6th Grade - Gateway 1

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. 

The assessments are aligned to grade-level standards. The instructional materials reviewed for this indicator were the Post-Tests, which are the same assessments as the Pre-Tests, both Form A and Form B End of Topic Tests, Standardized Practice Tests, and the Topic Level Performance Tasks. Examples include: 

• Module 5, Topic 2, Performance Task: 6.SP.1, 2, 3, 4, 5: Numerical Summaries of Data: Hours Playing Video Games: Students are given a scenario of a student collecting data about video game usage; however, she lost the data set. She still has information such as range, minimum value, median, and interquartile range; the student uses this information to create a data set that could represent the data that was lost. Work is scored on accurate numbers in the data set, summary of the data set, box-and-whisker plot representing the data, explanation of how the data set was generated, and a statement about data sets. 

• Module 1, Topic 1, End of Topic Test-Form A, 6.NS.4: Students find the greatest common factor using the distributive property to rewrite an expression. Questions 3 and 4  state, “Rewrite each sum in the form a(b + c) such that the integers b and c have no common factor: 82+30; 35+42” 

• Module 2, Topic 2, End of Topic Test Form A, 6.RP.3c: Students find a percent of a quantity when completing a fraction-decimal percent table with survey results. Question 6 states, “One hundred middle school students take a survey that asks them about their food preferences. Complete the table by representing the survey results as a fraction, decimal, and percent. Make sure your fractions are in lowest terms.” 

• Module 3, Topic 3, End of Topic Test Form A, 6.EE.9: Students represent distance and time in an equation. Question 5 states, “A hiker is climbing at a constant rate of 2.4 miles per hour. a. Write an equation to model the relationship between the hiker’s distance climbed and the time in hours.”

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The design of the materials concentrates on the mathematics of the grade. Each lesson has three sections (Engage, Develop, and Demonstrate) which contain grade-level problems. Each topic also includes a performance task. 

• In the Engage section, students complete one activity that will “activate student thinking by tapping into prior knowledge and real world experiences and provide an introduction that generates curiosity and plants the seeds for deeper learning.” For example, Module 3/Topic 3/Lesson 3 (500) has students work in pairs to determine the values of shapes represented as objects balancing in a mobile. The activity is designed to engage students in thinking about different representations of equality and equations and to stimulate students’ reasoning about solutions to equations. (6.EE.9) 

• In the Develop section, students do multiple activities that “build a deep understanding of mathematics through a variety of activities —real-world problems, sorting activities, worked examples, and peer analysis—in an environment where collaboration, conversations, and questioning are routine practices.” For example, Module 1/Topic 2/Lesson 5/Activity 4 (56) has students explore dividing fractions by dividing across the numerators and denominators and then rewriting the quotient. Students compare different strategies for dividing using analysis of peer work. (6.NS.1) 

• In the Demonstrate section, students “reflect on and evaluate what was learned.” An example of this is Module 2/Topic 2/Lesson 2 (278), “Talk the Talk: Brain Weights,” where students order the brain weights of different mammals given as percents in relation to the weight of a chimpanzee’s brain. They use benchmark percents to calculate the brain weights. Students also use benchmarks to reason about percents less than one percent and greater than 100 percent. (6.RP.3c) 

The end of each lesson in the student book includes Practice, Stretch, and Review problems. These problems engage students with grade level content. Practice problems address the lesson goals. Stretch problems expand and deepen student thinking. Review problems connect to specific, previously-learned standards. All problems, especially Practice and Review, are expected to be assigned to all students. 

After the lessons are complete, the students work individually with the MATHia software and/or on Skills Practice that is included. 

• MATHia - Module 1, Topic 1 (3B-3D): Students spend approximately 195 minutes In the MATHia software using the Commutative, Associative, and Distributive Properties to rewrite numeric expressions. Students practice calculating finding the prime factorization, GCF, LCM, multiplying and dividing fractions. 

• Skills Practice - Module 1, Topic 2 (63B-63D): Students find area, volume, and surface area of geometric figures.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the amount of time spent on major work, the number of topics, the number of lessons, and the number of days were examined. Review and assessment days were also included in the evidence. 

• The approximate number of topics devoted to major work of the grade (including supporting work connected to the major work) is 8.5 out of 13, which is approximately 65.3 percent. 

• The approximate number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 40 out of 51, which is approximately 78 percent. 

• The approximate number of days devoted to major work (including supporting work connected to the major work) is 108 out of 139, which is approximately 77.6 percent. 

The approximate number of days is most representative of the instructional materials because it most closely reflects the actual amount of time that students are interacting with major work of the grade. As a result, approximately 77.6 percent of the instructional materials focus on major work of the grade.

The materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Supporting standards/clusters are connected to the major standards/clusters of the grade. Examples include: 

• In Module 1, Topic 3, Lesson 1, Product Placement - Multiplying Decimals.  Using the area model, students represent the multiplication of two decimals less than one on a hundredths grid. They use estimation to reason about decimal point placement in multiplication problems and then analyze patterns to develop the algorithm for multiplying decimals. Students solve area and volume problems that require multiplying, adding, and subtracting decimals. Major Standard 6.NS.3 is connected to 6.G.1, 6.G.2.

• In Module 1, Topic 3, Lesson 4, Dividend in the House - Dividing Whole Numbers and Decimals.  Students learn the standard algorithm for long division with whole numbers. They demonstrate how the algorithm works for decimal dividends by relating it to a model, and they make sense of how to modify the algorithm for decimal divisors. Students solve area, surface area, and volume problems requiring decimal division.. Major Standards NS.2, 6.NS.3 are connected to supporting standards 6.G.1, 6.G.2.

• In Module 1, Topic 2, Lesson 1 All About That Base... and Height –  Area of Triangles and Quadrilaterals. Students progressively derive the formulas for the area of a parallelogram, triangle, and trapezoid by using composition and decomposition of polygons with known area formulas. They use their formulas to calculate the area of parallelograms, triangles, and trapezoids. Major cluster 6.EE is supported by 6.G.

• In Module 4, Topic 2, Lesson 2 Playing with Planes - Graphing Geometric Figures. Graphing geometric figures and finding their perimeter and area in the coordinate plane (6.G.3) supports the major work of 6.NS.8 involving graphing points in the four quadrants of a coordinate plane.

The materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Materials include problems and activities that connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples include: 

• In Module 1, Topic 3, Lesson 4, “Students learn the standard algorithm for long division with whole numbers. They demonstrate how the algorithm works for decimal dividends by relating it to a model, and they make sense of how to modify the algorithm for decimal divisors. Students solve area, surface area, and volume problems requiring decimal division.” Connecting 6.NS.2, 6.NS.3, 6.G.1, 6.G.2. 

• In Module 4, Topic 2, Lesson 2, “Students solve geometry problems using the coordinate plane. They conjecture about graphed polygons and prove their conjectures. Students graph triangles and quadrilaterals using given criteria and calculate distances to solve perimeter, area, and volume problems. They then label a parallelogram's coordinates without a coordinate grid and write an algebraic expression to solve for its area”, Standards 6.NS.8, 6.G.3 are connected as students Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane using coordinates and absolute value to find distances between points.

• In Module 5, Topic 2, Lesson 3, “Students compare the means of two data sets displayed with dot plots and discover the need for another measure of variation. They use the mean absolute deviation (MAD) to describe the spread of data. Students calculate and analyze the MAD to interpret data in context.” 6.SP.2, 6.SP.3, 6.SP.5a, 6.SP.5b, 6.SP.5c connected when students summarize numerical data sets in relation to their context. 

• In Module 1, Topic 1, Lesson 2, “Students create rectangles with given areas and relate their dimensions to factors and common factors. They use prime factorizations to determine the greatest common factor (GCF) and least common multiple (LCM) of two numbers. Students examine the rows and columns of an area model to identify multiples and the LCM. They describe the relationship between the product, GCF, and LCM.” 6.NS.4, 6.EE.1 are connected as students write, read, and evaluate expressions in which letters stand for numbers as they find the GCF and LCM.

The materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

The instructional materials clearly identify content from prior and future grade levels and use it to support the progressions of the grade level standards. The content is explicitly related to prior knowledge to help students scaffold new concepts. Content from other grade levels is clearly identified in multiple places throughout the materials. Examples include: 

• A chart in the Overview shows the sequence of concepts taught within the three grade levels of the series (FM-6 and 7). 

• The Family Guide (available online) presents an overview of each Module with sections that look at “Where have we been?" and "Where are we going?” which address the progression of knowledge. 

• The Teacher Guide provides a detailed Module Overview which includes two sections titled, “How is ____ connected to prior learning?” and “When will students use knowledge from ___ in future learning?” 

  • Module 5 Overview- How is Describing Variability in Quantities connected to prior learning? (614C): “Describing Variability in Quantities builds on students’ informal work with the statistical process and displaying categorical data. Students have displayed one-variable categorical data and quantitative data on dot plots (called line plots in grade 5). MATH REPRESENTATION The dot plot shows data about the number of hours a group of students spends on their phones each day. Students will use this experience to formalize the statistical problem-solving process. They will use their knowledge of symmetry and reflections as they describe the shapes of data distributions. By Module 5, students should be fluent in operating with whole numbers, fractions, and decimals and determining the distance between two numbers on a number line. Students will use their knowledge of absolute value to compute the mean absolute deviation” 

  • Module 5 Overview- When will students use knowledge from Describing Variability in Quantities in future learning? (614D): “This module supports future learning by providing the foundations of the statistical process, data displays, and numeric summaries of data. Students’ understanding of statistical questions and variability will continue to deepen as they work with random sampling and drawing inferences about data. In the next course, students will use graphical displays and summary statistics to compare populations. They will expand their knowledge of numeric summaries of data and analysis techniques in high school as they learn additional mathematics, including square roots and probability distributions.”

• At the beginning of each Topic in a Module, there is a Topic Overview which includes sections entitled “What is the entry point for students?” and “Why is ____ important?” 

  • Module 1, Topic 2- Area, Volume, and Surface Area (63D) - What is the entry point for students?: “ Students enter this topic with a conceptual understanding of area and experience in computing the perimeter and area of rectangles. Area, Volume, and Surface Area draws on this prior knowledge to develop strategies to calculate the area of new shapes based on the area formula for rectangles. Their work with composite areas prepares them to determine the surface area of solids using nets. This topic revisits the concept of volume that students first encountered in grade 5. The experiences students have had with calculating the volume of right rectangular prisms— packing them with unit cubes and using formulas—lays the foundation for calculating the volume of rectangular prisms with fractional side lengths.” 

  • Module 4, Topic 1- Signed Numbers (527D) - Why are Signed Numbers important?: “Just as they reflected the number line to include negative values, students will reflect the first quadrant of a coordinate plane to create the four-quadrant coordinate plane in the next topic. They will explore the coordinate plane in ways similar to their exploration of negative numbers on the number line. Students will also use absolute values to solve problems on a coordinate plane. In Module 5, Describing Variability of Quantities, students will use absolute value in their computation of mean absolute deviation. Students will operate on signed numbers and learn about irrational numbers in grade 8. In high school, they will broaden their knowledge of number systems to include complex numbers.” 

• The Topic Overview also contains a table called, How does a student demonstrate understanding? This is a detailed checklist of what the students should know and be able to do by the end of the topic. 

• Each “Lesson Resource” has mixed practice for the students to utilize with reminders of concepts taught previously.