8th Grade - Gateway 2

The materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. 

Materials include problems and questions that develop conceptual understanding throughout the grade level. Students develop understanding throughout “Engage” and “Develop” activities, which typically activate prior knowledge and use manipulatives to introduce and build understanding of a concept. Students also have the opportunity to independently demonstrate their understanding in the “Demonstrate” questions at the end of each lesson where they attempt to synthesize their learning. 

• In Module 1, Topic 1, Lesson 4, students gain an understanding of translations, reflections, and rotations by manipulating two dimensional figures on a coordinate plane. In Activity 4.1 Mirror, Mirror, students see an image of an ambulance appearing backward like it would in a mirror. Students discuss why the image looks as it does and write their name in a similar manner. Students then use patty paper to reflect pre-images across the x-axis and y-axis and explore how the reflection affects the coordinates. (8.G.A) 

• In Module 2, Topic 1, Lessons 1, students explore slope and similar triangles to understand the slope-intercept form for the equation of a line. In Activity 1.4 Comparing Depth of Color, using concrete visual representations, students develop connections between tables, graphs, and expressions for a given situation. (8.EE.6) 

• In Module 3, Topic 1, Lesson 3, students demonstrate an understanding of a function when they create an input/output table and an x/y table. In Activity 3.1 Functions as Mappings from One Set to Another, students see examples of mapping ordered pairs and look at the relationship of x and y coordinates. Students write the ordered pairs shown by mapping, create their own mapping and then represent the numbers in an input/output table and an x/y table to visualize the relationships between them. (8.F.A) 

• In Module 4, Topic 2, Lesson 1, students develop an understanding of the Pythagorean Theorem through visual models. In Getting Started and Activity 1.1 Introducing the Pythagorean Theorem, students square the length of each side of a right triangle and describe patterns they see. They summarize the patterns to discover that, in a right triangle, the hypotenuse must be opposite the right angle. (8.EE.2, 8.G.6, 8.G.7) 

• In Module 4, Topic 2, Lesson 1, students develop an understanding of the Pythagorean Theorem when using manipulatives to create a proof. In Activity 1.2, Proving the Pythagorean Theorem, students use manipulatives (grid paper, cut-outs) to create a geometric proof of the Pythagorean Theorem three different ways. (8.EE.2, 8.G.6, 8.G.7) 

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. 

• In Module 1, Topic 2, Lesson 3, students demonstrate an understanding of similarity by justifying their answer with a rationale. In Talk the Talk - Summing Up Similar Figures, students determine if statements about similarity are always, sometimes, or never true and justify their answer based on prior learning. (8.G.A) For example, “The same order for a sequence of transformations can be used to map between two similar figures, regardless of which figure is used as the pre-image.” 

• In Module 2, Topic 1, Lesson 2, students demonstrate an understanding of proportional relationships when providing examples of connections between concepts. In Talk the Talk - A Web of Connections, students summarize what they have learned by connecting the steepness of a line to the concepts of slope, rate of change, unit rate, and the constant of proportionality. Students provide illustrations and examples showing the connections among these concepts. (8.EE.5 &.6) 

• In Module 3, Topic 1, Lesson 3, students create various representations of a function to demonstrate their understanding. In Talk the Talk - Function Organizer, students represent a function in different ways (problem, situations, graph, and ordered pairs) in a graphic organizer. (8.F.A) 

• In Module 2, Topic 1, Lesson 2, students demonstrate an understanding of proportional relationships when solving real-world problems. In Practice Questions, students are given situations and use their prior knowledge to choose a strategy that best represents the information, solve it using that method, and explain their solution. (8.EE.B)

The materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The materials develop procedural skill throughout the grade level. They also provide opportunities to independently demonstrate procedural skill throughout the grade level. This is primarily found in two aspects of the materials: first, in the “Develop” portion of the lesson where students work through activities that help them deepen understanding and practice procedural skill; second, in the MATHia Software, which targets each student’s area of need until they demonstrate proficiency. 

The materials develop procedural skill and fluency throughout the grade level. 

• In Module 1, Topic 1, Lesson 4, students develop procedural skill when determining coordinates of the vertices of two-dimensional figures. In Activity 4.2, Reflecting Any Points on the Coordinate Plane, students reflect a point across the x-axis and y-axis and record the coordinates of the images. Next, they are given the coordinates to graph a triangle which they reflect and record the coordinates of the vertices of the images. Finally, they are given the coordinates of the vertices of a triangle, and without graphing, they determine the coordinates of images resulting from different reflections. (8.G.2 & 3) 

• In Module 2, Topic 1, Lesson 1, students develop procedural skill when working with proportional relationships. In Activity 1.2, Comparing Ratios and Graphs, students connect ratios with graphs to show proportional relationships. They draw additional lines of equations with ratios greater than or less than the ratios graphed, comparing ratio magnitude with steepness of the lines. (8.EE.5) 

• In Module 3, Topic 2, Lesson 1, Activity 4,students develop procedural skill when using the Pythagorean Theorem to calculate side lengths. Students are given four right triangles with a missing side length. Students substitute values into the Pythagorean Theorem and solve the equation for the missing value. (8.G.6)

The materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. 

• In Module 2, Topic 3, Lesson 3, Students determine when equations have one solution, no solutions, or infinitely many solutions. Finally, students write their own equations: one that has one solution, one that has no solutions, and one that has infinite solutions. (8.EE.7a)

• In Module 3, Topic 3, Lesson 2, students demonstrate procedural skill when analyzing and writing systems of linear equations. In The Road Less Traveled, Systems of Linear Equations, students write and analyze systems of linear equations. They informally calculate the solutions to systems of linear equations and then graph the systems of equations. Students conclude when parallel lines comprise the system the lines will never intersect, so there is no solution to the system. (8.EE.8a) 

• In Module 5, Topic 1, Lesson 1, students demonstrate procedural skills when solving expressions with exponents. In Activity 3, students solve expressions with exponents using the product of power rule and the power to a power rule. In Activity 4, students solve expressions with exponents using quotients of powers. (8.EE.1)

• In Module 5, Topic 2, students demonstrate procedural skill when calculating the volume of three-dimensional figures. In the MATHia Software, students have multiple opportunities to determine the volume of cylinders, cones, and spheres and use the volume of a cylinder or sphere to determine its radius. (8.G.9)

The materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. 

The materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. The materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. This is primarily found in two aspects of the materials: first, in the “Demonstrate” portion of the lesson where students apply what they have learned in a variety of activities, often in the “Talk the Talk” section of the lesson; second, in the Topic Performance Tasks where students apply and extend learning in more non-routine situations. 

The materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade level. 

• In Module 3, Topic 1, Lesson 4, students engage in the application of mathematical skills when using linear functions to solve real world problems. In Activity 3.4, students construct linear functions and analyze the graphical behavior of linear and nonlinear functions in situations such as Little Red Riding Hood’s journey, climbing cliffs, plant growth, bank accounts, etc. (8.F.5) 

• In Module 3, Topic 1, Lesson 5, students engage in the application of comparing functions to solve real-world problems. In Activities 5.1 & 5.2, students compare functions presented in different ways, such as tables, graphs, equations, and context, to explore the rate of change in real-world situations. (8.F.2) 

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

• In Module 1, Topic 1, students independently demonstrate the use of mathematics when working with transformations to solve real world problems. In the Performance task, students are given a piece of a pattern. Students use transformations to create an original complete pattern. This work is done within the context of creating a quilt. (8.G.2 &.3) 

• In Module 2, Topic 2, students independently demonstrate the use of mathematics when analyzing and writing equations to solve real-world problems. In the Performance Task, given pricing data for three health clubs, students generate equations and determine the best value for various lengths of memberships. (8.EE.7) 

• In Module 5, Topic 1, Lesson 2, students independently demonstrate the use of mathematics when using exponents to solve real world problems. In Talk the Talk - Organize the Properties, students organize their learning regarding six rules for exponents. For each rule, they write a definition, list facts and characteristics, develop and solve example problems, and then write a generalized rule for each property. (8.EE.1)

The materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

Within each topic, students develop conceptual understanding by building upon prior knowledge and completing activities that demonstrate the underlying mathematics. Throughout the series of lessons in the topic, students have ample opportunity to practice new skills in relevant problems, both with teacher guidance and independently. Students also have opportunities to apply their knowledge in a variety of ways that let them show their understanding (graphic organizers, error analysis, real-world application, etc.). In general, the three aspects of rigor are fluidly interwoven. For example: 

• In Module 1, Topic 3 Overview, “In Line and Angle Relationships, students use their knowledge of transformations, congruence, and similarity to establish the Triangle Sum Theorem, the Exterior Angle Theorem, relationships between angles formed when parallel lines are cut by a transversal, and the Angle-Angle Similarity Theorem for similarity of triangles. Students use hands-on tools to make and justify conjectures about the sum of the interior angles of a triangle, the relationship between triangle side and angle measures, and the value of exterior angles of triangles. They then apply their results to new problems. Next, students use patty paper and translations to form parallel lines cut by a transversal. They determine and informally prove the relationships between the special angle pairs formed when parallel lines are cut by a transversal and use these relationships to solve mathematical problems, including writing and solving equations. Finally, students use parallel line relationships and tools to establish the Angle-Angle Similarity Theorem and use the theorem to determine if triangles in complex diagrams are similar.” 

There are areas where an aspect of rigor is treated more independently, such as developing procedural skill and fluency in the MATHia software and Skills Practice or in the Performance Task where students work primarily with application. 

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Materials state that MP1 aligns to all lessons in the Front Matter of the MATHbook and Teacher’s Implementation Guide. Generally, lessons are developed with activities that require students to make sense of mathematics and to demonstrate their reasoning through problem solving, writing, discussing, and presenting. Overall, the materials clearly identify the MPs and incorporate them into the lessons. All the MPs are represented and attended to multiple times throughout the year. With the inclusion of the Facilitation Notes for each lesson in the Teacher’s Implementation Guide, MPs are used to enrich the content and are not taught as a separate lesson. 

MP1 - Make sense of problems and persevere in solving them. 

• In Module 4, Topic 1, Lesson 2, students have to make sense of a set of numbers and the size of that set of numbers in order to answer the following three questions. “The first set of numbers that you learned when you were very young was the set of counting numbers, or natural numbers. Natural numbers consists of the numbers that you use to count objects: {1, 2, 3, …}. Consider the set of natural numbers: 1. Why do you think people call this set of numbers the set of counting numbers? You have also used the set of whole numbers. Whole numbers are the natural numbers and the number 0, the additive identity. 2. Why is zero the additive identity? 3. Explain why having zero makes the set of whole numbers more useful than the set of natural numbers. Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses 4. What is the additive inverse of a number? 5. Represent the set of integers. Use set notation and remember to use three dots to show that the numbers go on without end in both directions.”

MP2 - Reason abstractly and quantitatively. 

• In Module 3, Topic 1, Lesson 4, Activity 4.2, students reason abstractly and quantitatively when they create equations from word problems, solve the equations, and then give their answers within the context of the original problem.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students are consistently asked to verify their work, find mistakes, and look for patterns or similarities. The materials use a thumbs up and thumbs down icon on their “Who’s Correct” activities, where students question the strategy or determine if the solution is correct or incorrect and explain why. These situations have students critique work or answers that are presented to them. 

Examples of students constructing viable arguments and/or analyzing the arguments of others include: 

• In Module 2, Topic 2, Lesson 2, Activity 1, “Analyze Cely’s calculation of the slope using the table of values. Cely: I used the ratios of \frac{y}{x} to calculate the slope of this line. \frac{y}{x}=\frac{2.3}{1}=2.3. Explain why Cely did not calculate the slope of this line correctly.”

• In Module 3, Topic 1, Lesson 4, “You and your friends are rock climbing a vertical cliff that is 108 feet tall along a beach. You have been climbing for a while and are currently 36 feet above the beach when you stop on a ledge to have a snack. You then begin climbing again. You can climb about 12 feet in height each hour. Does this situation represent a function? Explain your reasoning.” 

• In Module 4, Topic 2, Lesson 2, “Orville and Jerri want to put a custom-made, round table in their dining room. The tabletop is made of glass with a diameter of 85 inches. The front door is 36 inches wide and 80 inches tall. Orville thinks the table top will fit through the door, but Jerri does not. Who is correct and why?” 

• In Module 5, Topic 1, Lesson 3, “Kanye and Brock each tried to write the number 16,000,000,000 in scientific notation. Analyze each student’s reasoning. Who is correct?” 

The materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Throughout the teacher materials, there is extensive guidance with question prompts, especially for constructing viable arguments. 

• In Module 1, Topic 3, Lesson 1, teachers are prompted to ask, “Is there another way to arrange your three angles? Do you get the same result? Is the sum of the interior three angles the same for everyone’s triangle?" 

• In Module 2, Topic 3, Lesson 5,Getting Started, teachers are prompted to ask, “Why did you choose to write your equation in slope-intercept form? What is another equation that represents each line? In general, when two lines intersect at one point, do their slopes have to be opposites? Explain your thinking.”

• In Module 3, Topic 1, Lesson 5, Getting Started, teachers are prompted to ask, “How can you determine the greater rate of change without making any calculations? Will this method work if the scales on the axes are not the same? Why not? How can you calculate the actual slopes from the graphs?”

• In Module 5, Topic 1, Lesson 2, teachers are prompted to ask, “For expression B, what is the rule when rewriting negative exponents as positive exponents? For Expression A, what clarifications could you make to the rule so that Adam and Shane apply it correctly? For Expression B, why does the 2 remain in the numerator?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Each activity asserts that a practice or pair of practices are being developed, so there is some interpretation on the teacher’s part about which is the focus. In addition, what is labeled may not be the best example; i.e., using appropriate tools strategically (MP5) is sometimes weak where it’s labeled, but student choice is evident in Talk the Talk and Performance Tasks, which are not identified as MP5. Over the course of the year, the materials do attend to the full meaning of each mathematical practice. 

MP4 - Model with mathematics. 

• In Module 3, Topic 2, Lesson 3 Assignment, students model situations with equations. The materials state, “Analyze the scatter plot relating a male chihuahua’s weight with its age. a)Draw the line of best fit for the set of data values. Then determine the equation for the line of best fit. b)Interpret the meaning of the slope and y-intercept for your equation. c)Use your line of best fit to predict the weight of a 9-week-old male chihuahua.”

• In Module 5, Topic 5, Lesson 1, students complete a tree diagram detailing the puppy’s lineage back seven generations. They express the number of sires and dams for each generation in expanded notation and power notation and answer related questions. 

MP5 - Use appropriate tools strategically. 

• In Module 2, Topic 3, Lesson 4, Activity 2, students solve systems of equations using a graph, inspection, or substitution. Students, “Solve each linear system. State which elements of each system led to your chosen solution strategy.”

• In Module 5, Topic 1, Lesson 4, Activity 3, students choose a method to calculate each sum or difference. In Activity 3, problem 6 reads, “Calculate each sum or difference using any method. Write your answer in scientific notation. a. 3.7\times10^5+2.1\times10^6.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.  

Each Topic has a “Topic Summary” with vocabulary given with both definitions and examples (problems, pictures, etc.) for each lesson. There is consistency with meaning, examples, and accuracy of the terms. 

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

• In Module 1 Topic 1, Lesson 6, Talk the Talk, students justify their answer by using the term transformation. “Suppose a point (x, y) undergoes a rigid motion transformation. The possible new coordinates of the point are shown. Assume c is a positive rational number. (y,-x) (x,y - c) (x,-y) (x+c,y) (x-c,y) (-y,x) (-x,-y) (-x,y) (x,y+c) 1. Record each set of new coordinates in the appropriate section of the table, and then write a verbal description of the transformation. Be as specific as possible. 2. Describe a single transformation that could be created from a sequence of at least two transformations. Use the coordinates to justify your answer.” 

• In Module 4, Topic 1, Lesson 2, the term bar notation is defined for students. “A repeating decimal is a decimal with digits that repeat in sets of one or more. You can use two different notations to represent repeating decimals. One notation shows one set of digits that repeats with a bar over the repeating digits. You call this bar notation.” Students are also shown how to write repeating decimals as a fraction or with ellipses. 

The materials use precise and accurate terminology and definitions when describing mathematics and include support for students to use them. 

• In Module 3, Topic 1, Lesson 1, the teacher guide provides detailed definitions to help with explanations. “A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term in a sequence is an individual number, figure, or letter in the sequence.” 

• In Module 2, Topic 3, Lesson 1, Talk the Talk - How Do You Choose? “Throughout this topic, you have solved systems of linear equations through inspection of the equations, graphing, and substitution. How do you decide when each method is most efficient? Create a presentation or a poster to illustrate your decision-making process when you solve a system of linear equations. Consider these questions to guide the content of your presentation.  What methods do you know for solving systems of linear equations?  What visual cues about the equations in the system guide your decision?  How do the slope and y-intercept of the equations affect your decision?  Does the form of the equations in the system affect your choice? Use the systems of linear equations you solved throughout this lesson to support your reasoning and as examples of when you would choose each solution method. List at least three key points that you want to include in your presentation.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

MP7 - Look for and make use of structure. 

• In Module 4, Topic 1, Lesson 1, students look for and make use of structure when sorting numbers into different groups. “Searching for patterns and sorting objects into different groups can provide valuable insights. Cut out the 24 number cards located at the end of the lesson. Then, analyze and sort the numbers into different groups. You may group the numbers in any way you feel is appropriate. However, you must sort the numbers into more than one group. In the space provided, record the information for each of your groups; Name each group of numbers; List the numbers in each group; Provide a rationale for why you created each group.” 

MP8 - Look for and express regularity in repeated reasoning. 

• In Module 1, Topic 1, Lesson 3, students look for regularity in repeated reasoning when performing translations of figures. In Activity 3.1, students copy figures and the coordinates of their vertices onto patty paper and perform translations of the figures. They record the coordinates of the original and translated figures and explore how the translation affected the coordinates of the pre-image. Students make a general conjecture about the effect of a horizontal or vertical translation on an ordered pair.