High School - Gateway 2

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. Overall, the instructional materials develop conceptual understanding throughout the series as well as provide opportunities for students to demonstrate conceptual understanding independently throughout the series.

Examples that develop conceptual understanding and present students with opportunities to independently demonstrate conceptual understanding include:

• N-RN.1: In Mathematics I, Lesson 5-1, there is an explanation of why a power of $$\frac{1}{2}$$ must be equivalent to the square root of the number. In Mathematics III, Lesson 4-1, a similar explanation is provided for a fourth root and for a rational exponent of $$\frac{2}{3}$$. Then, in Problems 4, 6, 20, and 21, students explain problems aligned to this standard. In Lesson 5-1, Problems 20, 22, and 23, students explain concepts related to this standard.
• A-APR.B: In Mathematics III, Lesson 2-4, Example 3 provides an explanation of why the Remainder Theorem is true (A-APR.2). In exercises in the lesson, students explain their reasoning involving the Factor Theorem. Students further examine the relationship between factors and zeros in Mathematics II, Lesson 4-2. Students factor quadratic expressions to find the solutions of quadratic equations, and students further develop their understanding of the relationship between zeros and factors by finding factors when zeros of a quadratic function are given. In Mathematics III, Lesson 2-5, students extend their understanding of the relationship between factors and zeros to higher order polynomials.
• A-REI.A: In Mathematics I, Lesson 1-2, students create and solve simple linear equations. Students evaluate various methods for solving linear equations, including using Algebra Tiles, and determine which operations are needed to solve a variety of problems. Further, in Lesson 5-1, students solve equations with exponents and radicals. Students look for relationships between square roots and squaring and solve algebraically and graphically.
• G-SRT.2: In Mathematics II, Lesson 9-2, students examine examples of student work. The questions provided in the teacher edition promote reasoning and problem solving, such as “What is preserved with different types of transformations?” and “How might you use side length to help you determine whether there is a composition of transformations that maps one figure to the other?” Through these questions, students develop an understanding of a similarity transformation, which is the essential question for the lesson. Also, in Lesson 9-2, there are questions for the teacher to help students develop an understanding of the connection between congruence and similarity.
• S-ID.7: In Mathematics I, Topic 2, students understand that linear equations can be written in three forms. Students also develop an understanding that choosing a form for writing a linear equation depends on given information, and equivalent forms can be obtained using the properties of equality. Students further their understanding of linear equations by interpreting the meaning of the slope and y-intercept of each form used in the context of the problem posed.

One standard for which the materials do not fully develop conceptual understanding is:

• G-SRT.6: In Mathematics II, Lesson 9-7, Example 1, students answer, “How are the sine of two different angles with the same measure related?” After showing students the trigonometric ratios, students explore the sine ratios of two similar right triangles. The materials provide students with the definitions of trigonometric ratios, and the students’ independent practice develops skill with using the trigonometric ratios.

The instructional materials reviewed for enVision Integrated Mathematics 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 while also providing opportunities for students to independently demonstrate procedural skills throughout the series.

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

• A-SSE.2: Throughout the series, students build procedural skills in rewriting expressions of linear and exponential (Mathematics I), quadratic (Mathematics II), and additional (Mathematics III) types.
• A-APR.1: In Mathematics II, Lessons 2-1 and 2-2, students develop and practice procedural skills in adding, subtracting, and multiplying polynomials. They further develop these skills in the remainder of the series. For example, in Mathematics III, they add and subtract rational expressions.
•  A-APR.6: In Mathematics III, Lesson 2-4, students divide polynomials using long and synthetic division, and write answers as quotient plus remainder divided by divisor. Students practice division without context in Do You Know How, Problems 4 and 5, and Practice and Problem-Solving, Problems 15 through 22. Additionally, students analyze an error in polynomial long division in Problem 10, and students practice polynomial long division in a context in Problems 32 through 34.
• F-BF.3: Throughout the series, students build procedural skills in identifying transformed functions and transforming functions given an equation. Students work with linear and exponential functions in Mathematics I, quadratic functions in Mathematics II, and logarithmic, rational, radical, and trigonometric functions in Mathematics III.
• G-GPE.4: In Mathematics II, Lessons 11-1 and 11-2, students plan a coordinate geometry proof, prove theorems using coordinate geometry, derive the equation for a circle in the coordinate plane, and write equations for, and graph, circles.
• G-SRT.5: In Mathematics I, Lessons 9-3, 9-4, and 9-5, students develop procedural skills through determining if triangles are congruent, and students also develop procedural skills while determining congruent relationships between parts of triangles and other polygons. In Mathematics II, Lessons 8-3 and 8-4, students practice with properties of parallelograms and other quadrilaterals.

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting 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. Throughout the series, students engage in a variety of application-based problems through STEM projects for each topic, Apply in Practice & Problem Solving, Performance Tasks in Assessment practice, and Mathematical Modeling in 3 Acts embedded into topics.

Examples that demonstrate multiple opportunities for students to engage in routine and non-routine application of mathematics include:

• A-REI.2: In Mathematics III, Topic 3, Mathematical Modeling in 3 Acts, students use rational functions to compare scenarios in which a pool is being filled by one of two hoses or the two hoses combined.
• F-IF.4: In Mathematics I, Lesson 5-2, Practice & Problem-Solving, Problem 27, students are provided a scenario and a graph, and students formulate their own exponential function, compute various values for the function, and interpret the results to answer the given question. 
• G-SRT.8: In Mathematics II, Lessons 9-6 and 9-7, students use the Pythagorean Theorem and trigonometric ratios to solve many different types of problems. In each lesson, there is one application problem intended to be solved by groups of students and several application problems intended to be solved by individual students. The Topic 9 performance assessments provide additional opportunities for students to solve problems using both the Pythagorean Theorem and trigonometric ratios.
• S-ID.6a: In Mathematics II, Lesson 3-4, Practice & Problem-Solving, Problem 25, students are provided data for prices and profits of a company. Students determine if they agree or disagree with a statement about maximum profit and justify their response. Students create a scatter plot, formulate a quadratic expression to match the data, compute the maximum value, and interpret that result compared to the price suggested to validate their response.

Examples that include opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts include:

• A-SSE.3: In Mathematics I, Lesson 5-3, students work with exponential functions to model plant growth, population growth, and investments. In Topic 5, Mathematical Modeling in 3 Acts, students solve problems related to investing money.
• A-REI.11: In Mathematics II, Topic 9, students solve systems of linear and quadratic equations using different methods. Contexts for students to explain why the intersections of the two functions are solutions include: comparing cell phone sales, costs of ropes course facilities, and the number of individuals who prefer rock climbing to zip-lining.

The instructional materials reviewed for enVision Integrated Mathematics 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 of a single topic/unit of study throughout the materials. Each topic in this series includes a Topic Opener, STEM Project, and Mathematical Modeling in 3 Acts (which relates to the Topic Opener). Each lesson includes: Explore and Reason, Understand and Apply (which guides students through examples and problems to try on their own), Concept Summary, Practice and Problem Solving, and a formative quiz to determine understanding and mastery. This structure of the materials lends itself to balancing the three aspects of rigor.

The following are examples of balancing the three aspects of rigor in the instructional materials:

• Mathematics I, Topic 3 addresses Linear Functions. In the STEM Project, students explore how recycling can offset carbon dioxide production. Students use linear functions to determine recycling rates by planning a recycling drive at their school to increase the amount of trash that gets recycled. In the first three lessons, students develop procedural skill in using function notation, evaluating functions, graphing the lines described by functions, and graphing translations through applications, tables and graphs. In the Modeling in 3 Acts problem, students find a strategy for picking a checkout lane in the grocery store. In lesson 3-6, students apply linear functions through representing arithmetic sequences, determining a linear function from a scatter plot, and analyzing trend lines. Students use linear functions to solve real-world problems, such as the time to download a given file size and the number of hybrid cars sold in America over 16 years.
• Mathematics II, Topic 9 addresses similarity transformations, similar triangles, and proportional relationships in triangles. In the STEM Project, students use similarity to find the dimensions of an engine part while given the dimensions of a model of the part. Students calculate key values related to the 3D printing of the part and describe steps for its production. Students extend their conceptual understanding of transformations to include dilations and develop the understanding that two figures are similar if a similarity transformation occurs. Students develop skill in identifying a series of transformations used in mappings, and application problems (such as comparing blueprints to actual measurements, working with a surveying device to determine the distance from the lens to the target, and constructing supports for a roof) integrate all aspects of rigor. In the Modeling in 3 Acts problem, students make scale models of a building project.
• Mathematics II, Topics 3 through 5, address extending understanding of quadratic functions. In the STEM Project, Topic 5, students explore how the design of a ballpark influences the number and frequency of home runs. Throughout the three topics, students develop the conceptual understanding that all quadratic functions are transformations of the parent function. Students develop procedural skill in factoring quadratic expressions and solving quadratic equations through factoring. There are many opportunities to apply the understanding of quadratic equations to real life as seen in the Topic 5 3 Acts where students develop a conjecture to model kicking a soccer ball into a goal. Students also interpret key features of the graph of a quadratic function in terms of the context, which includes describing projectile motion, maximizing space of a rectangular patio, and determining maximum profits for a bike manufacturer.

There are two standards where multiple aspects of rigor within a standard are not balanced in the materials, and those standards are:

• A-APR.1: In Mathematics II, Topic 2, students have limited opportunities to develop understanding that polynomials form a system analogous to the integers that is closed under addition, subtraction, and multiplication, but they do develop their skills with adding, subtracting, and multiplying polynomials. The Topic also shows the classification of polynomials and how to write them in descending order of degree, but this is not aligned to the standard.
• G-C.5: In Mathematics II, Lesson 12-1, students do not independently derive the relationship between arc length and the angle that intercepts the arc, but students are shown examples relating arc length to the radius of the circle and the angle that intercepts the arc. Students also complete calculations to find the area of sectors.

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards. Overall, MP1 and MP6 are used to enrich the mathematical content and demonstrate the full intent of these mathematical practices across the series. The mathematical practices are identified in both the teacher and student editions.

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

• In Mathematics I, Lesson 5-3, students construct exponential growth and decay functions given a description of a relationship.
• In Mathematics I, Lessons 7-1 and 7-2, students explain the relationship between angles formed by the intersection of two parallel lines and a transversal.
• In Mathematics II, Lesson 1-1, students find entry points to problems involving exponential relationships by identifying given information and goals. Students write exponential growth or decay equations in a form determined by the content of the problem in order to find solutions.
• In Mathematics II, Lesson 11-1, students plan how to use the formulas for slope, distance, and midpoint to determine properties of figures.
• In Mathematics III, Lesson 3-1, students look for an entry point to a problem when they use what they know about inverse variation to mentally compute an approximate answer to a problem that uses an inverse variation model to find the frequency of a guitar string.
• In Mathematics III, Lesson 4-1, students plan solution pathways through problems when they use nth roots to solve equations involving exponents.

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 Mathematics I, Lesson 3-2, students understand the meanings of the symbols used in function notation and see the advantage of choosing letters that relate to the problem, such as choosing d(t) to represent distance as a function of time.
• In Mathematics I, Lesson 9-4, students determine what properties can be used to show that angles are congruent.
• In Mathematics II, Lesson 10-3, students explain independent or dependent events and if a table can be used to make a recommendation.
• In Mathematics III, Lesson 1-6, students elaborate on what the ordered pair solution of a system means as they answer, “Does the point of intersection represent a solution to both equations?”
• In Mathematics III, Lesson 4-5, students explain how the domains of (f+g)(x), (f-g)(x), (f*g)(x), and (f/g)(x) are related to the domains of f(x) and g(x).

The instructional materials reviewed for enVision Integrated Mathematics 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. There is an intentional development of MP2 and MP3 that reaches the full intent of the MPs. There are many examples in the instructional materials of MPs 2 and 3 where students reason abstractly and quantitatively and critique a solution to determine if it is correct or to find the mistake. Every lesson has at least one error analysis problem, and there are many occasions throughout the topics where students construct an argument to support their answers. 

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 Mathematics I, Lesson 4-1, students predict how substituting an exact solution into a system of equations would differ from substituting an approximate solution.

• In Mathematics I, Lesson 10-1, students explain how interpreting a histogram is similar to, and different from, interpreting a dot plot.

• In Mathematics II, Lesson 9-5, students recognize and apply the relationship between the segments formed by an angle bisector in a triangle and the side lengths of the other two sides of the triangles.

• In Mathematics II, Lesson 4-4, students contextualize the solutions of quadratic equations in real-world situations to determine when it’s appropriate to use the symbol, ±, and when only a positive or negative value applies.

• In Mathematics III, Lesson 3-5, students explain why you can’t average the individual rates to determine how long it will take to complete a job together. 

• In Mathematics III, Lesson 4-3, students make sense of quantities and their relationships when they identify the effect that a has on the graph of a parent square root function.

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 Mathematics I, Lesson 2-2, students construct mathematical arguments to explain how linear equations written in different forms can be equivalent.

• In Mathematics I, Lesson 6-2, students explain why a copy of the angle is always congruent to the original no matter its orientation. 

• In Mathematics II, Lesson 8-4, students determine which arguments about quadrilaterals are correct.

• In Mathematics II, Lesson 9-3, students prove triangle similarity theorems using similarity transformations.

• In Mathematics III, Lesson 2-1, students compare and contrast the end behavior of two exponential graphs, and students write a general statement that compares the end behavior when exponents are odd or when they are even. 

• In Mathematics III, Lesson 2-7, students identify the two errors made by a student when writing the function for the volume of a cube.

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards. The majority of the time, MP7 and MP8 are used to enrich the mathematical content and to reach the full intent of the MPs.

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 Mathematics I, Lesson 8-1, students recognize the significance of lines of symmetry in order to draw transformations.
• In Mathematics I, Lesson 10-5, students use two-way frequency tables to assess the reasonableness of inferences made about people’s preferences.
• In Mathematics II, Lesson 2-6, students consider how substitution can be used during the process of factoring.
• In Mathematics III, Lesson 1-3, students identify linear pieces of a piecewise function as increasing when the slope is positive and decreasing when the slope is negative.
• In Mathematics III, Lesson 5-7, students look for overall structure and patterns in exponents and logarithms, as well as apply general math rules to evaluate logarithms. 

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 Mathematics I, Lesson 5-3, students identify mathematical consistencies in the behavior of exponential growth and exponential decay functions as they approach the horizontal asymptote.
• In Mathematics I, Lessons 9-3 and 9-4, students use patterns and repeated reasoning in problems and proofs to create generalizations and shortcuts.
• In Mathematics II, Lesson 3-5, students use the rate of change to determine the type of function that best fits the data.
• In Mathematics III, Lesson 9-5, students write expressions for coordinates to facilitate shortcuts in calculations.
• In Mathematics III, Lesson 2-3, students recognize the relationship between Pascal’s Triangle and the Binomial Theorem.