8th Grade - Gateway 2

The instructional 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 2, Topic 3, 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 2, Topic 3, 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 3, Topic 2, Lesson 4, 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 instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional 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 instructional 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 2, Topic 3, Lesson 3, students develop procedural skill when they use an equation to generate a table. In Getting Started, What’s My Rule?, students are given tables with ordered pairs and determine the equation that generated those ordered pairs. Then students create their own table of ordered pairs based on an equation they generate and give it to a partner to determine the equation. (8.F.2)

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

• In Module 3, Topic 1, Lesson 2, students demonstrate procedural skill when solving and writing algebraic expressions. In MP3s and DVDs, Analyzing and Solving Linear Equations, given situations, students write algebraic expressions then use the expressions to write and solve equations. 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 2, 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 skill when solving expressions with exponents. In Activity 1.3, Multiplying and Dividing Powers, given information about megabytes, and kilobytes, students calculate the storage capacity of eBooks and jump drives, providing opportunities for students to perform multiplication and division on expressions with exponents. (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 instructional materials for Carnegie Learning Middle School Math Solution Course 3 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. The instructional 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 instructional 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 2, Topic 3, Lesson 3, 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 2, Topic 3, 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 instructional 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 3, Topic 1, 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 five party venues, students generate equations and determine the best value for various numbers of guests. (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 instructional materials for Carnegie Learning Middle School Math Solution Course 3 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

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 and use their drawings and real-world scenarios to identify transversals and special pairs of angles. 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 were students work primarily with Application.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

Standards for Mathematical Practice are referred to as Habits of Mind in this program. The Habits of Mind are identified in all lessons in both the teacher and student workbooks using an icon. There are four icons, only one represents a single MP, “attend to precision,” while the other three represent pairs of MPs, though generally one MP is the focus of the lesson. No icon is used for MP1, and it is stated in the Teacher’s Implementation Guide (TIG): “This practice is evident every day in every lesson. No icon used.” Each activity shows the practice or pair of practices being developed. Questions to facilitate the development of Habits of Mind are listed for both students and teachers throughout the program. The Habits are identified in the Overview in the Student and Teacher Editions, but not in the Family Guide that comes with the Topics. The icon appears within each lesson with questions listed in the Teacher Guide to facilitate the learning where they occur. 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 “Questions to Ask” in the Teacher Guide and the corresponding Facilitation Notes in each lesson, 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: a. Why do you think this set of numbers is sometimes referred to as the set of counting numbers? b. How many natural numbers are there? c. Does it make sense to ask which natural number is the greatest? Explain why or why not.”

MP2 - Reason abstractly and quantitatively.

• In Module 3, Topic 1, Lesson 2, Activity 2.2, 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.

MP3- Construct viable arguments and critique the reasoning of others.

• In Module 4, Topic 1, Lesson 3, students critique the reasoning of others when choosing which solution is correct. In Activity 3.1, students are asked to decide which student's reasoning is correct when determining the square root of 144, 14.4, 1.44, 1,440, and 14,400, and to explain their reasoning why.

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 30 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.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that the instructional materials carefully attend to the full meaning of each practice standard.

Each activity asserts that a 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.

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

• In Module 4, Topic 1, Lesson 2, students make sense of a set of integers. “Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses. What is the additive inverse of a number? 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. Does it make sense to ask which integer is the least or which integer is the greatest? Explain why or why not.”

MP2 - Reason abstractly and quantitatively.

• In Module 1, Topic 1, Lesson 1, students learn about the processes of conjecture, investigate, and justify by deciding which figures are congruent to other provided figures. Students make conjectures, investigate with patty paper, and then explain how they could slide, flip, or spin the original figure to obtain each congruent figure.
• In Module 2, Topic 4, Lesson 4, students reason about lines of best fit from data they collected in an experiment. “Interpret the meaning of the y-intercept in this situation. 1. Compare your results for the matching lists to the results for the non-matching lists. Do your results seem reasonable? Explain your reasoning. 2. Revisit the statistical question you asked at the beginning of the lesson. How did the results of the experiment help to answer this question? Explain your reasoning. 3. What conclusions do you think a cognitive psychologist might draw from your experiment results?”

MP3 - Construct viable arguments and critique the reasoning of others.

• In Module 5, Topic 1, Lesson 1, problems 9 and 10, students explain their reasoning when choosing a correct solution from several given solutions. This is practiced through “thumbs up and down” problems. “Who is Correct” allows students to construct an argument as well as critique the reasoning of others when choosing a correct solution.

MP4 - Model with mathematics.

• In Module 3, Topic 2, Lesson 3, students model situations with equations. "Janet was helping her mother make potato salad for the county fair and was asked to go to the market to buy fresh potatoes and onions. Sweet onions cost $1.25 per pound, and potatoes cost $1.05 per pound. Her mother told her to use the $30 she gave her to buy these two items. 1) Write an equation in standard form that relates the number of pounds of potatoes and the number of pounds of onions that Janet can buy for $30. Use x to represent the number of pounds of onions, and y to represent the number of pounds of potatoes that Janet can buy. 2) Janet's mother told her that the number of pounds of potatoes should be eight times greater than the number of pounds of onions in the salad. Write an equation in x and y that represents this situation."
• 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 3, Topic 2, Lesson 4, students create a system of equations and solve it. Students use a table, a graph, and algebraically solve their equations. They are reminded that “You can use a variety of strategies and representations to solve a system of linear equations. Inspection, table, graph, substitution.” Question 9 says, “Explain the advantages and disadvantages of using each strategy. Table, graph, substitution.”
• In Module 5, Topic 1, Lesson 4, students choose a method to calculate each sum or difference. In Activity 4.3, problem 6 reads, “Calculate each sum or difference using any method. a. 3.7105 + 2.1106.”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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 3, 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 3, Topic 1, Lesson 1, “Sandy and Sara each divided both sides of their equations by a factor and then solved. a. Explain the reasoning used by each. b. Do you think this solution strategy will work for any equation? 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, Corinne, Brock, and Daniel each tried to write the number 16,000,000,000 in scientific notation. Analyze each student’s reasoning.”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that the instructional 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 4, teachers are prompted to ask, “Does the time elapsed depend on the distance climbed, or does the distance climbed depend on the time elapsed? What is another way to write the equation? Does one equation make more sense than the other? If so, why? Are there points located between the points plotted on the graph? What is their relevance to the problem situation? Does the relation shown on the graph pass the vertical line test? What does this imply? If the graph is extended, will it pass through the origin? What does this imply?”
• In Module 3, Topic 1, Lesson 1, students critique the reasoning of others when analyzing different solutions to a problem involving variables. Teachers are prompted to ask, “What is the first step in Sandy’s solution? Why did Sandy decide to divide both sides of the equation by three first? What is the first step in Sara’s solution? Why did Sara decide to divide both sides of the equation by 21 first? How do you know when it is a good idea to use Sara’s method? Which student(s) subtracted the smaller x-value from both sides of the equation?”
• In Module 5, Topic 1, Lesson 2, teachers are prompted to ask, “What changed from the previous step to this step? What operation must have been performed in order to have this result? What property must have been used to justify that mathematical operation? How did you decide what mathematical operation to perform? What property justified you to perform that mathematical operation? Is there another way to solve the problem? Did anyone solve the problem another way?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 3 meet expectations that materials attend to the specialized language of mathematics.

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. This is called 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 2, Topic 3, 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 3, Topic 2, Lesson 1, students describe how to determine a point of intersection using a table alone. They then compare this process with the process using a graph and equations. Students conclude that in all three representations, the point of intersection is the common solution to the linear equations, represented by a single ordered pair.