7th Grade - Gateway 1

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 Test, and the Topic Level Performance Task.

For example:

• Module 1, Topic 3, End of Topic Test-Form B, 7.RP.2b,2c: Students represent a proportional relationship in an equation. Question 6 states, “The number of trees (n) that the average American uses in paper products varies directly with the time (t) in years. Assume that the constant of proportionality is 7. a. Write an equation to represent the proportional relationship between n and t using the information given. b. How many trees does the average American use in 6 years? c. In how many years would an average American use 133 trees?“
• Module 3, Topic 1, End of Topic Post-test Form B, 7.EE.3: Students solve multi-step equations. Question 2 states, “A theater charges a service fee of $4.50 plus a ticket fee based on the section of the theater.” A table is provided with only part of the information given. Students fill in the missing information; “Write an algebraic expression to represent the cost of x number of Orchestra tickets; Can the same algebraic expression be used for tickets in the Mezzanine and tickets in the Second Balcony? Explain your reasoning. If a group buys 4 Mezzanine tickets and 2 Orchestra tickets, what will be the total cost of the tickets? Explain your reasoning.”
• Module 4, Topic 1, Standardized Test, 7.SP.5: Students express the likelihood of a random event. Question 3 states, “Ilana drew a marble at random from a bag containing 4 blue, 3 red, 2 yellow, and 5 green marbles. What is the probability that she picked a marble that is not red?”
• Module 3, Topic 2, End of Topic Test Form A, 7.EE.4a: Students solve a word problem in the form px+q=r. Question 11 states, “Rachel ordered early learning software for her daughter from an online retailer. Each item costs $6.50, and there is a shipping fee of $8.50 for the entire order. a. What is the cost of Rachel’s order if she buys 7 items? b. Write a sentence to describe how you calculated the cost of Rachel’s order. c. Write an equation to describe this situation. Let s represent the number of software items ordered and c represent the total cost of the order.”
• Module 5, Performance Task, 7.G.5: Students use facts about angles to write multi-step problems to solve for unknown angle measures in a figure. In X Marks the Spot, given pairs of intersecting lines, students reason to “Explain how you could determine the measure of each of the marked angles made by the X; Calculate the measures of the marked angles; Show two different ways to determine the unknown values; Write equations and determine the measures of all four angles.”

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 assessments and supporting work connected to the major work) is eight out of 14, which is approximately 57 percent.
• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 31 out of 56, which is approximately 55 percent.
• The number of days devoted to major work (including assessments and supporting work connected to the major work) is 95 out of 139, which is approximately 68 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 68 percent of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Carnegie Learning Middle School Math Solution Course 2 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 1, Lesson 2, Activity 2.3: That’s a Spicy Pizza-Unit Rates and Circle Areas: Students use their knowledge of circles (7.G.4) and unit rates (7.RP.1) to determine which pizza is the better buy.
• In Module 1, Topic 4, Lesson 4, Activity 4.4: More Ups and Downs: Students solve problems involving volume and surface area (7.G.6) to apply percent increase and decrease concepts (7.RP.3).
• In Module 4, Topic 3, Lesson 4, students use random samples from two populations to draw conclusions (7.SP.1-4) and solve real-world problems involving the four operations with rational numbers (7.NS.3). Students create graphic displays to answer questions regarding means, medians, ranges, mean absolute deviation, and interquartile ranges.
• In Module 5, Topic 1, Lesson 2, students investigate special angle relationships, including complementary and supplementary angles (7.G.5) and solve real-world problems involving the four operations with rational numbers (7.NS.3). Practice question 5 states, “Suppose each street in the map shown represents a line. Provide an example of each angle relationship. a. Vertical angles, b. Supplementary angles, c. Linear pair, d. Adjacent angles, e. Vertical angles, f. Congruent angles. Calculate the measure of each unknown angle.” Questions 6-10 continue the task: “6) Angles C and D are complementary. The measure of angle D is 25 degrees greater than the measure of angle C. What is the measure of each angle? 7) If the supplement of an angle is 30 degrees more than the measure of the angle, what is the measure of the angle? 8) If the supplement of an angle is 12 degrees less than twice the measure of the angle, what is the measure of the angle? 9) If two angles form a linear pair and the measure of the first angle is one-fifth the measure of the second angle, what is the measure of each angle? 10) If two angles form a linear pair and the measure of the first angle is three times the measure of the second angle, what is the measure of each angle?”

Instructional materials for Carnegie Learning Middle School Math Solution Course 2 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 2 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 2 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 2 Overview- How is Operating with Signed Numbers connected to prior learning? (M2-1B): “Operating with Signed Numbers builds on students’ experiences with signed numbers in grade 6. Students revisit distance on a number line in terms of magnitude, building on their prior knowledge of absolute value, to model operations on the number line and to connect the model to an algorithm. The module also draws on students’ fluency with adding, subtracting, multiplying, and dividing whole numbers, decimals, and fractions, developed in grades K through 6. Students are expected to be fluent in these operations when operating with signed rational numbers. Finally, Operating with Signed Numbers uses properties of numbers that were informally developed in elementary school and formalized in grade 6.”
• Module 3 Overview- When will students use knowledge from Reasoning Algebraically in future learning? (M3-1B): “The underlying purpose of representing expressions on a number line and solving equations using a double number line—physically transforming expressions or equations and noting the corresponding changes in the symbolic representation—is to provide the underpinnings for geometric and algebraic transformations of objects and equations. In grade 8 and in high school, students will transform geometric objects and make conjectures about how the coordinates of geometric figures on the plane change after undergoing the transformation.”
• 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 1- Circles and Ratio (M1-3A) - What is the entry point for students?: “Throughout elementary school, students used and labeled circles and determined the perimeters of shapes formed with straight lines. In grade 6, students worked extensively with ratio and ratio reasoning. To begin Circles and Ratio, students draw on these experiences as they use physical tools to investigate a constant ratio, pi. They form ratios of the distance around circles to the distance across circles. As they engage in this investigation, students review basic ideas of ratios and proportional relationships."
• Module 1, Topic 3- Proportionality (M1-87B) - Why is Proportionality important?: “The characteristics of proportional relationships, their graphs, and their equations as developed in Proportionality provide the underpinnings of algebra and the study of functions. In grade 8, students are expected to “understand the connections between proportional relationships, lines, and linear equations” (8.EE.B) and “define, evaluate, and compare functions” (8.F.A), including nonlinear functions. In each domain, students are expected to compare relationships represented in different ways.”
• 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 activities 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.” An example of this is Module 1, Topic 4, Lesson 5 (M1-223), where students are given a triangle drawn on triangular pattern block paper and asked to draw a new triangle with side lengths 50% the length of the original. They determine the ratios that describe the relationship between the original and the new triangle in terms of side lengths, perimeters, and areas. (7.G.1)
• 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 2, Topic 2, Lesson 2, Activity 2.2 (M2-107) has students sort signed rational numbers into sets of equivalent rational numbers. The focus on this activity is understanding that if the quotient of two integers is negative, the negative sign can be placed in front of the representative fraction, in the numerator of the fraction, or in the denominator of the fraction. (7.NS.2b,d)
• In the Demonstrate section, students “reflect on and evaluate what was learned.” An example of this is Module 3, Topic 1, Lesson 1 (M3-7), where “students describe a strategy for evaluating an algebraic expression.” (7.EE.1-3)

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 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 4, Topic 3 (M4-3E): Students spend approximately one day in MATHia software comparing the characteristics of data displays, specifying which numerical characteristics can be determined from each display, then using data displays to compare populations by determining the visual overlap and describing the difference between the measures of centers in terms of measures of variability.
• Skills Practice - Module 5, Topic 1 (M5-3E): Students spend approximately one day classifying angles as complementary, supplementary, or vertical.

The instructional materials for Carnegie Learning Middle School Math Solution Course 2 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:

7. EE.A Use properties of operations to generate equivalent expressions.

• In Module 3, Topic 1, Lesson 2, the Lesson Overview states, “Students rewrite linear expressions using the Distributive Property. First, they plot related algebraic expressions on a number line by reasoning about magnitude. Students realize that rewriting the expressions reveals structural similarities in the expressions, which allows them to more accurately plot the expressions. They then review the Distributive Property. Students expand algebraic expressions using both the area model and symbolic representations, focusing on the symbolic. They then reverse the process to factor linear expressions. Students factor expressions by factoring out the greatest common factor and by factoring out the coefficient of the linear variable. Finally, students rewrite expressions in multiple ways by factoring the same value from each term of the expression.”

7.NS.A Apply and extend previous understandings of operations with fractions.

• In Module 2, Topic 1, Lesson 4, the Lesson Overview states, “Number lines and two-color counters are used to model subtraction of signed numbers. Through a series of activities, students will develop rules for subtracting integers. As in the lesson on adding signed numbers, the number line method is used to model the difference between two integers. Students then learn how to use zero pairs when performing subtraction using the two-color counter method. Students analyze real-world situations that require calculating the distance between two signed numbers. They build on what they already know about absolute value to determine the distance.”

7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems.

• In Module 1, Topic 3, Lesson 1, the Lesson Overview states, “Students explore tables and graphs that illustrate proportional relationships. First, students review equivalent ratios and that the graphs of equivalent ratios form straight lines that pass through the origin. They are then given three sets of scenarios, equations, and graphs to match, using any strategy. Each group illustrates a different type of relationship: linear and proportional, linear and non-proportional, non-linear. Students classify the groups of representations as linear and non-linear and use tables of values to classify the linear relationships as proportional or as non-proportional. They summarize the relationships between the terms linear relationship, proportional relationship, and equivalent ratios.”

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 4, Lesson 5, clusters 7.G.A and 7.RP.A are connected when students analyze scale drawings and determine which scale will produce the largest and smallest drawing of an object when different units of measure are given.
• In Module 2, Topic 2, Lesson 3, clusters 7.NS.A and 7.RP.A are connected when students solve multi-step real-world ratio and percent problems involving simplifying numeric expressions using the four operations and signed rational numbers.
• In Module 3, Topic 3, Lesson 3, clusters 7.NS.A and 7.EE.B are connected when students use rational numbers to write and analyze equations and inequalities.
• In Module 4, Topic 1, Lesson 1, clusters 7.SP.B and 7.NS.A are connected when students utilize knowledge of rational numbers to represent probability as a value between zero and one.