a statement and asks them, “Decide if the statement in the box is always true, sometimes true, or never true.” Students demonstrate conceptual understanding by providing evidence of why they chose always, sometimes or never. For example, Statement 4: “Use the remainder of the page to provide mathematical evidence that supports your decision. If a function in the form of y = mx + b has a slope of 0, it is linear.” (8.F.3)
• Lesson 4.6, Interpreting Graphs of Functions, students demonstrate conceptual understanding of graphs of functions. An example is in problem 18, “Sketch a graph that is made up of four connected line segments. The graph should include segments that are increasing and others that are decreasing. Write a story that could be modeled by your sketch.” (8.F.5)
• Lesson 6.1, Student Gems, there is a link for an Illustrative Mathematics Task called “Street Intersections.” In this activity, students “apply facts about angles (including congruence of vertical angles and alternate interior angles for parallel lines cut by a transverse) in order to calculate angle measures in the context of a map.” According to the Commentary from the task, “the goal is not to apply congruence of alternate interior angles for parallel lines cut by a transverse: rather it is to explain why this is true, in this particular setting.” (8.G.5)
• Lesson 5.6, Distributive Property, Online Practice, contains questions that require students to demonstrate conceptual understanding of transformations. The directions state, “Use the point T(4, −3). Write the ordered pair for the final location of the given point after completing the transformations in the order listed.” (8.G.3)

The instructional materials for EdGems Math Grade 8 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The materials include problems and questions that develop procedural skill and fluency and provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade. The materials develop procedural skills and fluencies in Student Gems, Lesson Examples, Student Exercises, and Teacher Gems. The materials provide opportunities for students to independently demonstrate procedural skills and fluencies in Proficient, Tiered, and Challenge Practice, Online Practice, Gem Challenges, and Exit Cards. Each unit provides additional practice with procedural skills in the Student Gems. Additional practice activities are specific to the standard(s) in each lesson. Included in each unit are links to: Khan Academy, IXL Practice, and Desmos Practice. Examples of developing procedural skill and fluency include:

• Lesson 5.4, Proficient Practice, students solve systems of equations using elimination, “Solve each system of equations using the elimination method. Check the solution. 1) 3x − 2y = 10 and 7x + 2y = 30” (8.EE.8b)
• Lesson 1.3, students solve linear equations, “Solve each equation. Describe the number of solutions (one, none or infinitely many). 2(x + 7) = 2x + 7” (8.EE.7)
• Lesson 8.3, students practice identifying large numbers by powers of ten in scientific notation with multiple problems. An example is, “Write each number in scientific notation. 0.000058” (8.EE.3) 
• Lesson 5.6, Proficient Practice, students practice converting repeating decimals into fractions with multiple problems. An example is, “Convert each repeating decimal into a fraction. 1.5̅” (8.NS.1)
• Lesson 8.1, Multiplication Properties of Exponents, Exit Card, students know and apply properties of exponents, “Simplify. 1. $$5^4 \times 5^3$$, 2. $$(p^3)^2$$, 3. $$(5w)^2$$, 4. $$(4x^5y)(3x^2y^2)$$.” (8.EE.1)

The instructional materials for EdGems Math Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the grade-level mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. Students engage with materials that support non-routine and routine applications of mathematics in the Explore! activities, Teacher Gems, Performance Tasks, and Rich Tasks. Some of the Student Pages and Proficient, Tiered, and Challenge Practice allow students to engage with problems including real-world contexts and present multiple opportunities for students to independently demonstrate application of grade-level mathematics. Examples include:

• Lesson 4.2, Writing Linear Equations for Graphs, the Explore! activity engages students in real-world application of constructing a function to model a linear relationship between two quantities (8.F.4). “Stacey and Mario like to go to the coffee shop before school. They decided to conduct an experiment to study the rate at which their coffees cool when left untouched on the table. The graph below and right shows the information they gathered.” A graph that models the situation is given. 
• Lesson 5.2, Solving Systems by Graphing, Exercises, students solve systems of linear equations by graphing. (8.EE.8) An example in , Exercise 18, “Sarah begins the year with $100 in her savings account. Each week, she spends $8. Martin begins the year with no money saved, but each week he puts $12 into an account. Let x represent the number of weeks since the beginning of the year and y represent the total money in the account.” Students respond to a series of questions and prompts to determine Sarah’s total money, Martin’s total money, graph both equations, determine intersection, and identify real-world meaning. 
• Unit 8, Exponent Properties, Performance Task, Ribbon for Sale, students independently demonstrate applying operations with numbers expressed in scientific notation (8.EE.4). The task provides the situation, “A fabric store has $$5.5 \times 10^6$$ inches of ribbon in stock.” Multiple situations are provided, for example, “1. How many spools of ribbon are in stock if each spool holds $$1.1 \times 10^2$$ inches of ribbon? Show all work necessary to justify your answer.”
• Unit 7, Transformations, Rich Task, Fractals: A Project Resource for 8th Grade Teachers, students apply transformations to create fractal patterns (8.G.A). Students are given images of several fractals and choose one to recreate. Students must answer a series of questions and tasks related to the image recreated. 
• Unit 2, The Pythagorean Theorem, Rich Task, students apply the Pythagorean Theorem to solve problems (8.G.7). Students determine how far a golfer is away from the hole based on a statement that the announcer in Act 1 makes, along with an aerial image of the golf course with measurements of three different holes that appear at the end of the Act 1 video.

The instructional materials for EdGems Math Grade 8 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 independently throughout the program materials. Examples include:

• In Lesson 5.2, Student Lesson, page 118, students develop procedural skill by solving multiple problems involving systems of equations, “Solve each system of equations using the substitution method. 9) x = y − 3 and 5x + 3y = 1” (8.EE.8b)
• In Lesson 7.2, Translations, students develop conceptual understanding of translations through visual representations and representing translations on a graph from context. An example is, “Describe the translation from the pre-image to the image.” “Graph the pre-image and the image under the given translation. Label the vertices correctly.” (8.G.3)
• In Lesson 5.5, Applications of Linear Systems, Student Lesson Examples, students engage in routine real-world applications of solving linear systems (8.EE.8c). Example 1, “Nai is trying to decide between two different cell phone plans. Plan A charges a flat fee of $22 per month plus $0.10 per minute of phone usage. Plan B charges $0.18 per minute with no flat fee. How many minutes would Nai have to use each month for the cell phone plans to cost the same amount? How much would it cost?” 

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include:

• In Lesson 2.2, Applying the Pythagorean Theorem, Teacher Gems, “Climb the Ladder”, students develop procedural skill within an application problem while using the Pythagorean Theorem (8.G.7). The instructions for the Climb the Ladder Activity state, “Climb the Ladder is an activity where students work through four ladders that increase in complexity and depth of knowledge levels for a given standard. A Climb the Ladder works best for standards that reach a variety of depths of knowledge. The first ladder focuses on the basic skills and concepts of the standard. Each ladder is more challenging, reaching higher depths of knowledge than the previous ladders.” In Ladder 1, students must use the Pythagorean Theorem to find the length of a missing side in each right triangle. In Ladder 4, students are applying the Pythagorean Theorem to solve problems: For example, problem 1, “ The diagonal of a square is 12 centimeters. How long is each side of the square? Round to the nearest tenth of a centimeter.”
• In Lesson 7.1, Reflections, students develop conceptual understanding and procedural skills in understanding and describing the effect of reflections on two dimensional figures. (8.G.3) In the Explore! activity, “Mirror, Mirror,” students create reflections using tracing paper (or patty paper) as well as on a grid. This activity allows for students to use prior knowledge about reflections from everyday life and scaffolds into using a coordinate plane. 
• In Lesson 5.3, Solving Systems Using Substitution, Student Gems, MARS Activity, students develop procedural skills within an application problem as they create a system of linear equations to represent the cost of two different baseball jerseys. Then students determine the price of a third jersey that has a cost that is between that of the original two jerseys. (8.EE.8c)

The instructional materials reviewed for EdGems Math Grade 8 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade level.

All 8 MPs are identified throughout the materials. Each lesson includes a Lesson Guide with a section titled, Mathematical Practices - A Closer Look that explains a few of the MPs that will be used within that lesson. The MP is identified and an explanation of how to address the MP within the lesson is provided. At times, the identification is targeted, and gives a specific problem where the MP is included, but often it is broad and provides a general statement of how to include the MP within the lesson.

Examples of MPs that are identified and enrich the mathematical content include:

• Lesson 3.1, Understand Functions, “MP4: Students create or examine a variety of common models (equations, graphs, tables, verbal contexts) in this lesson. Stress the importance of moving between the models as each highlights different information about the relationship.”
• Lesson 5.2, Solving Systems by Graphing, “MP6: Graphing systems of equations to find the point of intersection is not always the most accurate method (other methods will be introduced later in this unit) since the solution may not always be a “nice” number. Explain to students how this makes it important to check their answer by substituting the x- and y-values into the original system.”
• Lesson 9.1, Volume of Cylinders, “MP7: Students should recognize that each single unit of height adds a layer of volume to a cylinder. The area of the base is the first layer and each unit of height is stacked upon the base. Stacks of Petri dishes or coins make good visuals.”
• Lesson 8.3, Scientific Notation, “MP8: Prior to teaching the lesson, give students a variety of numbers (on station cards around the room or small cards in individual packets) that may or may not be in scientific notation. Have them sort the cards into numbers that are in scientific notation and numbers that are not. Have them explain their thinking to a partner and come up with a definition of scientific notation.”

The instructional materials reviewed for EdGems Math Grade 8 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Examples of the student materials prompting students to construct viable arguments and/or analyze the arguments of others include:

• One of the Teacher Gems activities is called “Always, Sometimes, Never.” The instructions for this type of activity state, “Always, Sometimes, Never is best used with concepts that allow for situations that create exceptions to the “rule” or require students to understand subcategories to fully understand the standard. Students are given the opportunity to create evidence to support whether a statement is always true, sometimes true or never true.” For example, in Lesson 1.2, Square Roots and Cube Roots, the student directions for the Always, Sometimes, Never Activity state, “Decide if the statement in the box below is always true, sometimes true, or never true. Use the remainder of the page to provide mathematical evidence that supports your decision. Statement #1: A decimal is a rational number.”
• In Lesson 2.1, The Pythagorean Theorem, Exercise 27, students solve the problem and construct an argument to explain their reasoning. “Julio leaned a ladder against the tree his clubhouse was built around. The ladder was 8.5 feet long. It leaned against the tree 7.5 feet off the ground. Julio placed the base of the ladder 4 feet from the trunk of the tree. Does the tree make a right angle with the ground? Explain your reasoning.”
• The Unit 8, Exponent Properties, Student Gem from Math Assessment Project, students create a viable argument on how to use different strategies to determine how much money will be made, “For this task, students need to calculate, using exponents, how much would be made through an email money-making scheme. They also need to provide one reason why this type of scheme could go wrong.”
• Lesson 1.1, Solving One- and Two-Step Equations, exercise 24: “Jordin solved three problems incorrectly. Describe the error she made in each problem; then find the correct answers.” (There are pictures of three two-step equations and incorrect worked out solutions to accompany the problem). Students analyze the errors of another student and provide correct answers. 
• In Lesson 3.3, Calculating Slope from Graphs, Problem 23, students analyze mathematical reasoning of another person and constructing a viable argument of where he made an error and how to fix it. “Owen incorrectly found the slope of the line on the graph at the right. He said the slope of the line is 23. Explain what he did wrong and give the correct slope.” 
• In Lesson 5.1, Parallel, Intersecting, or the Same Line, Exercise 19, students must analyze student work, critique it, and fix the mistake to find the correct solution. Exercise 19: “On her Unit 5 Test, Victoria was asked to give an example of two lines that are parallel but not the same line. She answered with the equations, y = 4x + 5 and y = 3x + 5. Did she get the question right? If not, what mistake did she make?”
• In Lesson 5.4, Solving Systems using Elimination, Problem 22, students analyze others reasoning and construct a viable argument on how one strategy may be more efficient than another:  “Nicolette likes to solve systems of equations using substitution rather than elimination. Her teacher gave her the system of equations below. Explain to Nicolette why solving this system using elimination may be easier than using substitution.”

The instructional materials reviewed for EdGems Math Grade 8 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. The Teacher/Lesson Guide and Teacher Gems within most lessons support teachers to engage students in constructing viable arguments and analyzing the reasoning of others. Examples include:

• In Lesson 1.6, Simplifying Roots, Teacher/Lesson Guide, teachers support students to analyze the reasoning of others and justify their thinking. “Exercise 18 can be used as a small group discussion item prior to releasing students to independent work time or a Teacher Gem activity. Have students locate the error and describe her error in thinking.”
• In Lesson 2.3, Distance on the Coordinate Plane, Teacher/Lesson Guide, teachers group students in pairs to work towards a solution and justify their answers. Students exchange papers with another partner pair to critique the reasoning of others. 
• In Lesson 3.1, Understanding Functions, Teacher/Lesson Guide, teachers support students to construct a viable argument and critique the reasoning of others. “Use Exercise 11 to give students an opportunity to construct a viable argument to support their response. Have students partner with a peer and critique each other’s reasoning to come up with the best possible argument as a pair."
• In Lesson 6.3, Angle Sum of a Triangle, Teacher/Lesson Guide, teachers engage students in constructing a viable argument using the Explore! Activity. “In the Explore! students work through the steps to make a conjecture in Step 4. In Step 5 students share their conjectures and listen to the reasoning of others to determine if their conjecture is correct.”
• In Lesson 8.2, Division Property of Exponents, Teacher Gem, Categories, teachers support students to conduct viable arguments. “Once sharing has been done informally in groups and through the use of the observers, the teacher may choose to ask students to share out what categories they created and how they knew what went in each category. Choosing a specific card and asking students which of their categories they would put it in and why allows students to construct arguments and attend to precision.”

The instructional materials reviewed for EdGems Math Grade 8 meet expectations for explicitly attending to the specialized language of mathematics. The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Throughout the materials, precise terminology is used to describe mathematical concepts, and each lesson includes a visual lesson presentation. In most of the lesson presentations, there is at least one slide dedicated to explicit teaching of vocabulary. The Teacher/Lesson Guide, Student Lesson, and Parent Guide all contain information about mathematical language. Examples include:

• In the Student Lessons, the font for vocabulary words is red and a definition is included. 
• Each unit includes a Parent Guide which contains “Important Vocabulary” related to the unit. 
• In Lesson 1.2, Solving Multi-Step Equations, the Lesson Presentation includes a slide to introduce vocabulary from the lesson and provide mathematical definitions. “Algebraic Expression - An expression that contains numbers, operations and variables. Term - A number or the product of a number and a variable in an algebraic expression; a number in a sequence. Constant - A term that has no variable. Coefficient - The number multiplied by a variable in a term.”
• In Lesson 2.1, The Pythagorean Theorem, the Lesson Presentation includes a slide to introduce vocabulary from the lesson and provide mathematical definitions. “Hypotenuse - The side opposite the right angle in a right triangle. Legs - The two sides of a right triangle that form a right angle. Pythagorean Theorem - In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.”
• Lesson 3.2, Proportional Relationships, Teacher/Lesson Guide, Teaching Tips includes information about mathematical language. “It is important to connect the idea that the rate of a proportional relationship is also called the constant of proportionality or unit rate. This rate will also be called the slope in the next lesson.”
• In Lesson 5.4, Solving Systems Using Elimination, students use precise mathematical terminology. “As students discuss systems of equations, reinforce the use of vocabulary. Have students describe systems using words such as “coefficient”, “equivalent equations” and “solution”. "
• In Lesson 6.2, Corresponding and Same-Side Interior Angles, students use correct names for angle pairs. “Emphasize the names for the angle pairs when working through the examples and listen to be sure students are using the correct names.”
• Lesson 10.1, Scatter Plots and Associations, Teacher/Lesson Guide, Explore! Summary and Suggestions includes opportunities for students to reinforce new vocabulary terms. “Learn Those Terms” is an Explore! activity focused on vocabulary acquisition. Students are given a variety of scatter plots and must use each term from a word bank one time in the activity to correctly fill in the blanks. Students are encouraged in the instructions of the activity to use their textbook, glossary, and online resources to look up the terms with which they are unfamiliar.