7th Grade - Gateway 2

The instructional materials for Carnegie Learning Middle School Math Solution Course 2 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 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 2.1 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 1 Lesson 2, In Activity 2.2 Mathematics Gymnastics - Applying 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 1.3 Slicing and Dicing - Slicing a Right Rectangular Prism, building from a previous cube activity, students build a right rectangular prism in clay that is not a cube and again slice their clay prisms to create six cross-sectional shapes (a square, a rectangle that is not a square, a triangle, a pentagon, a hexagon, and a parallelogram that is not a rectangle). They discuss with other students where and how they sliced the prism 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 - Strategies, students write and describe their strategies for evaluating expressions and explain how tables are helpful. (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 1.4 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 instructional materials for Carnegie Learning Middle School Math Solution Course 2 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 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 1, Lesson 2, 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 instructional 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 2, 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 instructional materials for Carnegie Learning Middle School Math Solution Course 2 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 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 calculate the total amount of smoothie that the recipe makes and then use what they have learned about fractional unit rates to calculate the amount of pumpkin-y ingredients per unit of smoothie. (7.RP.1)
• In Module 1, Topic 3, Lesson 1, 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. The paint problem is revisited, building bird houses, growing bamboo, and speed and car rates situations are used. (7.RP.2b)
• In Module 3, Topic 3, Lesson 2, students engage in the application of mathematical skills when analyzing and comparing expressions to solve real-world problems. In Getting Started, students examine the cost structures for two different limousine companies in order to create a competitive cost structure for a third company. (7.EE.4)

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

• In Module 1, Topic 4, 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 stem-and-leaf 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 stem-and-leaf 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 instructional materials for Carnegie Learning Middle School Math Solution Course 2 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 5, Topic 1, Lesson 1 Overview, “Students are introduced to geometry and geometric constructions. Measuring tools are distinguished from construction tools, and the concepts of sketch, draw, and construct are differentiated. Students sketch and draw the same figure, and then they compare the processes used to create each figure. Point, line, and plane are described as the essential building blocks of geometry. Students learn how to properly draw, sketch, and name each of these essential building blocks. Line segment and endpoints are also defined, and students learn how to name and use symbols to represent them. Finally, they use a compass to construct circles and arcs. The terms arc, congruent, congruent line segments, and intersection are given. Students conclude the lesson by duplicating 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 were students work primarily with Application.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 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 $$-(1/2)xy$$ 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.

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. The lesson specifically states, ”Students should have access to construction tools, measuring tools, and patty paper.”

MP6 - Attend to precision.

• In Module 1, Topic 3: Lesson 2, students attend to precision when working with proportional relationships. “In a proportional relationship, the ratio of all y-values, or outputs, to their corresponding x-values, or inputs, is constant. This specific ratio, y to x, is called the constant of proportionality. Generally, the variable k is used to represent the constant of proportionality. Suppose you want to determine the actual lengths of your favorite television shows, without commercials, if you know the total program length. Identify the input and output quantities in this scenario.”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 4, Lesson 4, students make sense of Percent Increase and Percent Decrease problems such as, “Casey added together all of the increases from 2000 to 2005 and subtracted the decreases. She concluded that there was about a 35 percent increase in gas price from 2000 to 2005. Is Casey correct? Explain why or why not.”

MP2 - Reason abstractly and quantitatively.

• In Module 1, Topic 4, Lesson 1, students reason about percent problems such as, “Dante has been shopping around for a new mountain bike. He found two bikes that he likes equally, one is sold at Mike’s Bikes for $300, and the other is sold at Cycle Center for $275. Dante has a coupon for 25 percent off any bike at Mike’s Bikes. However, the manufacturer of the bike at Cycle Center has included a $40 rebate after the purchase of the bike. Where should Dante purchase his mountain bike? Show all of your work and explain your reasoning.”

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

• In Module 1, Topic 1, Lesson 3, students use different strategies to determine the area of shaded regions inside geometric figures. They compare strategies from other students, explain the one they prefer, and use it to solve additional problems.

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 coins, number cubes, spinners, and 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 1, Topic 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.

MP6 - Attend to precision.

• In Module 5, Topic 1, Lesson 1, students are introduced to basic geometry vocabulary. New terms are defined by building on the understanding of the initial terms. Students use the mathematical notation, language, and definitions accurately. They also use tools to create constructions and draw conclusions from those constructions. In Talk the Talk, the problems state: “Use the given sides and angles to complete each construction. 1. Construct and label a segment twice the length of segment PQ. 2. Construct and label an angle twice the measure of angle P. 3. Identify all the points, lines, line segments, rays, and angles that you can in the two figures in Questions 1 and 2.”

MP7 - Look for and make use of structure.

• In Module 3, Topic 1, Lesson 2, 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 5, 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.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Topic 3, Lesson 1, 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 2, Lesson 3, “Vernice is told that the DC to Boston flight took 10 minutes longer than estimated. She calculated the percent error and got 10.3 percent. She later learns that she had been given the wrong information. The flight took 10 minutes less than estimated. Vernice thinks that the percent error should just be -10.3 percent. Is she correct? Explain why or why not.”
• 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 instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 2, Lesson 3, teachers are prompted to ask, “How are John David’s and Parker’s methods different? How did you verify Natalie’s method? Does using all three methods to solve this proportion always give you the same answer? Is one method easier to use than the other methods? How are the three methods similar to each other? How are the three methods different from each other? What method did you use to solve each proportion? Did the position of the unknown quantity factor into your method for solving the proportion? How can you check your solution to make sure it is correct?”
• In Module 1, Topic 4, Lesson 4, Activity 4.1, teachers are prompted to ask, “Does it matter which number is divided by which number? Will you get the same answer? What is the difference between computing the percent increase and the percent decrease? How do you know when to compute the percent increase or the percent decrease in a problem situation?”
• In Module 2, Topic 2, Lesson 3, Activity 3.2, teachers are prompted to ask, “How is calculating sums with two-color counters similar to calculating sums with a number line? What pattern(s) are you noticing?”
• In Module 3, Topic 1, Lesson 3, Combining Like Terms, teachers are prompted to ask, “Why is x+5 located to the left of x+10? Why is x+10 located to the right of x+5? What is an example of an expression that is located between x+5 and x+10? Is the value of x always a positive number in this situation? Can the value of x be a negative number? Can the variable x represent any number?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 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 ∠. 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. “Variables are used to represent unknown quantities and each quantity corresponds to a specific location on a number line. Values are substituted for the variable to validate the correct placement of the variables on the number lines.” 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.