6th Grade - Gateway 1

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that they assess grade-level content.

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.

For example:

• Module 5, Topic 2, Performance Task: 6.SP.2, 3, 4, 5c, 5d: 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 15 and 16 state, “Rewrite each sum in the form a(b + c) such that the integers b and c have no common factor: 82 + 30”
• 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 instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations for spending a majority of instructional time on major work of the 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 nine out of 13, which is approximately 69 percent.
• The approximate number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 35 out of 50, which is approximately 70 percent.
• The approximate number of days devoted to major work (including supporting work connected to the major work) is 98 out of 139, which is approximately 70.5 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 70.5 percent of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 1 meet expectations that supporting work 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.

For example:

• In Module 1, Topic 3, Lesson 1, Activity 1.2 Volume of Rectangular Prisms: Solving problems involving volume given fractional side lengths (6.G.2) supports the major work of 6.NS.1, dividing fractions.
• In Module 1, Topic 3, Lesson 1, Activity 1.3 Volume Formulas: Solving problems involving volume using volume formulas (6.G.2c) supports the major work of 6.EE.2, evaluating expressions in which letters stand for numbers.
• In Module 1, Topic 3, Lesson 3, Activity 3.1 Nets of Rectangular Prisms: Solving problems involving surface area given fractional side lengths (6.G.4) supports the major work of 6.NS.1, dividing fractions.
• In Module 4, Topic 2, Lesson 2 It’s a Bird, It’s a Plane...It’s a Polygon on the Plane!: 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.
• In Module 2, Topic 2, Lesson 1, Activity 1.3 Many Ways to Measure: Students identify equivalent representations between fractions, decimals, and percents. (6.RP.3c,d and 6.NS.3).

Instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that the amount of content designated for one grade-level is viable for one year. The suggested amount of time and expectations for teachers and students of the materials are viable for one school year as written and would not require significant modifications.

Carnegie Learning provides explicit pacing information in several places:

• The most concise is the Content Map on page FM-15 in the Teacher’s Implementation Guide in both Volumes 1 and 2. There are 139 days of instructional material. This document also provides the information that one day is 50 minutes, facilitator notes offer suggestions for changing the pacing if appropriate, and that allowing 25 assessment days would bring the total to 164 days.
• The Course 1 Standards Overview on pages FM-18 and 19 in the Teacher Implementation Guide provides a chart of all standards covered in each lesson indicating that students would be able to master all grade-level standards within one school year. All of the standards for each grade-level are taught at least once in the curriculum, and most are addressed more than once.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations for the materials being consistent with the progressions in the Standards.

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-15).
• The Family Guide (included in the student book) 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? (M5-1B): “Describing Variability in Quantities builds on the elementary grades’ Measurement and Data standards, specifically students’ informal work with the statistical process and displaying categorical data (e.g., 1.MD.4, 2.MD.10, 3.MD.3, 4.MD.4, 5.MD.2). Prior to grade 6, students should have a basic understanding of displaying one-variable categorical data and of displaying one-variable quantitative data on a dot plot (called a line plot in grade 5). Students will use their knowledge of these graphs to formalize the statistical process and as an introduction to the formal study of statistics.”
• Module 5 Overview- When will students use knowledge from Describing Variability in Quantities in future learning? (M5-1C): “This module supports future learning by providing the foundations of the statistical process, data displays, and numerical summaries of data. Students’ understanding of statistical questions and variability will continue to develop as they work with random sampling and drawing inferences about data (7.SP.A), and students will use graphical displays and summary statistics to compare populations (7.SP.B). In high school, students will expand their knowledge of numerical summaries of data and analysis techniques 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 3- Decimals and Volume (M1-111A) - What is the entry point for students?: “Students began learning about decimals in grades 4 and 5. They have experience in using concrete models and place-value strategies to operate with decimals to the hundredths place. In Decimals and Volume, students formalize the operations with standard algorithms. In grade 5, students learned how to calculate the volume of a right rectangular prism by packing it with unit cubes and using the formulas V = lwh and V = Bh. Decimals and Volume begins by revisiting volume and the terminology associated with solids. From there, students move to determining the volume of right rectangular prisms with fractional edge lengths.”
• Module 4, Topic 1- Signed Numbers (M4-3B) - Why are Signed Numbers important?: “Signed Numbers provides students with a comprehensive view of the number system with which they will primarily operate in the next few years of their mathematical journey. The focus in grade 6 is on understanding and positioning rational numbers. Students will operate on signed numbers beginning in grade 7. The foundation provided in this topic will enable students to develop strategies for operating with signed numbers. To contrast with rational numbers, students will learn about irrational numbers such as pi in grades 7 and 8. As students enter high school, they will broaden their knowledge of number systems to include complex numbers, including imaginary numbers. Developing a formal understanding of nesting number systems will prepare students to study additional number systems.”
• The Topic Overview also contains a table called “Learning Together” that identifies the standards reviewed from previous lessons and grades called “Spaced Review.”
• Each “Lesson Resource” has scaffolded practice for the students to utilize with reminders of concepts taught previously.

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 (M3-193B) 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 3/Activity 3.3 (M1-93B) 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 (M2-123B), “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 (M1-1D): Students spend approximately four days In the MATHia software using the Commutative, Associative, and Distributive Properties to rewrite numeric expressions. Students practice calculating the areas of parallelograms, trapezoids, triangles, and composite figures in mathematical and real-world situations.
• Skills Practice - Module 1, Topic 2 (M1-67E): Students spend approximately two days expressing fraction multiplication and division relationships represented in bar models.

The instructional materials for Carnegie Learning Middle School Math Solution Course 1 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings, including:

6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.

• In Module 2, Topic 3, Lesson 3, the Lesson Overview states, “Students use what they know about unit rates to further develop flexible thinking and problem solving with unit rates in different situations using a variety of representations, including tables and graphs. The lesson begins with students investigating a speedometer as a double number line. Students then reason with unit rates in various mathematical and real-world situations, including measuring the diagonals of a Golden Rhombus and investigating the speed of the Duquesne Incline. Finally, students demonstrate their learning by creating a situation of their own to represent the graph of equivalent rates.”

6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions.

• In Module 3, Topic 1, Lesson 4, the Lesson Overview states, “Students begin by reviewing the properties of arithmetic and algebra that they have formally or informally studied in the past. This allows students to use properties as they rewrite algebraic expressions in equivalent forms. Students analyze pairs of expressions. They use properties, tables, and graphs to show that the expressions are or are not equivalent. Students compare the algebraic expressions and are asked to use tables and graphs to determine if they are equivalent. This opens the discussion that one non-example is necessary to disprove a claim, while an infinite number of examples are necessary to prove a claim.”

6.SP.A Develop understanding of statistical variability.

• In Module 5, Topic 1, Lesson 1, the Lesson Overview states, “Students consider a variety of questions and determine which are statistical and which are not. They learn about the statistical process: formulating a question, collecting data, analyzing data, and interpreting the results. Students organize data into two types, categorical and quantitative. Then, they determine the best method of data collection to answer each question. Students decide if conducting a survey, performing an experiment, or using an observational study is the best method. Students then conduct a survey in their classroom, interpret bar graphs and circle graphs for categorical data, and create a bar graph or circle graph for their survey data. Finally, students interpret the results, stating conclusions they can make from their data displays. In subsequent lessons, students will interpret histograms and line plots for discrete quantitative data and histograms, stem-and-leaf, and box-and-whisker plots for continuous quantitative data.”

6.SP.B Summarize and describe distributions.

• In Module 5, Topic 1, Lesson 3, the Lesson Overview states, “Students analyze a histogram. They discuss intervals and interpret information from the histogram. Students then convert information from the histogram to a grouped frequency table and compare the two representations. The process is reversed, and students create two histograms beginning with two tables of information. For each table, they convert the information to a grouped frequency table and finally to a histogram. At the end of each problem, students summarize the data from the data displays.”

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.

For example:

• In Module 1, Topic 3, Lesson 2, students find the volume of a three-dimensional shape with measurements given in decimals, thus connecting clusters 6.G.1 and 6.NS.B.
• In Module 3, Topic 3, Lesson 4, clusters 6.NS.C and 6.RP.A are connected when students create tables and graph points on a coordinate plane in order to solve real-world problems involving distance and time.
• In Module 4, Topic 1, Lesson 3, standards 6.NS.5, 6.NS.6 and 6.RP.1 are connected when an understanding of proportional relationships is used to sort and order given rational numbers.
• In Module 5, Topic 2, Lesson 4, clusters 6.SP.A and 6.SP.B are connected when students determine whether the mean or median most appropriately represents a typical value in a data set and relate the choice of measures of center and variability to the context.