7th Grade - Gateway 2

The materials for Carnegie Learning Middle School Math Solution Course 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content 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 2, Topic 1, Lesson 1, students show their understanding of adding and subtracting rational numbers using a visual representation. In Talk the Talk, Mixing Up the Sums, students create addition problems from a given sum using number lines to explain and demonstrate their understanding. (7.NS.A) 

• In Module 2, Topic 1, Lesson 2, students show their understanding of adding and subtracting rational numbers using a visual representation. In Activity 1 Walking the Number Line, students walk along a number line to develop a conceptual understanding using physical and visual representations. (7.NS.1b) 

• In Module 3, Topic 2 Lesson 1, In Activity 2.2 Mathematics Gymnastics - Rewriting Expressions Using the Distributive Property, students develop their understanding of the distributive property by drawing area models to represent expressions. (7.EE.A) 

• In Module 5, Topic 2, Lesson 1, students are introduced to and explore the concept of a cross-section when they create different shapes by slicing a three-dimensional figure. In Activity 1. Slicing a Cube - students build a cube in clay and slice their clay cubes to create six cross-sectional shapes (a triangle, a square, a rectangle that is not a square, a parallelogram that is not a rectangle, a hexagon, and a pentagon). They discuss with other students where and how they sliced the cubes to make the cross-sections. (7.G.3) 

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

• In Module 2, Topic 2, Lesson 1, students interpret models and write explanations to demonstrate an understanding of multiplication of rational numbers. In Activity 1.1 Equal Groups - Modeling the Multiplication of Integers, knowing that multiplication can be represented as repeated addition, students are shown an example of 3 x 4 with both a number line and circles with positives in them. Students explain how these represent the multiplication problem. For each integer problem, students explain their understanding. (7.NS.2, 7.NS.3) 

• In Module 3 Topic 1, Lesson 1, students demonstrate conceptual understanding of expressions when writing an explanation of their solution. In Talk the Talk - Business Extras, students write and evaluate expressions for given real-world problems. (7.EE.1, 7.EE.2) 

• In Module 5, Topic 2, Lesson 1, students demonstrate an understanding of a cross-section by categorizing each sliced cross-section. In Activity 3 Cross-Sections of Right Rectangular Prism, students create a graphic organizer of cross-sections they sliced in a rectangular prism. They cut out diagram cards and description cards related to the different cross-sections and tape them into the appropriate rows of the organizer. (7.G.3)

The materials for Carnegie Learning Middle School Math Solution Course 2 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 2, Lesson 1, students develop procedural skill when solving problems involving unit rate. In Getting Started, students complete given tables and include the unit rate of lemon-lime for each cup of punch for each recipe. Students then draw a graph for each recipe on the coordinate plane. Then students label each graph with the person's recipe and the unit rate. (7.RP.1) 

• In Module 3, Topic 2, Lesson 1, students develop procedural skill when writing expressions using the distributive property. In Mathematics Gymnastics, students rewrite algebraic expressions with rational coefficients using the distributive property. They also expand linear expressions. Students factor linear expressions in a variety of ways, including by factoring out the greatest common factor and the coefficient of the variable. (7.EE.1) 

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

• In Module 2, Topic 2, students demonstrate procedural skill when they make conjectures about the rules for multiplying and dividing integers. In the MATHia Software, students fill in the blanks to show the understanding of multiplying and dividing integers. (7.NS.A) 

• In Module 3, Topic 1, students demonstrate procedural skill when solving two-step equations through technology. In the MATHia Software, students determine unknown values and enter values into tables to recognize patterns. Students express the patterns in two-step expressions and use the solver tool to solve two-step equations. (7.EE.4a) 

• In Module 4, Topic 1, students demonstrate procedural skill when determining the probability of an event. In the MATHia Software, students work to build probability models and determine probabilities of simple and disjoint events and use proportions to make predictions based on samples and theoretical probabilities. (7.SP.6 & 7)

The materials for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 2, Lesson 1, students engage in the application of mathematical skills when calculating unit rates to solve real world problems. In Talk the Talk - Getting Unit Rate-ier, students are asked to determine whether each statement is true or false, and provide an example to justify their answer. (7.RP.1) 

• In Module 1, Topic 2, Lesson 2, students engage in the application of mathematical skills when using proportional relationships to solve real-world problems. In Activity 1.3, Proportional or Not?, students explore tables and graphs to discover that graphs of proportional relationships are straight lines and tables have a constant ratio. (7.RP.2b) 

•  (7.EE.4) 

• In Module 3, Topic 1, Lesson 4, Activity 2, students engage in the application of mathematical skills when creating equations to solve real-world problems. The materials provide the following scenario, “Felicia’s Pet Grooming charges $15 for each dog washed and groomed on the weekend. The cost of the dog shampoo and grooming materials for a weekend’s worth of grooming is $23.76. Felicia wants to know her weekend profits.”(7.EE.4a)

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

• In Module 1, Topic 3, Lesson 1, students independently demonstrate the use of mathematics when using proportional relationships to solve real-world problems. Percent models, proportions, and the constant of proportionality are revisited to solve percent problems with simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error. (7.RP.3) 

• In Module 2, Topic 1, students independently demonstrate the use of mathematics when addition and subtraction of rational numbers is used to solve real-world problems. In Performance Task, students represent rational numbers as a sum and a difference of two rational numbers, then develop their own real-world problems that model their representations. (7.NS.3) 

• In Module 4, Topic 3, Lesson 3, students independently demonstrate the use of mathematics when describing how to use a dot plot or box-and-whisker plot to determine variation of data and the mean. In Talk the Talk, students write one to two paragraphs to summarize the key points in the lesson by explaining how it is possible to determine the mean and the variation of data for two populations from a dot plot or box-and-whisker plot. They include answers to questions such as: “How can you compare the mean and the spread of data for two populations from a dot plot? If the measures of center for two populations are equivalent, how can the mean absolute variation show the differences in variation for two populations?” (7.SP.B)

The materials for Carnegie Learning Middle School Math Solution Course 2 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 5, Topic 1, Lesson 1 Overview, “Students distinguish measuring tools from construction tools as they differentiate the concepts of sketch and construct. They learn how to correctly draw, sketch, and name each of the essential building blocks of geometry. Students use a compass to construct circles and arcs. They then duplicate line segments and angles using only construction tools.” 

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 2 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 2, Topic 2, Lesson 2, students make sense of problems when classifying numbers. “What types of numbers are the quotients in Question 1? Use the definitions of the different number classifications to explain why this makes sense.” 

MP2 - Reason abstractly and quantitatively. 

• In Module 3, Topic 1, Lesson 1, students reason quantitatively when looking at various expressions. “The expressions $$3x^2 + 5$$ and −(\frac{1}{2})xy are examples of expressions that are not linear expressions . Provide a reason why each expression does not represent a linear expression.”

• In Module 1, Topic 1, Lesson 1, students reason abstractly when they use a formula to compute circumference. In Activity 1.3, students create a formula for the circumference of any circle and use it to compute unknown values.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 2, Lesson 2, problems include: “Dontrell claims that the number of bird feeders Bob builds is proportional to the number of bird feeders Jake builds. Do you agree with Dontrell’s claim? Explain your reasoning.” and “Vanessa thinks that there are only two: one with a width of two inches and a length of six inches, and another with a width of three inches and a length of four inches. Is she correct? Explain your reasoning.” 

• In Module 2, Topic 1, Lesson 5, Activity 1, students are given a table showing when some elements reach freezing point. Then, students are given the following questions, “Patricia and Elliott are trying to figure out how much temperature would have to increase from the freezing point of hydrogen to reach the freezing point of phosphorus. Patricia says the temperature would have to increase 545.7\degree F, and Elliott says the temperature would have to increase 322.3\degree F. Who is correct? Explain your reasoning.”

• In Module 4, Topic 1, Lesson 3, students construct viable arguments when evaluating the probability of the landing position for a cup in a game two friends are designing called Toss the Cup. “1. Predict the probability for each position in which the cup can land. 2. List the sample space for the game. 3. Can you use the sample space to determine the probability that the cup lands on its top, bottom, or side? Explain why or why not. 4) Do you think all the outcomes are equally likely? Explain your reasoning.” 

• In Module 5, Topic 1, Lesson 3, “Sarah claims that even though two segment lengths would form many different triangles, she could use any three segment lengths as the three sides of a triangle. Sam does not agree. He thinks some combinations will not work. Who is correct? Remember, you need one counterexample to disprove a statement.” 

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 2. Lesson 3, Activity 2, teachers are prompted to ask, “Why did Parker multiply each side by 3200? How did Nora get the fraction \frac{1}{40}? What is similar and different between the two solution strategies? How can you solve the problem using Nora’s method? How could you use scaling to solve this proportion?”

• In Module 1, Topic 3, Lesson 4, Activity 4.1, teachers are prompted to ask, “What is the difference in the steps to calculate the percent increase and percent decrease? Which value goes in the denominator? Provide an example of a 50% decrease.” 

• In Module 2, Topic 1, Lesson 3, Activity 2, teachers are prompted to ask, “How is using two-color counters related to using a number line to represent integer addition?” 

• In Module 3, Topic 1, Lesson 3, Activity 1, teachers are prompted to ask, “Why do you think Brent started this way? What is another way you could have solved the equation? How can you check your solution?”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 4, Topic 1, Lesson 4, Talk the Talk, students develop a simulation to model different situations and describe one trial. They then conduct the simulations and answer related questions. Supplies such as note cards for the experiments are provided for students to decide how they will simulate the situation to answer the questions. 

MP5 - Use appropriate tools strategically. 

• In Module 5, Topic 1, Lesson 4, students use appropriate tools when constructing triangles with given angles. In Activity 4.1 A Triangle Given Three Angles, students construct various triangles. Materials list for this lesson include: Compasses, Patty paper, Protractors, Rulers, and Straightedges. 

• In Module 1, Topic 1, Lesson 1: Talk the Talk, students are asked to use what they have learned during previous activities to now draw two circles, one with a radius length of 3 centimeters and one with a diameter length of 3 centimeters and compare their characteristics. No guidance is given on what tools are to be used when doing this.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 5, Topic 1, Lesson 1.3, The terms angle, sides of an angle, vertex, and ray are defined for students. “An angle is formed by two rays that share a common endpoint. The angle symbol is \angle. The sides of an angle are the two rays. The vertex of an angle is the common endpoint the two rays share.” Students copy an angle using a compass and a straightedge. They then identify the angles and vertices of angles in their constructions. 

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 variable represents an unknown quantity. You can model the relationship between variable expressions with the same variable on a number line.” The student book condenses this definition to a more student-friendly version. “In algebra, a variable is a letter or symbol that is used to represent an unknown quantity.” 

• In Module 4, Topic 1, Lesson 3, students explain the difference between experimental and theoretical probability in their own words.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 3, Topic 2, Lesson 1, students analyze an expression in order to write equivalent expressions using the distributive property. This lesson states, “Often, writing an expression in a different form reveals the structure of the expression. Meaghan saw that each expression could be rewritten as a product of two factors.” 

• In Module 5, Topic 2, Lesson 4, students develop a strategy for calculating the areas of regular polygons, generalize it by decomposing regular polygons, and determine that they can calculate the area of one of the n congruent triangles in the n-gon and multiply the area by n to calculate the area of the regular polygon. They transfer their work to pentagons and any regular polygonal base of a prism or pyramid. 

MP8 - Look for and express regularity in repeated reasoning. 

• In Module 2, Topic 2, Lesson 1, students look for and express regularity in repeated reasoning, when they notice repeated calculations to understand algorithms and make generalizations. In Activity 2, students describe and extend two different patterns of equations, students then make generalizations that can be applied to different pairs of integers and eventually describe a rule that can be used to multiply any two integers.