High School - Gateway 2

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Throughout the series, the instructional materials develop conceptual understanding and provide opportunities for students to independently demonstrate conceptual understanding.

Examples that show the development of conceptual understanding throughout the series include:

• A-SSE.1b: In Math 2, Module 4, Topic 1, Activity 5.2, students interpret how the first term of each root obtained from the quadratic formula is represented graphically. Students then consider why the second term of each root obtained from the quadratic formula is the same except for the sign and how this is represented graphically. At the conclusion of the activity, students generalize their findings by labeling the vertex, axis of symmetry, roots, and distance each root lies from the axis of symmetry for quadratics of the form $$y = ax^2 + bx + c$$ for a > 0 and a < 0 with two real roots and double real roots.
• F-IF.1: In Math 1, Module 1, Topic 1, Lesson 3, students develop their understanding of functions. In Activity 3.1, students analyze relations represented as an ordered pair, written description, graph, table, mapping, and equation to determine whether the relations are functions. In Activity 3.2, students identify the domain and range of functions by writing the domain and range in words or using inequality notation.
• G-CO.7: In Math 1, Module 5, Topic 3, Activity 2.1, students use a worked example to explain why a segment can be mapped onto itself in at most two reflections. In Activity 2.2, students use that criteria to demonstrate two triangles are congruent using the SSS congruence theorem.

Examples that show the materials providing an opportunity for students to independently demonstrate conceptual understanding throughout the series include:

• A-APR.2: In Math 3, Module 2, Topic 1, Activity 2.3, students divide polynomials and observe relationships between factors, divisors, dividends, and remainders to develop their understanding of the Remainder Theorem.
• G-SRT.6: In Math 2, Module 2, Topic 2, Activities 1.1 and 1.2, students use properties of similar right triangles to compare side length ratios for 45-45-90 and 30-60-90 triangles. These activities set the foundation for students to develop the definitions of trigonometric ratios later in the topic.
• S-ID.3: In Math 1, Module 4, Topic 1, Lesson 2 Assignment, students create a data set of 15 numbers where the mean and median are both 59 and the standard deviation is between 10 and 11. Students add an outlier to the data set and explain how the center and spread are affected.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. Opportunities for students to independently demonstrate procedural skills across the series are provided in activities and assignments in the student materials, the online skills practice workbook, and MATHia software. 

Examples that show the development of procedural skills across the series include:

• A-SSE.1b: In Math 2, Module 3, Topic 2, Activities 2.2 and 2.3, students interpret a and b in exponential functions of the form $$f(x) = a(b^x)$$ within the context of population growth and decay. There are several additional practice problems in the Math 2, Module 3, Topic 2 skills practice workbook.
• A-SSE.2: In Math 3, Module 2, Topic 3, students rewrite rational expressions when graphing rational functions to determine asymptotes or discontinuities and when performing operations with rational expressions. Procedural skill practice is included in activities and assignments throughout the topic as well as in the Math 3, Module 2, Topic 3 skills practice workbook and MATHia.
• F-IF.7b: In Math 2, Module 3, Topic 1, Lessons 3 and 4, students graph piecewise-defined functions, including absolute value and step functions. In Math 3, Module 3, Topic 1, Activities 2.1 and 2.2, students graph square root functions. In Math 3, Module 3, Topic 1, Activity 2.3, students graph square root and cube root functions. The skills practice workbooks provide additional opportunities for students to independently demonstrate their procedural skills in graphing these functions.
• G-GPE.4: In Math 1, Module 2, Topic 4, Lesson 1, students use the distance formula and the slope criteria for parallel and perpendicular lines to classify triangles and quadrilaterals on a coordinate plane. In Math 2, Module 4, Topic 3, Lesson 2, students determine whether a given point lies on a circle on the coordinate plane. There are additional problems in the Math 1, Module 2, Topic 4, and Math 2, Module 4, Topic 3 skills practice workbooks.
• G-SRT.5: In Math 2, Module 2, Topic 1, Lesson 5, students solve problems using triangle similarity. Both MATHia and the skills practice workbook provide additional opportunities for students to independently demonstrate their procedural skills in using triangle similarity to solve problems.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. The instructional materials include multiple opportunities to engage students in routine and non-routine applications of mathematics throughout the series. Applications are included in single activities in each course and span several lessons and topics throughout a module. Performance task assessments corresponding to each topic often engage students in an application of mathematics in a real-world context.

Examples where students engage in the application of mathematics throughout the series include:

• N-Q.2: In MATHia, Math 1, Searching for Patterns, Identifying Quantities Workspace, students watch an animation of a skier and decide on appropriate quantities to model the scenario.
• A-CED.1: In Math 1, Module 2, Topic 2, Activity 3.1, students write a linear equation to represent total sales as a function of the number of boxes of popcorn sold for a fundraiser. In Topic 2, Lesson 3 assignment, students write a linear equation to represent the cost of a trip as a function of the number of gallons of gas for the trip. In Topic 2, Lesson 4 assignment, students write an inequality to represent the cost to produce t-shirts in a month making a profit of at least $2,000 but no more than $10,000.
• F-IF.4: In Math 2, Module 3, Topic 1, Activity 3.2, students interpret key features of a graph of a piecewise function that models the percent of charge remaining on a cell phone battery over time. Students write a possible scenario that models the graph and explain what the slope, x-intercepts(s), and y-intercept represent in terms of the problem context. Students also write a scenario to model their own cell phone use during a typical day, graph the scenario, and determine the equation of the piecewise function.
• G-SRT.8: In the Math 2, Module 2, Topic 2, Performance Task, students use trigonometric ratios to compare the height of two drones, determine the distance between the two people controlling the drones, and calculate the angle of elevation from one person to their flying drone.
• S-ID.4: In Math 3, Module 5, Topic 1, Getting Started and Activity 3.1, students recognize that a data set representing the fuel efficiency for a sample of hybrid cars is normally distributed. Students use the Empirical Rule and a table to estimate areas under the normal curve.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the program materials. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding. Each topic includes: activities and assignments that develop students’ conceptual understanding and procedural skills in the student materials, skills practice worksheet that allows students additional practice to develop procedural skills, and performance tasks that assess students’ conceptual understanding and/or procedural skills often times in the context of a real-world scenario. 

Examples of where multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In Math 3, Module 2, Topic 1, Activity 1.2, students use chunking as a method to factor quadratics that have common factors in some of the terms, but not all the terms. Students use their procedural knowledge of chunking to identify and factor perfect square trinomials. In Module 2, Topic 1, Activity 2.2, students use their procedural knowledge of chunking to rewrite the difference of two cubes and build their conceptual understanding when determining the formula for factoring the difference of two cubes. 
• In Math 2, Module 4, Topic 2, Performance Task, students examine a table relating ticket prices and number of tickets sold for a baseball game, and students determine the linear regression equation that models the data. Students add a column to the table for the total amount of money earned from ticket sales and determine the quadratic equation that models the data. Students identify key features of the two graphs, describe these key features in the context of the problem, and explain why the data sets are modeled by different functions.
• In Math 1, Module 1, Topic 2, Performance Task, the materials provide two scenarios (written description and diagram) for how to track the growth of a tree. Students recognize that one scenario represents an arithmetic sequence, whereas the other scenario represents a geometric sequence. Students represent the two sequences using a table, graph, and equation. Students’ conceptual understanding of the differences between arithmetic and geometric sequences and their procedural knowledge of how to write recursive and explicit formulas are assessed in this performance task.
• In Math 3, Module 1, Topic 2, Activity 5.2, the materials provide a context involving cylindrical planters for city sidewalks and storefronts that come in a variety of sizes with specific height and radius requirements. Students generate a cubic function to model the volume $$V(x)=(\pi x^2)(2x)$$ from the base area function $$A(x) = \pi x^2$$ and the height function $$h(x) = 2x$$.

Examples where the instructional materials attend to conceptual understanding and procedural skills independently include:

• In Math 2, Module 3, Topic 2, Activity 1.4, students develop their procedural skills in rewriting expressions involving radicals and rational exponents using the properties of exponents.
• In Math 2, Module 4, Topic 2, Activity 3.3, Talk the Talk, students develop their conceptual understanding of systems of equations when considering the number of possible solutions for a system of equations consisting of a linear equation and a quadratic equation and two quadratic equations.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. MP1 and MP6 are used to enrich the mathematical content and demonstrate the full intent of these mathematical practices across the series.

The materials state that MP1 is “evident every day in every lesson” and is not explicitly identified in either the teacher or student materials. The materials provide students opportunities to explain the meaning of problems, look for entry points when problem solving, plan solution pathways, and make conjectures about the form and meaning of a solution.

Examples of where and how the materials use MP1 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 3, Topic 2, Activity 3.2, students choose an appropriate function to model data of carbon dioxide concentration in the Earth’s atmosphere over time. Students consider what information would help them make a decision as to whether a linear or exponential function is best to model this context and data.
• In Math 2, Module 3, Topic 1, Activity 3.1, students develop a piecewise function from a scenario of pizza sales during the day. Students determine which piece should be used to determine the y-intercept of the function. 
• In Math 3, Module 1, Topic 2, Activity 5.1, students identify what x-intercepts represent and whether their values make sense within the context of a volume formula relating height, length, and width of a planter box.

Materials use a target icon to explicitly identify MP6 in teacher and student materials. Examples of where and how the materials use MP6 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 4, Activity 1.2, students classify triangles on the coordinate plane as acute, right, or obtuse, and scalene, isosceles, or equilateral by calculating distances and slopes.
• In Math 2, Module 1, Topic 1, Activity 1.1, students construct a circle, a diameter of a circle, and perpendicular bisector, and identify radii, arcs, central angles, chords, and secants using definitions of each geometric term.
• In Math 3, Module 1, Topic 1, Activity 5.3, students use the quadratic formula to determine how long it takes for a t-shirt to land on the ground after being launched and consider whether an exact solution or approximate solution is more appropriate for the context.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. MP2 and MP3 are used to enrich the mathematical content, and the materials demonstrate the full intent of these mathematical practices across the series. The materials use a puzzle piece icon to explicitly identify MP2 and MP3 in teacher and student materials. 

Examples of where and how the materials use MP2 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 1, Activity 5.3, students identify the quantities represented in an equation and table and determine which representation is converting Farenheit to Celsius and which representation is converting Celsius to Farenheit. Students compare the slope and y-intercept for each function within the context of the problem.
• In Math 2, Module 3, Topic 1, Activity 5.1, students use a table to show the conversion between the U.S. dollar and the Turkish lira. Students convert Turkish lira to the U.S. dollar and consider how the quantities for this conversion relate to the original conversion as an introduction to inverses.
• In Math 3, Module 3, Topic 3, Activity 5.2, students examine how many social media followers a quarterback and running back for a professional football team have. Students decontextualize the situation in order to think about the functions that represent the situation, and students contextualize the situation in order to interpret both equations to answer follow-up questions.

Examples of where and how the materials use MP3 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 1, Topic 1, Lesson 2, Getting Started, students analyze 19 graphs, sort them into at least two different groups, and provide a rationale for how their groups were created. In Activity 2.1, students consider four different groupings students created and determine the “rule” used to create their groupings, justify why a grouping is correct based on a provided rationale, or justify why a grouping is not correct based on a provided rationale. 
• In Math 2, Module 1, Topic 2, Activity 2.3, students analyze two students’ proof plans and determine which proof plan is correct.
• In Math 3, Module 5, Topic 2, Activity 2.1, students explore different types of biased samples as they consider who is correct in identifying a sampling procedure as a convenience sample or a subjective sample. Students also consider a student response regarding biased samples and explain why the student’s statement is correct based on their knowledge of sampling definitions.

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. MP4 and MP5 are used to enrich the mathematical content, and the materials demonstrate the full intent of these mathematical practices across the series. Materials use a wrench icon to explicitly identify MP4 and MP5 in teacher and student materials. 

Examples of where and how the materials use MP4 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 3, Lesson 5 Assignment, students use information about how much a baker makes when she sells decorated cookies and cupcakes, her minimum profit goal, and the maximum hours she’d like to work to create a system of linear inequalities that model the constraints. Students modify their existing constraints to account for running out of supplies.
• In Math 2, Module 3, Topic 2, Activity 4.1, students use written descriptions of two methods of saving money. Students write a function to model each situation and add the functions. Students graph all three functions and make a connection to what students have previously learned about transformed functions.
• In Math 3, Module 4, Topic 1, Activity 2.1, students model the height of a rider on a Ferris wheel with a periodic function. Students create a graph and a table to represent the height of a rider above the ground as a function of the number of rotations of the Ferris wheel. 

Examples of where and how MP5 is used to enrich the mathematical content and demonstrate the intentional development of the full intent of MP5 across the series include:

• In Math 1, Module 1, Topic 3, Activity 2.2, students use technology to construct a scatter plot, determine the linear regression equation, and compute the correlation coefficient for a data set.
• In Math 2, Module 1, Topic 1, Lesson 4, students use constructions to determine the appropriate place for building an information kiosk at a zoo. Students choose from dynamic software, a compass, or other appropriate tools. Students are not told which tool to use and are expected to choose based on availability and/or appropriateness.
• In Math 3, Module 5, Topic 1, Activity 3.1, students use technology to find z-scores and percentiles for situations that can be modeled by normal distributions.
  

The instructional materials for the Carnegie Learning Math Solutions Integrated series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. MP7 and MP8 are used to enrich the mathematical content, and the materials demonstrate the full intent of these MPs across the series. The materials use a box icon to explicitly identify MP7 and MP8 in teacher and student materials.

Examples of where and how the materials use MP7 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 1, Topic 2, Lesson 2, Getting Started, students consider several sequences on cut out cards, determine the unknown terms of each sequence, and describe the pattern of each sequence. Students use the patterns they observed to group the sequences and provide a rationale as to why they created each group. Students re-group the sequences as arithmetic, geometric, or neither.
• In Math 2, Module 3, Topic 3, Activity 2.1, students use a table to calculate first and second differences for two linear equations and two quadratic equations and graph each equation. Students notice patterns to discern a relationship between the first differences for a linear function and whether the graph is increasing or decreasing as well as a relationship between the second differences for a quadratic function and whether the parabola opens up or down. 
• In Math 3, Module 4, Topic 1, Activity 4.2, students explore patterns in the unit circle coordinates and use their knowledge of unit circle coordinates in the first quadrant and symmetry to label the coordinates in quadrants II, III, and IV.

Examples of where and how the materials use MP8 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• In Math 1, Module 2, Topic 1, Activity 1.4, students express regularity in repeated reasoning to verify that constant differences in arithmetic sequences written explicitly are equivalent to the slope of the arithmetic sequence written in function notation of f(x) = ax + b.
• In Math 2, Module 3, Topic 2, Lesson 2, Talk the Talk, students express regularity in repeated reasoning by differentiating exponential growth and decay when identifying equations that are appropriate exponential models to represent a growing population.
• In Math 3, Module 2, Topic 3, Activity 2.1, students express regularity in repeated reasoning to sketch transformed rational functions based on the general form of transformed functions.