6th Grade - Gateway 2

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 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.

For example:

• In Module 2, Topic 1, Lesson 6, students develop an understanding of ratios by using various mathematical models to solve real-world problems. In Talk the Talk - “In Goes the Kitchen Sink,” students are given a ratio and use multiple representations (scale up/scale down, table, graph, double number line) to show equivalent ratios. (6.RP.A)
• In Module 2, Topic 1, Lessons 5, students develop an understanding of the proportional relationship needed to solve a problem by using a table and a coordinate graph. In Activity 5.1 Analyzing Rectangle Ratios, students cut out and sort rectangles. They group and stack the rectangles according to the ratio of the side lengths. Then, they attach their rectangles to a coordinate grid. Students learn that graphs of equivalent ratios form a straight line that passes through the origin. (6.RP.3a, 6.RP.3b)
• In Module 3, Topic 1, Lesson 3, students explore the use of properties of arithmetic in expressions and understand that these properties apply to expressions with variables. In Activity 3.2 Algebra Tiles and the Distributive Property, students use algebra tiles to multiply expressions using the Distributive Property, Order of Operations, and combining like terms. (6.EE.3)
• In Module 4, Topic 1, Lesson 1, students develop an understanding of positive and negative numbers through the use of number lines to solve real-world problems. In Activity 1.1 Investigating Time on a Number Line - Human Number Line, students create a human number line and use it to show locations of time in the past, present, and future. Students analyze number lines and discuss the meaning of zero in the context of their number line. Negative numbers are described as the numbers to the left of zero on the number line. (6.NS.C)
• In Module 4, Topic 1, Lesson 1, students engage in the application of mathematical skills when explaining how two integers are compared. In Talk the Talk, students communicate understanding through a situation: “Your sixth grade cousin goes to school in a different state. His math class has not yet started comparing integers. Write him an email explaining how to compare any two numbers. Be sure to include one or two examples and enough details that he will be able to explain it to his class.” (6.NS.7)

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

• In Module 1, Topic 3, Lesson 2, students show conceptual understanding of ratio reasoning and relationships when solving real-world unit rate problems. In Activity 2.2 Writing Unit Rates, students write unit rates that compare the unit rates in two ways: the number of objects per dollar and the number of dollars per one object. Then students write two different unit rates for situations that do not involve money. They decide which unit rates are useful, both in general and to answer specific questions. (6.RP.3b)
• In Module 2, Topic 1, Lessons 4, students demonstrate an understanding of ratios by creating and using ratio tables to solve real-world problems. In Activity 4.3 Parts and Wholes in Ratio Tables, students use ratio tables to answer questions about mixing paint. Five pints of bluish green paint is made by using two pints of yellow paint and three pints of blue paint. Students use a ratio table to analyze student thinking about mixing paint and to determine various amounts of paint needed. (6.RP.3)
• In Module 2, Topic 1, Lessons 5, students demonstrate an understanding of ratios by using various mathematical models to solve real-world problems. In Activity 5.2 Graphing Equivalent Ratios, students analyze a time-to-distance rate scenario: Stephanie drives her car at a constant rate of 50 miles per hour. Students use a table, double number line, and the coordinate plane to determine the number of miles Stephanie drives over a period of time, and then they compare these different representations. (6.RP.3a, 6.RP.3b)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. They also provide opportunities to independently demonstrate procedural skill and fluency 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 and fluency; 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 3, students develop fluency with multiplication and division of fractions. In Activity 3.1, Fractional Fact Families, students investigate area models that show fraction-by-fraction, multiplication-division fact families. Students use cut-out diagrams from Lesson 1 to write multiplication-division fact families involving fractions. (6.NS.1)
• In Module 1, Topic 3, Lesson 4, students develop fluency when computing volume and surface area. Students learn and practice the standard algorithm for division, including division of decimals, in the context of volume and surface area. (6.NS.2, 6.NS.3)
• In Module 3, Topic 1, Lesson 2, students develop procedural skill and fluency when evaluating algebraic expressions. In Activity 2.2, Matching Algebraic and Verbal Expressions, students play Expression Explosion to practice matching verbal and algebraic expressions. (6.EE.2)
• In Module 3, Topic 2, Lesson 1, students develop procedural skill and fluency as they determine whether a number makes an equation or inequality true. In Activity 1.2, Solutions from a Set, students create equations from a list of given expressions. Then students decide which values from a set make each equation true. They investigate true and false equations and equations with no solutions and infinite solutions. (6.EE.5)

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

• In Module 1, Topic 2, students independently demonstrate fluency and procedural skill with multiplication and division of fractions through technology. In the MATHia Software, students calculate products and quotients of fractions, including mixed numbers and improper fractions. (6.NS.1)
• In Module 1, Topic 3, students independently demonstrate fluency and procedural skill with addition, subtraction, multiplication, and division of decimals through technology. In the MATHia Software, students review Adding and Subtracting Decimals, Decimal Sums and Differences, Exploring Decimal Facts, Multiplying and Dividing Decimals, and Decimal Products and Quotients. (6.NS.3)
• In Module 4, Topic 1, students independently demonstrate fluency with addition, subtraction, multiplication, and division of multi-digit decimals. In Skills Practice Section A, problems 1-8, students are provided with a number line that has determined points and answer a series of questions on which point is greater or less than a given value. (6.NS.3)
• In Module 4, Topic 1, Lesson 2, students independently demonstrate fluency in solving mathematical problems involving absolute value. In Activity 2.2, Interpreting Absolute Value Statements, students complete tables of situations, absolute value statements, and numeric values described in given and student-generated situations. The tables include statements of equality and inequality. (6.NS.7)
• In Module 5, Topic 2, students independently demonstrate the procedural skill of calculating Mean Absolute Deviation through technology. The MATHia Software provides multiple opportunities for the students to calculate and compare the mean absolute deviations with the spread of similar data sets. (6.SP.5c,d)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 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 1, Lesson 6, students engage in the application of mathematical skills when analyzing ratios to solve real-world problems. In Activity 6.2, Choosing a Strategy to Solve Ratio Problems, students are given ratio situations and choose a strategy such as graph, scaling, or addition to solve the problem and analyze how their strategy worked. (6.RP.3)
• In Module 5, Topic 1, Lesson 2, students engage in the application of mathematical skills when analyzing and writing equations to solve real-world problems. In Activity 2.1, Creating and Analyzing Dot Plots, students are given information about the medals won at the 2014 Winter Olympics and analyze the data, create a dot plot and a stem and leaf plot, as well as describe the distribution. Students take skills they’ve practiced and apply them to real data in order to analyze and report out on the questions that were generated such as, “What is the typical number of gold medals won by a country?” (6.SP.B)

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

• In Module 2, Topic 2, Lesson 3, students independently demonstrate the use of mathematics when solving real-world problems involving percent. In Talk the Talk, students demonstrate two different ways to determine the answer to questions given such as, “Leah’s goal is to score a 90 percent on the upcoming science test. If there are 40 questions on the test, how many does Leah need to answer correctly?” They then plan a presentation of the two solutions, making sure to talk about how they are the same and how they’re different. (6.RP.3)
• In Module 3, Topic 3, students independently demonstrate the use of mathematics when analyzing and writing equations to solve real-world problems. In Performance task, Graphing Quantitative Relationships - Throw it in Reverse, students reverse the dependent and independent variables on a graph and analyze the impact that has. They discuss proportionality, rate of change, and generate questions that the new graph could answer. They also create tables and equations for both graphs. Finally, they compare and contrast the two versions of the data. (6.EE.9)

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 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 4, Topic 2 Overview: “In The Four Quadrants, students explore the four quadrant coordinate plane. They use reflections of the first quadrant on patty paper and their knowledge of the rational number line to build their own four quadrant coordinate plane. They look for patterns in the signs of the ordered pairs in each quadrant and for ordered pairs that lie along the vertical and horizontal axes. After developing a strong foundation for plotting points and determining distances on the coordinate plane, students analyze and solve problems involving geometric shapes on the coordinate plane. They identify geometric shapes defined by given coordinates and determine perimeters and areas of geometric shapes in mathematical and real-world situations. Finally, students use the knowledge gained throughout the course to solve a wide range of problems on the coordinate plane, using scenarios, graphs, equations, and tables. Throughout this topic, students continue to develop their fluency with whole numbers, fractions, and decimals.”

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 instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 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. (Side note: persevere is spelled incorrectly throughout the program.) 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.

MP2 - Reason abstractly and quantitatively.

• In Module 2, Topic 3, Lesson 3, students reason abstractly when completing a table and constructing a graph to represent a unit rate. Using the graph, two unit rates are identified, one in which the x-value is 1 and one in which the y-value is 1, and the student explains the meanings of the unit rate in terms of the situation.

MP4 - Model with mathematics.

• In Module 4, Topic 2, Lesson 3, students model with mathematics when creating a presentation of their solution. In Activity 3.8 The Diver, students “train” as a freediver to beat William Trubridge’s record from 2016 of freediving almost 407 feet into the ocean. Students decide what type of questions they would ask Trubridge, how long they would need to hold their breath in order to beat Trubridge, and then make a presentation for a similar problem.

MP6 - Attend to precision.

• In Module 2, Topic 2, Lesson 1, students attend to precision when completing several columns within a table involving ratios, fractions, and decimals. “The sixth grade class is planning a field trip to Philadelphia. To decide which historical site they will visit, the 100 sixth-graders completed a survey. The results of the survey are provided in the table. Complete the Ratio, Fraction, Decimal, and Grid columns with these representations of the survey results: a ratio using colon notation, a fraction in lowest terms, a decimal, a shaded grid, an equivalent percent.”

MP7 - Look for and make use of structure.

• In Module 1, Topic 1, Lesson 1, students make use of structure when writing equivalent equations. “Rewrite the original equation 5 x 27 = 135 with an equivalent equation to represent the model you drew. a. How can you rewrite the original product by substituting the sum of the two lengths making up the split side?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 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 3, students sort and classify rational numbers. They investigate how many rational numbers can fit between between two other rational numbers on a numberline.

MP2 - Reason abstractly and quantitatively.

• In Module 4, Topic 1, Lesson 1, students explain the meaning of zero, positive numbers, and negative numbers in a variety of contexts. The teacher is directed to ask: “What representation can I use to solve this problem?; How can this problem be represented with symbols and numbers?”

MP4 - Model with mathematics.

• In Module 2, Topic 3, Lesson 3, students identify statements that are true from a given set of information about the Duquesne Incline as commuter transportation. The teacher is directed to ask: “What expression or equation could represent this situation?; What representations best show my thinking?; How does this answer make sense in the context of the original problem?”
• In Module 3, Topic 3, Lesson 3, students create graphs, tables of values, and write algebraic equations to describe the relationship between two quantities. They also answer relevant questions about two real-world scenarios.

MP5 - Use appropriate tools strategically.

• In Module 2, Topic 1, Lesson 6, given a scenario, students choose to solve problems by using the graph, by scaling, or by using an addition strategy.
• In Module 3, Topic 3, Lesson 3, students choose standard and non-standard tools to measure length. “Use standard and non-standard tools to measure the lengths of the diagonals using six different units of measure and record them in the table. Be sure to include inches and centimeters as two of your units.” The teacher is directed to ask: “What tools would help me solve this problem?”

MP6 - Attend to precision.

• In Module 3, Topic 3, Lesson 1, students solve a real-world problem based on a pool party involving variables. “a. What two quantities are changing in this situation? b. Which quantity depends on the other? c. Define variables for each quantity and label them appropriately as the independent and dependent variables.”

MP7 - Look for and make use of structure.

• In Module 1, Topic 2, Lesson 1, students are given two strategies to order rational numbers, then look for and make use of structure to identify and evaluate the most efficient strategies for a solution.
• In Module 2, Topic 1, Lesson 2, students decompose a parallelogram to create a rectangle and conclude the two shapes have the same area. The same formula can be used to determine the area of either figure. In the process of reconstructing the rectangle from the parallelogram, students make use of the structure of rectangles to discover the relationship between the parallelogram and rectangle.
• In Module 3, Topic 2, Lesson 3, students use bar models and formal properties to solve equations. The teacher is directed to ask: ”What characteristics of this expression or equation are made clear through this representation?; How can I use what I know to explain why this works?”

MP8 - Look for and express regularity in repeated reasoning.

• In Module 3, Topic 1, Lesson 3, students use the Distributive Property to factor algebraic expressions, rewriting expressions as a product of two factors, including expressions where the coefficients of the original terms do not have common factors. Students answer questions such as: “Use your example to explain what is meant by the phrase, 'multiply or divide an expression by a given value.' Use your example to explain what is meant by the phrase, 'product of two factors.' The teacher is directed to ask: How can I use what I know to explain why this works?; Can I develop a more efficient method?; How could this problem help me to solve another problem?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 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 2, Lesson 2, “Noah and Dylan were assigned the numbers 0.06 repeating and 0.1 percent, but they disagreed on which was larger. Noah says that 0.06 repeating is less than 0.1, so 0.06 repeating is less than 0.1 percent. Dylan says that since 0.1 percent is the same as as 0.001 and 0.001 is less than 0.06 repeating, 0.1 percent is less than 0.06 repeating. Who is correct? Explain your reasoning.”
• In Module 3, Topic 2, Lesson 1, “Identify your equations that are always true, never true, and those equations where you don’t yet know whether they are true or false. Explain your reasoning.” and “Write an equation with variables that has no possible solution. Explain why the equation has no solution.”
• In Module 5, Topic 1, Lesson 2, “Jessica asked, 'How many medals did the United States win? How many of those were gold?' Maurice thought a better set of questions would be, 'What is the typical number of medals won? What is the typical number of gold medals won by a country?' Who’s correct? Explain your reasoning.”
• In Module 5, Topic 1, Lesson 3, “Bella says, 'There are 5 buildings represented in the histogram since there are 5 bars.' Do you agree or disagree with Bella’s statement? If you do not agree with Bella, estimate how many buildings are represented in the histogram.”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 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 3, the teacher is prompted to ask, “How is the area of a face of a cube measured? Analyze the two responses and explain why Leticia is incorrect in her reasoning.”
• In Module 2, Topic 1, Lesson 1, students critique the reasoning of others when analyzing two predictions of a solution to a problem. “Robena and Eryn each predicted the final score of a basketball game between the Crusaders and the Blue Jays. Analyze each prediction. Describe the reasoning that Robena and Eryn used to make each statement.” Teachers are provided questions to ask, such as: "How can both of these be thumbs up if their predictions are different? How is Robena’s reasoning different than Eryn’s reasoning?”
• In Module 3, Topic 2, Lesson 1, the teacher is prompted to ask, “Why is Rylee’s work incorrect? What feedback would you give Rylee about her strategy? What property did Clover use? Did anyone use the Reflexive Property of Equality as part of their strategy? Did Clover and Fiona use the same strategy? Is there more than one solution completing this equation?”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 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 3, Topic 3, Lesson 3, Talk the Talk, students complete a graphic organizer describing the advantages of using verbal, tabular, graphical, and equation representation.
• In Module 4, Topic 2, Lesson 3, students accurately represent absolute value equations to calculate temperature changes, for example, calculating the temperature change between two points: |15.6| + |-10.6| = 26.2 degree drop.

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

• In Module 1, Topic 1, Lesson 1, key terms are identified as numeric expression, equation, and Distributive Property. Students describe the expression 4(2+15) in different ways. Teachers are prompted to ask, “What is the difference between an expression and an equation?” A tip for helping ELLs differentiate expression and equation can be used as part of the lesson.
• In Module 2, Topic 3, Lesson 3, the Topic Overview describes how students deepen their understanding of converting units of measurement using ratio reasoning and strategies for determining equivalent ratios. The term convert is defined, and students use approximate conversion rates to estimate measurement conversions before engaging with formal methods of converting. Converting among units of measurement in the same system is recast in terms of conversion ratios, which can also be called conversion rates.