8th Grade - Gateway 2

The instructional materials for Eureka 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. For example:

• In Module 2, Lesson 1, students develop conceptual understanding of coherence through a series of hands-on activities with shapes and transparencies. Students move shapes around (transformations) to check congruence. (8.G.1)
• In Module 4, Topic D, students develop conceptual understanding of systems of equations simultaneously through contexts, tables, graphs and equations. (8.EE.8b)
• In Module 5, Topic A, students develop conceptual understanding when using non-linear data to make predictions. The predictions develop an understanding that a function is a rule that for every input there is only one output. (8.F.1)

The materials provide opportunities for students to demonstrate conceptual understanding independently throughout the grade level. For example:

• In Module 2, Lesson 2, students independently demonstrate understanding of translations. Students name a pictured vector, choose and name the vector along which a translation of a plane would map a point to its image, draw a vector that would translate one segment onto another segment, and explain why the lengths of the segments must be the same. (8.G.1a)
• In Module 2, Lesson 12, students independently demonstrate understanding when creating a table of solutions to a two-variable equation and plotting the solutions on a graph in order to see that the solution to that equation will be an ordered pair. (8.EE.5)
• In Module 5, Lesson 1, students independently demonstrate understanding when they make predictions of whether the given situation is linear or nonlinear. Students make predictions then analyze real-life data to determine if it is correct and identify what makes a situation non-linear. (8.F.A)

The instructional materials for Eureka Grade 8 meet expectations that they attend to those standards that set an expectation of procedural skill.

The instructional materials develop procedural skill throughout the grade level. For example:

• In Module 4, Lesson 4, students develop procedural skill of using properties of equality to solve linear equations void of any context. Classwork Exercise 1, Problem 1 states, “Solve the linear equation x + x + 2 + x + 4 + x + 6 = -28. State the property that justifies your first step and why you chose it.” (8.EE.7b)
• In Module 4, Lesson 6, students develop procedural skill of transforming mathematical equations into simpler forms using the distributive property in order to find a solution or solutions. (8.EE.7b)
• In Module 4, Lesson 28, students develop procedural skill when solving systems of equations void of any context by examining the equations, by sketching the graphs of the system, and by algebraic means. The teacher is prompted to ask the following questions in Classwork Example 1, “Use what you noticed about adding equivalent expressions to solve the following system by elimination: 6x - 5y = 21, 2x + 5y = -5. Notice that terms -5y and 5y are opposites; that is, they have a sum of zero when added. If we were to add the equations in the system, the y would be eliminated. Just as before, now that we know what x is, we can substitute it into either equation to determine the value of y. The solution to the system is (2, -9/5). (8.EE.8b)

The instructional materials provide opportunities to demonstrate procedural skill independentlythroughout the grade level. For example:

• In Module 4, Lesson 1, students independently demonstrate procedural skill of linear equations when writing statements using symbols to represent numbers. Problem Set Question 4 states, “One number is six more than another number. The sum of the squares is 90.”(8.EE.7)
• In Module 4, Lesson 25, students independently demonstrate procedural skill of solving systems of linear equations when completing the Exit Ticket activity. Lesson 25 Exit Ticket states, “Sketch the graphs of the linear system on a coordinate plane: 2x - y = -1, y = 5x -5” (8.EE.8b)
• In Module 5, Lesson 10, students independently demonstrate procedural skill using formulas to find the volume of cones and cylinders. (8.G.9)

The instructional materials for Eureka Grade 8 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the 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. For example:

• In Module 4, Lesson 5, students engage in grade-level mathematics when applying knowledge of angle relationships, congruence, and the triangle sum theorem to write and solve linear equations. These problems are non-routine because they rely on a broad range of mathematical knowledge, and can be solved in a variety of ways. Classwork Example 2 states, “Given a right triangle, find the degree measure of the angles if one angle is ten degrees more than four times the degree measure of the other angle and the third angle is the right angle.” (8.G.A, 8.EE.7b)
• In Module 5, Lesson 3, students engage in grade-level mathematics when writing a linear function that describes the volume of water in a bathtub as a function of time, and solve the equation to find how long it will take to fill the bathtub. Classwork Example 4 states, “Water flows from a faucet into a bathtub at the constant rate of 7 gallons of water pouring out every 2 minutes. The bathtub is initially empty, and its plug is in. Determine the rule that describes the volume of water in the tub as a function of time. If the tub can hold 50 gallons of water, how long will it take to fill the tub?” (8.F.B, 8.EE.7b)
• In Module 6, Lesson 1, students engage in grade-level mathematics when using knowledge of linear relationships to solve problems involving rate plans. Problem Set Question 2 states, “A shipping company charges a $4.45 handling fee in addition to $0.27 per pound to ship a package. Using x for the weight in pounds and y for the cost of shipping in dollars, write a linear function that determines the cost of shipping based on weight. Which line (solid, dotted, or dashed) on the following graph represents the shipping company’s pricing method? Explain.” (8.EE.8)
• In Module 7, Lesson 22, students engage in grade-level mathematics when solving for the average rate of change in various situations involving the rate of water filling a cone at a constant rate. Classwork Discussion Exercise states, “The height of a container in the shape of a circular cone is 7.5 ft., and the radius of its base is 3 ft., as shown. What is the total volume of the cone? Water flows into the container (in its inverted position) at a constant rate of 6 ft. cubed per minute. Approximately when will the container be filled?” (8.G.9)

The instructional materials provide opportunities for students to demonstrate independently the use of mathematics flexibly in a variety of contexts. For example:

• In Module 5, Lesson 11, students independently demonstrate the use of mathematics by calculating the volume of various choices of ice cream (1-, 2-, 3-scoops in a cup or cone) and determining which choice is the best value. Problem Set Question 6 states, “Bridget wants to determine which ice cream option is the best choice. The chart below gives the description and prices for her options. Use the space below each item to record your findings. A scoop of ice cream is considered a perfect sphere and has a 2-inch diameter. A cone has a 2-inch diameter and a height of 4.5 inches. A cup, considered a right circular cylinder, has a 3-inch diameter and a height of 2 inches. Determine the volume of each choice. Use 3.14 to approximate pi. Determine which choice is the best value for her money. Explain your reasoning.”(8.G.9)
• In Module 6, Lesson 2, students independently demonstrate the use of mathematics by writing a linear function to model the relationship between the number of songs downloaded and the total monthly cost, use the function to determine the cost of downloading various numbers of songs, graph the function on a coordinate plane containing the graph of a function representing a different music subscription site, and use the graphs to determine the site that offers the better deal. Problem Set Question 2 states, “Recall from a previous lesson that Kelly wants to add new music to her MP3 player. She was interested in a monthly subscription site that offered its MP3 downloading service for a monthly subscription fee plus a fee per song. The linear function that modeled the total monthly cost in dollars (y) based on the number of songs downloaded (x) is y = 5.25 + 0.30x. The site has suddenly changed its monthly price structure. The linear function that models the new total monthly cost in dollars (y) based on the number of songs downloaded (x) is y = 0.35x + 4.50. Explain the meaning of the value 4.50 in the new equation. Is this a better situation for Kelly than before? Explain the meaning of the value 0.35 in the new equation. Is this a better situation for Kelly than before? If you were to graph the two equations (old versus new), which line would have the steeper slope? What does this mean in the context of the problem? Which subscription plan provides the better value if Kelly downloads fewer than 15 songs per month?” (8.EE.8, 8.F.4)
• In Module 6, Lesson 8, students independently demonstrate the use of mathematics by informally fitting a straight line to data displayed in a scatter plot and making predictions based on the graph of a line that has been fit to data. Problem Set Question 2 states, “The scatter plot below shows the results of a survey of eighth-grade students who were asked to report the number of hours per week they spend playing video games and the typical number of hours they sleep each night. What trend do you observe in the data? What was the fewest number of hours per week that students who were surveyed spent playing video games? The most? What was the fewest number of hours per night that students who were surveyed typically slept? The most? Draw a line that seems to fit the trend in the data, and find its equation. Use the line to predict the number of hours of sleep for a student who spends about 15 hours per week playing video games.” (8.SP.2-3)

The instructional materials for Eureka Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

Conceptual understanding is addressed in Classwork. During this time, the teacher guides students through a new concept or an extension of the previous day’s learning. Students practice solving procedural problems in problem sets. The materials provide engaging applications of grade-level concepts throughout each lesson. The program balances all three aspects of rigor in every lesson.

All three aspects of rigor are present independently throughout the program materials. For example:

• In Module 7, Lesson 6, students develop conceptual understanding of irrational numbers as they examine the decimal expansion of numbers. Classwork Example 1 states, “Consider the fraction 5/8. Write an equivalent form of this fraction with a denominator that is the power of 10, and write the decimal expansion of this fraction.” (8.NS.1-2)
• In Module 4, Lesson 12, students develop procedural skill when finding solutions for linear equations. The Exit Ticket states, “Is the point (1,3) a solution to the linear equation 5x - 9y = 32? Explain. Find three solutions for the linear equation 4x - 3y = 1, and plot the solutions as points on a coordinate plane.” (8.F.4).
• In Module 7, Lesson 18, students engage in the application of mathematics when using the Pythagorean Theorem to solve real-world problems. Problem Set Question 1 states, “A 70 inch TV is advertised on sale at a local store. What are the length and width of the television?” (8.G.7).

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

• In Module 3, Lesson 12, students develop conceptual understanding of the properties of similar triangles when solving problems in real-world contexts. Problem Set Question 1 states, “The world’s tallest living tree is a redwood in California. It’s about 370 feet tall. In a local park, there is a very tall tree. You want to find out if the tree in the local park is anywhere near the height of the famous redwood. Describe the triangles in the diagram, and explain how you know they are similar or not. Assume Triangle ESO is similar to Triangle DRO. A friend stands in the shadow of the tree. He is exactly 5.5 feet tall and casts a shadow of 12 feet. Is there enough information to determine the height of the tree? If so, determine the height. If not, state what additional information is needed. Your friend stands exactly 477 feet from the base of the tree. Given this new information, determine about how many feet taller the world’s tallest tree is compared to the one in the local park.” (8.G.4)
• In Module 1, Lesson 8, students practice procedural skill when computing scientific notation to solve real-life situations. Classwork Example 1 states, “In 1723, the population of New York City was approximately 7,248. By 1870, almost 150 years later, the population had grown to 942,292. We want to determine approximately how many times greater the population was in 1870 compared to 1723. ” (8.EE.4)

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

All of the eight MPs are identified within the grade-level materials. The Standards for Mathematical Practice are identified at the beginning of each module under the Module Standards. The tab named “Highlighted Standards for Mathematical Practice” lists all of the MPs that are focused on in the module. Each MP is linked to the definition of the practice as in which lessons throughout the series that practice can be found.

Each Module Overview contains a section titled, “Focus Standard for Mathematical Practice.” Every practice that is identified in the module has a written explanation with specific examples of how each practice is being used to enrich the content of the module. For example:

• In Module 4, the explanation for MP 1 states, “Make sense of problems and persevere in solving them. Students analyze given constraints to make conjectures about the form and meaning of a solution to a given situation in one-variable and two-variable linear equations, as well as in simultaneous linear equations. Students are systematically guided to understand the meaning of a linear equation in one variable, the natural occurrence of linear equations in two variables with respect to proportional relationships, and the natural emergence of a system of two linear equations when looking at related, continuous, proportional relationships.”

Each lesson specifically identifies where MPs are located, usually within the margins of the teacher edition.

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

Student materials consistently prompt students to construct viable arguments and analyze the arguments of others.

• In Module 3, Lesson 2, students construct a viable argument when working with dilations. Problem Set Question 5 states, “Angle GHI measures 78 degrees. After a dilation, what is the measure of angle G’H’I’? How do you know?”
• In Module 3, Lesson 8, students work with similarity. Exercise 3 states, “Are the two triangles shown below similar? If so, describe a sequence that would prove triangle ABC is similar to triangle A’B’C’. If not, state how you know they are not similar.”
• In Module 4, Lesson 13, students construct viable arguments and analyze the arguments of others when graphing linear equations with two variables. Exercise 5 states, “Joey predicts that the graph of -x + y = 3 will look like the graph shown below. Do you agree? Explain why or why not.”
• In Module 5, Lesson 9, students construct viable arguments and analyze the arguments of others when determining the equation of a line. Exercise 4 states, “Several students decided to draw lines to represent the trend in the data. Consider the lines drawn by Sol, Patti, Marrisa, and Taylor, which are shown below. For each student, indicate whether or not you think the line would be a good line to use to make predictions. Explain your thinking.”

The instructional materials reviewed for Eureka Grade 8 meet expectations for explicitly attending to the specialized language of mathematics.

In each module, the instructional materials provide new or recently introduced mathematical terms that will be used throughout the module. The mathematical terms that are the focus of the module are highlighted for students throughout the lessons and are reiterated at the end of most lessons.

Each mathematical term that is introduced has an explanation, and some terms are supported with an example. The terminology that is used in the modules is consistent with the terms in the standards.

The materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams and symbols. For example:

• In Module 3, Lesson 9, the materials guide teachers through the teaching process of properties of similarity. The Concept Development states, “Expect students to respond in a similar manner to the response for part (d). If they do not, ask questions about what similarity means, what a dilation does, and how figures are mapped onto one another. For any two figures $$S$$ and $$S'$$, if $$S\backsim S'$$, then $$S'\backsim S$$. This is what the statement that similarity is a symmetric relation means.”
• In Module 5, Lesson 2, the materials guide teachers through a class discussion on functions. The Discussion notes state, “Let’s examine the definition of function more closely: For every input, there is one and only one output. Can you think of why the phrase one and only one (or exactly one) must be included in the definition? Most of the time in Grade 8, the correspondence is given by a rule, which can also be considered a set of instructions used to determine the output for any given input. For example, a common rule is to substitute a number into the variable of a one-variable expression and evaluating. When a function is given by such a rule or formula, we often say that function is a rule that assigns to each input exactly one output.”

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. For example:

• In Module 2, Lesson 1, the materials precisely define transformation in the Lesson Summary, and students use new vocabulary as they describe transformations. Problem Set Question 2 states, “Describe, intuitively, what kind of transformation is required to move Figure A on the left to its image on the right.”
• In Module 5, Lesson 3, students use words to describe a function in terms of area mowed and time. Exercise Question 1b states, “Hana claims she mows lawns at a constant rate. The table below shows the area of lawn she can mow over different time periods. Describe in words the function in terms of area mowed and time.”