High School - Gateway 2

The instructional materials for the Carnegie Learning High School Math Solution Traditional series meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The instructional materials develop conceptual understanding throughout the series and provide opportunities for students to independently demonstrate conceptual understanding. Examples include, but are not limited to: 

• In Algebra I, Module 5, Topic 1, students determine from an equation whether a function has an absolute maximum or minimum and explain their decision. In answering the questions, students demonstrate an understanding of absolute maximum and minimum and other characteristics of quadratic functions. (A-APR.B) 

• In Algebra I, Module 2, Topic 2, students complete steps to create equivalent equations by exploring equal statements and applying a series of arithmetic steps to both sides. Students work collaboratively to use properties of equality to justify how equivalent equations were created through the solution process. Students solve multi-step equations providing a justification for each step, and when justifications are provided, students complete the steps. (A-REI.A) 

• In Algebra I, Module 2, Topic 1, Lesson 1, students explore the concept of arithmetic sequences and build them into linear functions. In Module 3, Topic 1, Lesson 1, students explore geometric ratios and graph the terms of the geometric sequences before working with exponential growth. Students write explicit geometric formulas and exponential functions from the common ratios. In Module 3, Topic 2, Lesson 1, students compare the average rate of change between common intervals of a linear and an exponential relationship in contextual problems while justifying their thinking and processes. (F-LE.1) 

• In Geometry, Module 3, Topic 1, in addition to calculating ratios and angle measures to determine similar figures, students answer a series of questions to develop conceptual understanding. Students answer questions to explain or justify their answers using measurements or transformations, for example: “Explain why this similarity theorem is Angle-Angle instead of Angle-Angle-Angle.” In answering the questions, students demonstrate an understanding of similarity and the characteristics that make two figures similar. (G-SRT.2) 

• In Geometry Module 3, Topic 2, Lesson 1, students explore trigonometric ratios as measurement conversions and analyze the properties of similar right triangles. Starting with two parallel lines, students pick a point on one line and draw a line to another line and create two triangles. Students verify the two triangles are similar by measuring all sides and comparing the ratio of the lengths of the corresponding sides. Students use the triangles to find the ratio of the lengths of the sides that later are defined as sine, cosine, and tangent. Students use the ratios throughout the lesson to develop an understanding of the ratios before the formal definitions are given at the end of the lesson. (G-SRT.6) 

• In Algebra II, Module 3, Topic 1, Lesson 4, students review exponent rules and explore tables and graphs for power functions with integer exponents. Students write conclusions based on this exploration and use their conclusions with rational exponents and radical expressions. Students analyze other student work with rational exponents by comparing, contrasting, and justifying different approaches to rewriting radical expressions. (N-RN.1)

The instructional materials for the Carnegie Learning High School Math Solution Traditional series meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills and provide opportunities to independently demonstrate procedural skills throughout the series. The series includes practice problems in the lessons, MATHia software, and the Skills Practice workbook. 

Examples include, but are not limited to: 

• In Algebra II, Module 1, Topic 1, Lesson 2, students develop procedural skills with algebraic and geometric patterns. Students compare models and calculations and verify the equivalence of expressions. This topic has additional MATHia workspaces and Skills Practice on patterns, algebraically and graphically, modeled by linear, exponential, and quadratic equations. Students explore and analyze patterns, compose and decompose functions, sketch 3rd and 4th degree polynomials, and explore average rate of change. (A-SSE.1b) 

• In Algebra I, Module 5, Topic 2, Lesson 2, students rewrite a quadratic function of the form f(x)=ax^2-c as the product of two linear factors and use them to determine zeros. This topic has additional MATHia workspaces and Skills Practice on solving quadratic equations to determine the zeros and rewriting expressions using difference of squares. (A-SSE.2) 

• In Algebra II, Modules 1-4, students practice transformations with a variety of parent functions. MATHia workspaces offer additional practice with inverse functions, represented graphically and numerically, writing inverses of quadratic functions as square root functions, identifying transformations of quadratic functions, and exponential functions, both growth and decay. The Skills Practice includes additional problems on transformations, rewriting and solving radical and exponential functions, and properties of exponential graphs. (F-BF.3) 

• In Geometry, Module 1, Topic 1, Lesson 5, students use area and perimeter in a coordinate plane and the distance formula as they work with composite shapes. Students demonstrate procedural skills by finding the perimeter and area of polygons. (G-GPE.7)

The instructional materials for the Carnegie Learning High School Math Solution Traditional 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 for students to engage in routine and non-routine application of mathematics while providing opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts throughout the series. 

Examples where students engage in routine and non-routine application of mathematics include, but are not limited to: 

• In Algebra I, Module 1, Topic1, Lesson 1, students read scenarios and determine the independent and dependent quantities. Students then match each scenario to its corresponding graph. For each graph, students label the axes with the appropriate quantity and a reasonable scale, and then interpret the meaning of the origin. Students draw conclusions from the scenarios. (N-Q.A) 

• In Algebra II, Module 1, Topic 2, Lesson 1, students are given a scenario where a drain is built. Students make physical models from paper, compare with classmates, then complete a chart of possible height/width combinations for a certain size of sheet metal to determine the dimensions that produce the most water flow. Next, they write a function to model a cross sectional area and use technology to graph the function. Students interpret points on the graph and describe what that point represents. Then students work on a new scenario with larger dimensions of sheet metal. (A-REI.11) 

• In Geometry, Module 3, Topic 2, Lesson 4, Activity 2,  students engage in applications related to the cosine ratio. Questions 5 and 6 represent traditional inverse trigonometric problems where students are asked to find angles. However, these questions are contextualized in a way that is appropriate for high school courses. Question 7 represents a less traditional inverse trigonometric problem. This problem requires students to evaluate the given information and solve for more than one triangle in order to arrive at the appropriate answer. (G-SRT.8)

The instructional materials for the Carnegie Learning High School Math Solution Traditional 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 and multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. 

Examples of where the instructional materials attend to conceptual understanding, procedural skills, and application independently throughout the grade level include: 

• Conceptual Understanding: In MATHia, students must see the connections between algebraic representations of functions, graphical representations of different types of functions, and determine from a context the type of function that is being described. In Algebra I, students are exposed to different types of functions (e.g., cubic, polynomial) but asked to categorize them as “Other.” Students must understand the relationships between contextual situations, algebraic representations, and graphs. 

• Procedural Skills: In MATHia Algebra II, opportunities exist for students to simplify radicals with negative radicands, simplify powers of i, adding and subtracting complex numbers, multiplying complex numbers, and solving quadratic equations with complex numbers. 

• Application: In Algebra I, Module 5, Topic 1, Lesson 1, students write quadratic functions to model contexts. Specifically, Activity 2 includes a handshake activity where students must record and organize data and write a function that models that data. Students are then asked application questions related to minimums, domain, and range. They are also asked to compare the orientation of the graph to a previous problem in the lesson. 

Examples where two or more of the aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials include: 

• All sections related to G.SRT.6 represent a balance between solving simple proportions related to trigonometric ratios in right triangles, contextualized word problems involving different trigonometric ratios, and conceptual understanding related to the different trigonometric functions. For example, in Geometry, Module 3, Topic 2, Lesson 2, Activity 3, students are asked to extend their knowledge of the tangent ratio and apply it to different similar triangles and more abstract representations of angle measures. 

• In Algebra I, Module 2, Topic 1, End of Topic Test A, students demonstrate all aspects of rigor by rewriting equations in different forms and transforming figures on a coordinate grid. Students are required to recall arithmetic and linear functions and use knowledge of both to solve problems and answer questions. Students also apply knowledge of linear equations by reading a scenario and answering a series of questions involving the function. 

• In Algebra II, Module 3, Topics 1-4, radical, exponential, and logarithmic functions are developed, practiced, and utilized. In Topic 1, students develop conceptual understanding of a radical being the inverse of a power and develop procedural skills with practice in simplifying, rewriting, and graphing radical expressions and functions. Application occurs in real world scenarios. In Topic 2, students algebraically and graphically analyze and transform exponential and logarithmic functions expanding upon the concepts from Topic 1. In Topic 3, students use these skills to solve and develop procedural skills with the properties of logarithms. In Topic 4, students model with exponential functions and solve real-world scenarios of growth.

The instructional materials for the Carnegie Learning High School Math Solution Traditional series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. The majority of the time MP1 and MP6 are used to enrich the mathematical content, and there is intentional development of MP1 and MP6 across the series.

Standards for Mathematical Practice are referred to as Habits of Mind in this program. The Habits of Mind are first introduced for teachers and students in the Front Matter of the MATHbook and Teacher’s Implementation Guide. For each practice or pair of practices, students are provided a list of questions they should ask themselves as they work toward developing the habits of mind of a productive mathematical thinker throughout the series. Each activity within MATHbook explicitly denotes the practice or pair of practices intentionally being developed using a box labeled “Habits of Mind,” with the exception of MP1. Materials state that MP1 aligns to all lessons in the Front Matter of the MATHbook and Teacher’s Implementation Guide. MPs are identified for activities, but not for specific problems or exercises. 

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 Algebra I, Module 2, Topic 3, Lesson 5, Activity 3, students are provided information on two different job offers for a sales position. Students write and solve a system of equations to make a recommendation on which job offers better compensation. 

• In Geometry, Module 1, Topic 1, Lesson 1, students determine if the size of three adjacent and congruent squares will affect the sum of the measures of three labeled angles and use a protractor to test their predictions. Students copy each of the angles on a piece of patty paper and determine how to manipulate the three angles to show that their sum is 90 degrees. The materials state how there are different methods to verify sums. 

• In Algebra II, Module 2, Topic 2, Lesson 2, students explore the Binomial Theorem and Pascal’s Triangle. They complete a series of exercises that help them in future lessons related to the Binomial Theorem. Students make sense of the patterns in the triangle and answer questions related to the relationships between the numbers. 

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 Algebra I, Module 1, Topic 1, Lesson 3, Activities 1 and 2, students attend to precision as they compare functions and non functions as well as identify domain and range for the functions. Students use the precise definition of a function, and in Activity 2, they use appropriate notation involving inequalities to describe domain and range. 

• In Geometry, Module 3, Topic 2, Lesson 3, Activity 4, students use the appropriate trigonometric ratios and appropriately use a calculator to find the most precise angle measures for problems both in and out of contexts. 

• In Algebra II, Module 1, Topic 3, Lesson 2, Activity 2, students apply the mathematical principles related to rigid motions and accurately find points on a transformed graph. Students understand precise vocabulary and the operations required to find the new points.

The instructional materials for the Carnegie Learning High School Math Solution Traditional series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards. The majority of the time MP2 and MP3 are used to enrich the mathematical content, and there is intentional development of MP2 and MP3 across the series. 

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 Algebra I, Module 4, Topic 1, Lesson 3, Activity 1, students compare data and use the numerical evidence to reason and determine answers to additional questions. For example, a teacher is asked to use data to support the choice for who should compete in a spelling bee based on test scores, and the students must reason quantitatively to reach a reasonable conclusion. 

• In Geometry, Module 1, Topic 2, Lesson 2, students write equations that represent different translations and discuss the similarities and differences between geometric translation functions and algebraic equations which show the translations. 

• In Algebra II, Module 2, Topic 3, Lesson 6, students solve contextualized problems related to Work, Mixture, Distance, and Cost. Students write equations to solve specific problems and then answer specific questions. By having to generate the equations to solve, students reason abstractly and quantitatively. 

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 Algebra I, Module 3, Topic 2, Lesson 1, Activity 1, students critique Chloe’s reasoning about a statement comparing output values of linear and exponential functions. Students use examples to justify their thinking and determine whether Chloe is correct or incorrect.

• In Geometry, Module 2, Topic 1, Lesson 4, Activities 1 and 2, students critique the conjectures of other students. Students also use mathematical strategies to determine whether or not the conjectures are correct. 

• In Algebra II, Module 5, Topic 1, Lesson 3, Activity 1, students explain why DMitrius’ reasoning is incorrect when he estimated the percent of hybrid cars that get less than 57 miles per gallon using the Empirical Rule. The Teacher’s Implementation Guide includes the following questions to support discourse and encourage students to demonstrate the full intent of the mathematical practice: “How can you use the area under the normal curve to show DMitrius the error in his thinking?” “Is Dmitrius’s estimate too high or too low? Explain.”

The instructional materials for the Carnegie Learning High School Math Solution Traditional 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. The majority of the time MP4 and MP5 are used to enrich the mathematical content, and there is intentional development of MP4 and MP5 across the series. 

Examples of where and how MP4 is used to enrich the mathematical content and demonstrate the intentional development of the full intent of MP4 across the series include: 

• In Algebra I, Module 2, Topic 3, Lesson 3, students write systems of inequalities (with constraints), graph the system, and interpret results based on a contextualized problem. 

• In Geometry, Module 4, Topic 1, Lesson 3, students use what they know to describe the volume of two figures. Students use the Cavalieri’s Principle to draw conclusions about the volumes of two prisms. 

• In Algebra II, Module 3, Topic 2, Lesson 1, Activity 3, students complete a table to determine the amount of caffeine at each time interval. In Question 11, students complete a second table for the half life. Students use both tables to answer questions based on the scenario given at the beginning. 

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 Algebra I, Module 2, Topic 3, Lesson 1, Activity 2, students choose and use their own method, along with technology, to solve systems of equations. Students also use graphing tools to verify algebraic solutions to systems of equations. 

• In Geometry, Module 1, Topic 2, Lesson 3, students perform constructions. 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 Algebra II, Module 2, Topic 1, Lesson 4, Activities 1 and 2, students use graphing technology to connect algebraic solutions of polynomial inequalities to the graphs of the polynomial functions. The graphs created by the graphing technology also relate to the number line/interval representation of the solutions for the polynomial inequality.

The instructional materials for the Carnegie Learning High School Math Solution Traditional series meet expectations for supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards. The majority of the time MP7 and MP8 are used to enrich the mathematical content, and there is intentional development of MP7 and MP8 across the series. 

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 Algebra I, Module 1, Topic 2, Lesson 3, Activity 1, students use patterns revealed in a data set to write recursive and explicit formulas for arithmetic sequences. In Activity 2, students repeat this type of activity and write both explicit and recursive formulas and examine the connection between the two for geometric sequences. 

• In Geometry, Module 2, Topic 2, Lesson 3, Activity 2, students use triangles having interior angles that sum to 180 degrees to determine the sum of the interior angles of a polygon. Students examine patterns and make conjectures based on the idea of the sum of the interior angles of a triangle. 

• In Algebra II, Module 2 Topic 1, Lesson 2, Activity 2, students factor special binomials building on their understanding of factoring difference of squares. Students also factor the sum and difference of cubes. By examining long division examples, students determine the structure for factoring the sum and difference of cubes. 

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 Algebra I, Module 1, Topic 1, Lesson 4, Activity 1, students use repeated reasoning from scenarios to identify domain characteristics of the function families (linear, exponential, quadratic, absolute value) and compare the graphical behaviors within a family. 

• In Geometry, Module 1, Topic 1, Lesson 3, Activity 3, students use prior knowledge of parallel lines, perpendicular lines, and slopes to answer questions about horizontal and vertical lines. 

• In Algebra II, Module 3, Topic 4, Lesson 1, Activity 1, students express regularity in repeated reasoning to determine a rule for the sum of a geometric sequences.