## enVision A/G/A

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Our Review Process

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## Report for High School

### Overall Summary

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for alignment to the CCSSM for high school, Gateways 1 and 2. In Gateway 1, the instructional materials meet the expectations for focus and coherence by being coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM). In Gateway 2, the instructional materials meet the expectations for rigor and balance by reflecting the balances in the Standards and helping students meet the Standards' rigorous expectations, and the materials meet the expectations for mathematical practice-content connections by meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice.

##### High School
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus & Coherence

Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for focus and coherence. The instructional materials attend to the full intent of the mathematical content contained in the high school standards for all students, spend the majority of time on the CCSSM widely applicable as prerequisites, engage students in mathematics at a level of sophistication appropriate to high school, and make meaningful connections in a single course and throughout the series. The instructional materials partially attend to the full intent of the modeling process, let students fully learn each non-plus standard, and explicitly identify and build on knowledge from Grades 6-8.

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The materials focus on the high school standards.*
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The materials attend to the full intent of the mathematical content contained in the high school standards for all students.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials include few instances where all aspects of the non-plus standards are not addressed across the series.

A few examples of non-plus standards that are fully addressed in this series include:

• N-CN.2: In Algebra 2 Lesson 2-4, students perform arithmetic operations with complex numbers as well as apply the distributive, commutative, and associative properties.
• A-REI.4b: Students solve quadratic equations by factoring in Algebra 1 lesson 9-2. Students solve quadratic equations by inspection and taking square roots in Algebra 1 lesson 9-4. In Algebra 1 lesson 9-5, students complete the square to solve quadratic equations. In Algebra 1 lesson 9-6 and Algebra 2 lesson 2-6, students solve quadratic equations using the quadratic formula.
• F-IF.6: In Algebra 1 Lessons 5-1 and 8-1, students calculate the average rate of change over a specified interval. In Algebra 2 lesson 3-1, students compare the average rate of change of a polynomial function over two different intervals.
• G-CO.4: In Geometry Lesson 3-1, students draw and describe the reflection of a figure across a line of reflection. In lesson 3-2, students describe properties of a figure before and after a translation, and in lesson 3-3, students draw and describe the rotation of a figure about a point for a given angle of reflection.
• S-ID.6a: Students use models that represent linear, quadratic, and exponential functions. In Algebra 1 lesson 3-6, students fit a linear function to data and a quadratic function to data in Algebra 2 lesson 2-2. Exponential functions are addressed in Algebra 2 lesson 6-2.

The non-plus standards that are are partially addressed in this series include:

• N-Q.1: In Algebra 1 Lesson 1-4, students use units as a way to understand problems as seen in finding average rate (miles/hour). Students also choose units consistent with formulas in example 3, as students choose inches and inches squared while calculating perimeter and area. No evidence was found where students choose and interpret the scale and the origin in graphs and data displays.
• F-LE.1a: In Algebra 1 Lesson 6-2, students identify functions as linear or exponential using tables and graphs. However, no evidence was found proving linear functions grow by equal differences, and exponential functions grow by equal factors.
• F-TF.8: In Algebra 2 Lesson 7-3, students use the Pythagorean identity to find sine, cosine, and tangent, but there is no evidence found for proving the Pythagorean identity.
• G-CO.12: In Geometry Lesson 1-2, students use a straightedge and compass to make formal geometric constructions. However, no evidence was found regarding the construction of a parallel line to a given line through a point not on the line.
• G-GMD.1: In Geometry Lesson 11-2, students use the properties of prisms and cylinders to calculate volume. No evidence was found where students make informal arguments about circumference and area of circles.

There was no evidence found in this series for A-REI.10 and G-SRT.7.

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The materials attend to the full intent of the modeling process when applied to the modeling standards.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials omit the full intent of the modeling process for more than a few modeling standards across the courses of the series.

Each topic of the enVisionMath A/G/A series contains “Mathematical Modeling in 3 Acts” and STEM projects. In each lesson, students are posed a problem, usually by watching a video. Students develop questions of their own, formulate a conjecture, and explain how they arrived at the conjecture. In most of the tasks, the needed information is not given, and students determine what information is essential. Students compute a solution for the problem and interpret their results. Students are guided through validating their conjecture and considering reasons why their answers might differ. Students engage in the full modeling process within the “Mathematical Modeling in 3 Acts” and STEM projects. However, several modeling standards are not addressed within these 3 Acts and STEM projects.

Some of the modeling standards for which the full intent of the modeling process has been omitted include:

• N-Q.2: In Algebra 1 Lesson 1-3, there are several application problems, but none allow the student to complete the full modeling cycle. For example, on page 23 problem 50, a performance task is presented in which two individuals paint a wall. Although students determine when the painters have painted the same amount, students are not given the opportunity to create a conjecture and defend it as the rates are provided.
• A-SSE.1a: In Algebra 1 Lesson 7-5, problem 37, students use a quadratic expression to represent the area of a swimming area. Students factor the expression to determine possible dimensions of the swimming area. They interpret the factors as expressions that give the length and width of the swimming area. However, this problem does not include the full modeling cycle because students do not formulate a model, the formula is given to them. Students are not provided an opportunity to validate their results.
• A-SSE.1b: In Algebra 1 Lesson 6-3, students are shown the relationship between the growth rate and the growth factor in exponential growth. In problem 27, students determine when a particular plant will become invasive. Students are given all the necessary information and told to write an exponential growth formula. Students do not have any choice of which type of model to use, nor to identify the variables for this particular model. Students are not given the opportunity to validate their results. Other examples of parts of the modeling cycle being omitted with this modeling standard can be found in Algebra 1 Lesson 7-5, problems 37 and 41. These problems require students to compute and interpret but not formulate or validate.
• A-SSE.3a: In Algebra 1 Lesson 9-2, students solve quadratic equations by factoring. Students are given three problems (35, 36 and 39) that contain components of the modeling cycle. However in each problem, students are told to use a quadratic equation to describe the situation. In problem 35, students are given a graphical model and asked to write a quadratic equation. In problems 36 and 39, students are given a context with a diagram that is labeled with variables. Students write a quadratic equation using those variables to answer the questions. All three problems missed opportunities to have students formulate needed information and validate their findings. In Algebra 2 lesson 2-3, this topic is revisited. In problem 41, students use an equation that models the height of a drone in terms of time. This problem does not allow students to formulate a solution method because the equation is given, and students are directed how to solve the problem using the equation. Students do not validate their answer.
• A-SSE.3c: In Algebra 2 Lesson 6-2, example 1, students are shown how to use the properties of exponents to transform expressions for exponential functions; however, there are no problems where students do this as part of a modeling cycle.
• A-SSE.4: In Algebra 2 Lesson 6-7, students are shown how to derive the formula for the sum of a finite geometric series. Students use that formula to calculate a monthly payment on a loan. Students are given the formula to use and all the necessary information to solve the problem. The students compute and interpret within the problem. In problem 39 where students are not directed to use the formula for the sum of a geometric series and must formulate the correct model, then compute and interpret their result. However, the opportunity for students to validate their results is missing.
• G-GPE.7: In Geometry Lesson 9-1, students use coordinates to compute the perimeter of different polygons. In Homework problems 26-28, students use parts of the modeling process while formulating an equation and computing the perimeter of the polygon, but they are not provided the opportunity to validate their model nor interpret their results within the context of the problem.
• G-MG.2: In Geometry Lesson 11-2, problem 27, students are given information for making candles and are prompted to determine the weight of a specific order of candles. Students do not identify the needed variables as the dimensions of each candle is presented. The problem is specific and does not allow for interpreting individual findings.
• F-IF.5: In Algebra 1 Lesson 8-1, students identify key features of a quadratic function. There are several application problems for students to practice determining the average rate of change over a specific interval. Problems 26-28 provide students with aspects of the modeling process, such as evaluating a reasonable rate of change and defending that conclusion. However, students do not interpret or evaluation the solution. The questions do not provide multiple access points or various solutions.
• F-IF.6: In Algebra 2 Lesson 1-1, problem 29, students explain the meaning of the rate of change in the context of students jumping. There is only one possible solution, missing the opportunity for students to defend their solution and/or validate their conjecture.
• S-ID.6b: In Algebra 1 Lesson 3-6, students graph residuals from a linear model of data, and in lesson 8-4, students graph residuals from a quadratic model of data. Students do not complete the entire modeling process with this standard. In both lessons, students work with residuals in context, but students do not validate models or analyze results.

Examples where the materials intentionally develop the full intent of the modeling process across the series to address modeling standards include:

• In Algebra 1 Topic 2, students are presented with a situation in which height is measured in unconventional ways. Students watch a video that shows the height of a basketball player in terms of various objects. Students see a stack of cups being built next to him. As students attempt to figure out the basketball player’s height in foam cups, they have to formulate what information they would need. Data is given about smaller stacks of cups. Students utilize the information previously taught about linear functions. When students develop a plan, they complete computations. After they compute, students validate their findings when they view the final video which shows all the cups falling into place. Students report their findings compared to the final solution. This task addresses A-CED.1,3,4.
• In Algebra 2 Topic 6, students explore and apply concepts related to exponential equations and functions. Students watch a video of an athlete performing a running drill. They extrapolate both the time and distance for a certain round of the drill. Students formulate how to determine how far the athlete runs in the twentieth round and how long it will take. Students determine what information they need and consider how their ideas might relate to exponential functions. Students interpret their findings, validate them with each other, and view the final video which reveals the total time and distance. Students report their findings compared to the final solution. This task addresses F-LE.5, S-ID.6a.
• In Geometry Topic 11, students explore and apply concepts related to surface area and volume. Students are presented with different packaging options for candles. The problem is to determine the packaging option with the least surface area for a constant volume. Students watch a video which shows 24 individually-boxed candles. The smaller boxes are then packed inside one cardboard box. Students determine the dimensions of the package that has the least surface area. Students formulate a solution as they speculate how they could analyze the differences in surface area among the packages to find the one with the least surface area. Students compute a solution and think strategically to make sure they have found every possible set of dimensions for the packaging. They validate their results with each other to include ones not seen in the video. The final video shows the dimensions and surface area of each box. Students approach this solution in a variety of methods and report to each other. This task addresses G-GMD.3,4.
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The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
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The materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for, when used as designed, spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs and careers (WAPs). Overall, the majority of the Algebra 1 materials address the WAPs, the majority of the Geometry materials address other non-plus standards, and the Algebra 2 materials spend less than a majority of time on the WAPs.

• In Algebra 1 the materials address the WAPs in the conceptual categories of Algebra, Functions, and Statistics and Probability. The majority of the lessons in Algebra 1 address the WAPs, and there were only a few lessons that did not include a WAP.
• In Geometry, the materials spend less than a majority of time on the WAPs, but the lessons that do not address the WAPs address other non-plus standards. Topics 1, 2, 4, 5, 7 and 8 include lessons in which the majority of the work was related to the WAPs.
• In Algebra 2 Topics 1, 2, 3 and 5 included lessons in which the majority of the work was related to the WAPs, and Topic 11 included lessons in which the majority of the work was related to non-plus standards. Topics 8, 10 and 12 of the Algebra 2 materials spend the majority of time on plus standards.

Examples of students engaging in the WAPs include:

• In Algebra 1 Topic 1, students solve linear equations and inequalities. Students solve quadratic equations in one variable in Algebra 1 Topic 9 through graphing and completing the square (A-REI.B). Students explore the structure of an expression (A-SSE.B) throughout Algebra 1 and Algebra 2, and in Algebra 2 Topic 2, students build on solving quadratics by completing the square and the quadratic formula.
• In Algebra 1 Topic 3, students determine whether a relation is a function as they explore domain and range (F-IF.A). Throughout the topic, students transform linear functions and recognize sequences as functions. In Algebra 2 Topics 1-4, students address arithmetic sequences as functions.
• In Algebra 1 Topics 3-5, students are given data with a context, calculate a trend line, and interpret the slope and y-intercept of the trend line (S-ID.7).
• In Algebra 1 Lesson 11-1, students examine center and spread of a dataset (S-ID.2), and in the next lesson, students examine the shape of a dataset. Students also examine different data sets displayed in different ways.
• In Geometry, Topics 4, 6 and 7 address triangle congruence and applying congruence and similarity of triangles to solve problems involving polygons. Students prove theorems about triangles and write a proof of the Triangle Midsegment Theorem in Geometry lesson 7-5. In lesson 8-1, students prove the Pythagorean Theorem using similarity (G-SRT.B).
• In Geometry, Topics 2, 4 and 7 address theorems about lines, angles, and triangles (G-CO.9,10). Various theorems are proven across several lessons, including the Triangle Sum Theorem in Topic 2, base angles of isosceles triangles are congruent in Topic 4, and the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length in Topic 7.
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The materials, when used as designed, allow students to fully learn each standard.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for, when used as designed, letting students fully learn each non-plus standard. Overall, the series addresses many, yet not all, of the standards in a way that would allow students to fully learn the standards.

The non-plus standards that would not be fully learned by students across the series include:

• N-Q.3: In Algebra 1 Lesson 6-3, students find the population in one example. The “Common Error” box explains rounding to whole numbers when referring to people. In lesson 9-6, students round to the nearest hundredth but are given no context as to why to round to this particular number. Students are not given sufficient opportunities to practice choosing a level of accuracy when reporting quantities throughout the series.
• A-SSE.1: In Algebra 1 Lesson 7-5, students factor polynomials and interpret the factors in terms of the context; however, no opportunity exists to practice interpreting terms and coefficients in terms of the context.
• A-SSE.3c: In Algebra 1 Lesson 6-2, problems 25 and 28, students use properties of exponents to transform exponential functions. No other problems were given to help support students' learning of this standard.
• A-APR.4: In Algebra 2 Lesson 3-3, students prove the difference of cubes' identity. There is not an opportunity to prove any other identities nor use them to describe numerical relationships.
• A-REI.1: In Algebra 1 Lesson 1-2, students explain each step of solving a simple equation in an example, but there are no other opportunities for students to practice constructing a viable argument with other simple equations. In Algebra 2 lesson 5-4, students do not practice constructing arguments to justify solution methods.
• A-REI.5: In Algebra 1 Lesson 4-3, students are provided opportunities to practice solving systems of equations by elimination; however, students are not provided an opportunity to prove, given a system of two equations in two variables, replacing one equation by the sum of that equation, and a multiple of the other produces a system with the same solutions.
• F-IF.3: In Algebra 1 Lesson 1-4, example 1 shows a sequence being a function. However, there are no other opportunities found for students to recognize that sequences are functions. In addition, sequences are not defined using function notation. In Algebra 2 lesson 3-4, students determine whether each sequence is arithmetic, but students do not have the opportunity to develop a connection between sequences and functions.
• F-LE.3: In Algebra 1 Lesson 8-5, students compare linear and exponential graphs to determine which function will exceed the other in one problem. There are no other opportunities for students to address this standard.
• G-SRT.2: Geometry Lesson 7-2 addresses similarity transformations, but students do not explain similarity for triangles in terms of proportionality of all corresponding pairs of sides.
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The materials require students to engage in mathematics at a level of sophistication appropriate to high school.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts, use various types of real numbers, and provide opportunities for students to apply key takeaways from Grades 6-8.

Some examples where the materials illustrate age-appropriate contexts for high school students include:

• In Algebra 1 Lesson 2-2, students solve problems involving the rental of pedal boards and making payments for a cell phone.
• In Algebra 1 Lesson 4-3, students represent how many students a bus holds and the cost of the tickets as a system of equations in the context of a school field trip to an amusement park.
• In Algebra 1 Lesson 8-2, students apply the quadratic function to model the path of a volleyball in order to determine where the ball will go.
• In Geometry Lesson 2-2, students use properties of parallel lines and transversals to solve a problem involving a downhill skier maximizing their speed through a gate.
• In Geometry Lesson 5-4, students use distances on a track to find the measurements of the angles of the track.

Some examples where the materials represent key takeaways from Grades 6-8 include:

• In Algebra 1 Topic 2, students use solving linear equations to write linear equations in three different forms. Students also apply operations with rational numbers and use these properties to solve equations.
• In Algebra 1 Topic 4, students use a solving system of equations from Grade 8 to explore different methods for solving systems of equations. Students also apply their understanding of graphing linear equations in two variables to create and graph linear inequalities.
• In Algebra 1 Topic 8, students analyze graphs of functions to determine the effect of changing the values of the coefficients of quadratics and to model area and vertical motion problems.
• Geometry Topic 1 extends students’ Grade 8 learning of lines, segments, rays, and angles to learn formal definitions of the objects and use them in proofs.
• In Geometry Topic 3, students apply their understanding of congruence and similarity through rotations, reflections, translations, and dilations to learn about compositions of rigid motions.
• In Geometry Topic 7, students apply their knowledge of angle sums, exterior angles of triangles, and angle pairs formed by parallel lines cut by a transversal to learn the criteria that are sufficient to show that two triangles are similar.
• In Algebra 2 Topic 1, students apply their knowledge of solving equations from Grades 6-8 to identify key features of the graphs of linear functions.
• In Algebra 2 Topic 11, students apply their understanding of dot plots, box plots, and histograms from Grades 6-8, as well as calculate measures of center and spread, to analyze data. Students explore the requirements for a question to be considered a statistical question.
• In Algebra 2 Topic 5, students use their understanding of square roots and cube roots from Grade 8 to graph and solve radical equations. Students expand upon this skill to isolate the variable when solving radical equations.

Some examples where the instructional materials use various types of real numbers include:

• In Algebra 1 Lesson 4-2, students solve problems involving a lawn-mowing business and surfing lessons. The solutions to these problems are often decimal answers instead of whole numbers.
• In Algebra 1 Lesson 9-6, many of the problems working with the quadratic formula have irrational solutions.
• In Algebra 1 students write irrational solutions as decimal approximations. Later, in Algebra 2, students express these answers as radical expressions in simplest form. The level of sophistication increases as seen in Algebra 2 lesson 2-6, which introduces complex solutions to quadratic equations.
• In Geometry Lesson 7-3, students use decimals when determining the similarity between two triangles.
• In Geometry Lesson 8-1, students use rational and irrational numbers when finding missing sides' lengths of right triangles.
• In Algebra 2 Lesson 1-7, students have one practice problem with a decimal coefficient and another one with a rational coefficient.
• In Algebra 2 Lesson 3-5, students find the zeros of polynomial functions including integers, terminating decimals, irrational values, and imaginary numbers.
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The materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series, where appropriate and where required by the standards.

Some examples where the materials foster coherence through meaningful mathematical connections in a single course include:

• In Algebra 1 Topic 3, the materials introduce relations and functions (F-IF.1). The materials also develop domain and range as they relate to graphs and the quantitative relationships the graphs represent (F-IF.5). In Lessons 4 and 5, students investigate fitting functions to graphs (S-ID.6).
• In Geometry lesson 6-3, students use congruence and similarity criteria for triangles (G-SRT.5) to prove theorems about parallelograms (G-CO.11).
• In Algebra 2 Lesson 7-4, students graph and model with trigonometric functions and are introduced to terms such as amplitude, period, and midline. These terms are connected to transformations from lesson 1-2 (F-BF.3, F-TF.5).
• Algebra 2 Lesson 2-5 connects A-SSE.3b (completing the square to find the maximum or minimum) with F-IF.7a (graphing a quadratic function and show intercepts, maxima, and minima).

Some examples where the materials foster coherence through meaningful mathematical connections throughout the series include:

• In Algebra 1 Topic 3, students use function notation to express the relationship between x- and y-values as well as write, graph, and transform linear functions. In Topic 6, students continue to learn about exponential functions. In Algebra 2, students use arithmetic sequences to solve real-world problems, and in Algebra 2 Topic 6, students identify the key features of exponential functions and extend that knowledge with logarithms. (A-CED.1, A-CED.2, F-IF.7e, F-LE.4)
• In Algebra 1, students solve quadratic equations by completing the square, factoring, and using the quadratic formula. In Algebra 2, this knowledge is extended to factor higher degree polynomials and solve equations that have complex solutions. (A-SSE.3a, A-SSE.3b, A-APR.3, A-REI.4b, N-CN.7)
• In Algebra 1 Topic 10, students learn about transformations of functions, and in Algebra 2 lesson 1-2, students review these transformations. These transformations are revisited each time a new type of function is studied throughout Algebra 2. (F-BF.3)
• In Geometry Topic 8, students study trigonometric ratios of right triangles. In Algebra 2 Topic 7, students extend that knowledge to explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle (F-TF.2, G-SRT.6).
• In Geometry Topic 5, students use perpendicular bisectors, angle bisectors, medians, and altitudes to examine the relationships between the angle measures and side lengths within a triangle (G-CO.9, G-CO.10). Students also examine angle measures and side lengths of two triangles. In Algebra 2 Topic 7 examines the relationship between triangles and circles to make connections to trigonometric ratios (F-TF.1,2).
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The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School CCSSM Standards. The instructional materials do not explicitly identify content from Grades 6-8.

Some examples where the materials do not explicitly identify content from Grades 6-8 include:

• At the beginning of each topic in the teacher editions, the Topic Overview Math Background Coherence standards from middle school are identified as being from Grades 6-8, but specific standards are not listed. Concepts from middle school are explicitly identified but not connected to any standard.
• The Algebra 1 Teacher Edition, page 2B states how students will use their understanding of rational and irrational numbers from Grade 8 to compare and order rational and irrational numbers in Topic 1 of Algebra 1. The materials state that students will use their understanding of solving inequalities in the form $$px + q \gt r$$ or $$px + q \lt r$$ from Grade 7 to solve compound inequalities.
• In the Geometry Teacher Edition, page 2B, informal geometric arguments from Grade 8 are referenced and connected to building formal proofs in high school, but no specific standards are identified.

Some examples where the materials make connections between Grades 6-8 and high school concepts and allow students to extend their previous knowledge include:

• Algebra 1 Teacher Edition, page 86B describes how using functions to model relationships is extended to understanding domain and range. The materials state, “In Grade 8, students began to explore linear and nonlinear functions. Students learned about the key features of linear functions, including slope and rate of change.”
• Algebra 1 Topic 8 Quadratic Functions, page 312B references, “In Grade 8, students compared linear and nonlinear functions, learned about increasing and decreasing intervals, and sketched functions from a verbal description. Students explored key features of linear functions including slope and rate of change” as a precursor to the learning of quadratics.
• Geometry Teacher Edition, page 342B references ratios and proportions and connects that to understanding trigonometric ratios (G-SRT.C). Teachers use this resource to make the connection as it is not present in the student edition.
• Geometry Teacher Edition, page 462B refers to formulas for volume of cones, cylinders, and spheres from middle school, and these are used to solve problems aligned to G-GMD involving those shapes in applications of prisms, cylinders, pyramids, cones, and spheres.
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The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.

The instructional materials reviewed for enVisionMath A/G/A explicitly identify the plus standards and use them to coherently support the mathematics which all students should study in order to be college and career ready. All plus standards are taught at various places throughout the series.

The teacher editions explicitly identify the plus standards. At the beginning of each lesson is a section called Mathematics Overview which lists the Content and the Practice Standards that are addressed in that lesson. All plus standards are identified with the (+) symbol. The (+) standards are explicitly identified in the materials and coherently support the mathematics which all students should study in order to be college and career ready.

The (+) standards that are fully addressed include:

• N-CN.3: In Algebra 2 Lesson 2-4, students are introduced to complex conjugates and then find the quotients of complex numbers. Further, in Algebra 2 lesson 8-4, students find the modulus of a complex number.
• N-CN.4: Algebra 2 Lesson 8-5 uses the polar form of complex numbers to calculate products and powers. Students also explain the relationship between rectangular and polar forms of complex numbers.
• N-CN.5: In Algebra 2 Lessons 8-4 and 8-5, students use the complex plane to show complex numbers and operations of them.
• N-CN.6: In Algebra 2 Lesson 8-4, students find the distance and midpoint between two complex numbers.
• N-CN.8,9: Algebra 2 Lesson 3-6 states the Rational Root Theorem, the Fundamental Root Theorem, and the Conjugate Root Theorem. Students use the roots of a polynomial equation to find other roots, including complex roots.
• N-VM.1-5: In Algebra 2 Lesson 10-3, students find the magnitude and the direction of vector quantities. Students interpret and use vectors for addition, subtraction, and scalar multiplication, and they are presented with application problems involving vectors. Students solve problems involving velocity such as the course of a flying plane and the speed of a thrown ball.
• N-VM.6-8:In Algebra 2 Lesson 10-1, students use matrices to represent data such as inventory of a store and data from races, and they apply scalar multiplication to create a new matrix. Students also represent and manipulate data using matrices, multiply a matrix by a scalar, and add and subtract matrices.
• N-VM.9,10: In Algebra 2 Lesson 10-2, students examine multiplication of square. They understand the Commutative Property does not hold for all square matrices, but the Distributive Property and the Identity Matrix do hold. In lesson 4, students calculate the determinant and inverse of a 2x2 matrix.
• N-VM.11: In Algebra 2 Lesson 10-3, students use matrix multiplication to transform each vector by a given transformation.
• N-VM.12: In Algebra 2 Lesson 10-4, students work with finding the determinant of a 2x2 matrix and recognize they must take the absolute value of the determinant when calculating an area.
• A-APR.7: In Algebra 2 Lesson 4-4, students add and subtract rational expressions and multiply and divide rational expressions in Algebra 2 lesson 4-3. Students also have an introduction to closure of these operations in Algebra 2 lesson 4-4.
• A-REI.8: Students use matrices to represent a system of equations in Algebra 2 lesson 10-5.
• A-REI.9: In Algebra 2 Lesson 10-1, students use matrices to represent coordinates of pixels of images and matrix operations to represent the movement of the images. Students find the inverse of matrices in lesson 10-5.
• F-BF.1c: In Algebra 2 Lesson 5-5, students practice combining functions by adding, subtracting, and multiplying, and they also compose functions.
• F-BF.4b: In Algebra 2 Lesson 5-6, students prove two functions are inverses using compositions.
• F-BF.4c: Algebra 2 Lesson 6-4, example 1 shows how to use values from a table of $$y=2^x$$ to find coordinates on the graph of $$y=log_2x$$. Problem 7 asks students to determine graphically if two functions are inverses of each other. Students can solve this problem by reading values from the graph of one function and comparing those to the graph of the other function.
• F-BF.4d: In Algebra 2 Lesson 5-6, example 3, students are shown if a function is not invertible how they can restrict its domain to make a function that is invertible.
• F-BF.5: Algebra 2 Lesson 6-3 contains several examples showing how to solve exponential equations by converting to logarithmic form. Lesson 6-6, example 3 shows how to solve exponential equations by taking the logarithm of both sides.
• F-TF.3: Algebra 2 Lesson 7-1, example 4 reviews the special right triangles and shows how to find the trigonometric ratios for these special angles. In lesson 7-2, students are introduced to the unit circle and radian measure, and in lesson 7-3, students use the unit circle, radian measure, special right triangles, and reference angles to find the values of sine, cosine, and tangent for radian measures around the unit circle.
• F-TF.4: Algebra 2 Lesson 8-3 shows why the sine function is an odd function; in question 2, students determine if the cosine function is an even or odd function. The periodicity of trigonometric functions is addressed in lesson 7-4, example 1.
• F-TF.6: Algebra 2 Lesson 8-1, example 1 shows that the domain of the graph of y=sin x can be restricted to create an invertible function. Students then determine how the domain of y=cos x should be restricted so that it is also invertible.
• F-FT.7: Algebra 2 Lesson 8-1, example 4 shows how to solve a trigonometric equation using inverse trigonometric functions. Example 5 provides an example of modeling monthly temperature in New Zealand using a trigonometric function. The question asks in what months the average temperature will be less than 15 degrees Celsius. The solution requires using inverse trigonometric functions to solve the problem. In questions 32-36, students also practice using inverse trigonometric functions.
• F-TF.9: Algebra 2 Lesson 8-3, example 3 proves the cosine difference formula and the cosine sum formula. In “Try It,” students prove the sum and difference formulas for sine, and in problems 25 and 26, students prove the sum and difference formulas for tangent.
• G-SRT.9: In Geometry Lesson 5-8, students derive the formula for the area of a triangle using sine and use the formula to find the area of a given triangle.
• G-SRT.10: In Geometry Lessons 8-3 and 8-4, students prove the Law of Sines and Cosines for acute triangles. In Algebra 2 Lesson 8-2, students prove the Law of Sines and Cosines with obtuse triangles. Students also use the Law of Sines and Cosines to solve problems.
• G-SRT.11: In Geometry Lessons 8-3 and 8-4 and Algebra 2 Lesson 8-2, students apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.
• G-C.4: In Geometry Lesson 10-2, example 5, students construct a tangent line from a point outside a given circle.
• G-GMD.2: In Geometry Lesson 11-2, students use Cavalieri’s principle to solve a problem involving the volume of a cylinder, cone, and rectangular prism in exercises 12, 14, and 17. In Lesson 11-3, example 1, students apply the principle to relating volumes of pyramids and cones. In Lesson 11-4, example 1, students develop a formula for the volume of a hemisphere using Cavalieri’s principle by subtracting the areas of corresponding sections of a cone from a cylinder.
• S-CP.8: In Geometry or Algebra 2 Lesson 12-2, students use conditional probability to solve problems and interpret the answers in terms of the model.
• S-CP.9: In Geometry or Algebra 2 Lessons 12-3, 12-4, and 12-6, students use permutations and combinations to compute probabilities of compound events and solve problems.
• S-MD.1: In Geometry or Algebra 2 :esson 12-4, students use calculations and graphing techniques to define and graph a probability distribution.
• S-MD.5: In Geometry or Algebra 2 Lesson 12-5, students find expected payoffs and expected values within the contexts of restaurant sales, insurance policies, and warranty purchases.
• S-MD.6: In Geometry or Algebra 2 Lesson 12-6, students use probability to determine the fairness of a game.
• S-MD.7: In Geometry or Algebra 2 Lesson 12-6, students use probability concepts to analyze decisions and strategies in several activities throughout the unit. Students apply probability concepts to find solutions for the activities.

Plus standards that are partially addressed include:

• F-IF.7d: In Algebra 2 Lessons 4-1 and 4-2, students find asymptotes of rational functions and graph rational functions by plotting the asymptotes and several points on either side of the asymptotes. Students are not asked to determine the zeros of a rational function.
• G-GPE.3: In Algebra 2 Lesson 9-3, example 1, students derive the equation of an ellipse using the sum of the distances from the points (0, 5) and (0, -5) with foci at (0, 4) and (0, -4). The results are then stated in generalized form for the equation of an ellipse in “standard position” with its center at the origin and when the major axis is vertical or horizontal. In Algebra 2 Lesson 9-4, example 1, students derive the equation of a hyperbola using the difference of the distances from the points (-3, 0) and (3, 0) with foci at (-5, 0) and (5, 0). The results are then stated in generalized form for the equation of a hyperbola centered at the origin that opens horizontally or vertically. Students do not generalize their results to derive the general equation of an ellipse or a hyperbola.
• S-MD.2-4: In Geometry or Algebra 2 Lesson 12-5, students calculate expected values but do not connect them to probability distributions.

### Rigor & Mathematical Practices

##### Gateway 2
Meets Expectations

#### Criterion 2.1: Rigor

Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for rigor and balance. The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.

##### Indicator {{'2a' | indicatorName}}
Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The instructional materials for enVisionMath A/G/A meet expectations that the materials develop 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.

Some examples across the series that develop conceptual understanding and present students with opportunities to independently demonstrate conceptual understanding include:

• A-REI.A: In Algebra 1 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. In Algebra 2 Topic 4, students solve rational equations. Students analyze and critique various methods and check for extraneous solutions. Further, in Lesson 5-4, students solve equations with exponents and radicals. Students look for relationships between square roots and squaring and solve algebraically and graphically.
• N-RN.1: In Algebra 1 Lesson 6-1, there is an explanation of why a power of 1/2 must be equivalent to the square root of the number. Then in Algebra 2 Lesson 5-1, a similar explanation is provided for a fourth root and for a rational exponent of 2/3. In Lesson 6-1, problems 20, 22 and 23, students explain concepts related to this standard. In Algebra 2 Lesson 5-1, problems 4, 6, 20 and 21, students also explain problems aligned to this standard.
• A-APR.2: Algebra 2 Lesson 3-4, example 3 provides an explanation of why the Remainder Theorem is true. In exercises in the lesson, students explain their reasoning involving the Factor Theorem.
• S-ID.7: In Algebra 1 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.
• A-APR.B: Students examine the relationship between factors and zeros in Algebra 1 Lesson 9-2. 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 Algebra 2 Lesson 3-5, students extend their understanding of the relationship between factors and zeros to higher order polynomials.
• G-SRT.2: In Geometry Lesson 7-2, the lesson starts with examining examples of student work. The questions posed for the teacher 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 lesson 2. Also, in Lesson 7-2, there are questions for the teacher to help students develop an understanding of the connection between congruence and similarity.
##### Indicator {{'2b' | indicatorName}}
Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The instructional materials for enVisionMath A/G/A meet expectations that the materials provide 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 included in each lesson. An additional resource is available to provide problems for extra practice.

Some examples that show the development of procedural skills across the series include:

• A-APR.6: In Algebra 2 Lesson 3-4, students divide polynomials using long division and synthetic division. Students are also provided additional practice in Lessons 4-2 and 4-3.
• A-APR.1: In Algebra 1 Lesson 7-1, there are examples of simplifying, adding, and subtracting polynomials. The practice and problem solving set contains several practice problems for this skill. The next lesson provides examples of multiplying polynomials and has many problems for students to practice.
• A-REI.6: In Algebra 1 Lessons 4-1 and 4-2 and Algebra 2 Lessons 1-6 and 1-7, students work with systems of linear equations. Students solve systems of linear equations exactly and approximately.
• F-BF.3: Students transform different types of functions. For example, in Algebra 1 Lesson 3-3, students transform linear functions, then piecewise and absolute value functions in Lesson 5-4. In lesson 6-5, students transform exponential functions, and in Lessons 8-1 and 8-2, students work with transformations of quadratic functions. In Algebra 2 Lesson 3-7, students extend their understanding of transformations to higher degree polynomials, and in Lesson 5-4, they transform square root functions. In Lesson 6-4, students have the opportunity to use their understanding of transformations to work with logarithms, and in Lesson 7-6, students continue their work with transformations of trigonometric functions.
• G-GPE.4: In Geometry Lessons 9-2 and 9-3, 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-GPE.7: In Geometry Lesson 9-1, students use coordinate geometry to classify triangles and quadrilaterals. Students solve problems with polygons on the coordinate plane. Students use the distance formula, the midpoint formula, and the slope formulas to analyze polygons. Students are given multiple opportunities to compute perimeters and areas of polygons.
##### Indicator {{'2c' | indicatorName}}
Attention to Applications: 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 for enVisionMath A/G/A 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 throughout the series. Additionally, the instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts.

The materials provide multiple opportunities for students to engage in application of mathematics throughout the series. For example:

• G-SRT.8: In Geometry Lessons 8-1 and 8-2, 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. Lesson 8-5 primarily consists of application problems related to angles of elevation and depression in tandem with trigonometric ratios.
• F-IF.7a,c: Students solve application problems that involve the graphs of various types of functions. In Algebra 1 Lesson 2-1, students solve problems involving graphs of linear equations, and in lesson 9-1, students solve application problems involving graphs of quadratic equations.
• F-IF.7e,9: In Algebra 2 Lessons 6-1 and 6-4, students create and solve exponential and logarithmic application problems. In Lesson 6-1 example 3 and practice problems 12 and 24, students solve problems involving growth rates of populations. In Lesson 6-4, example 5 and practice problem 25, students solve problems using logarithmic functions to approximate the altitudes of airplanes.

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

• A-REI.4 In Algebra 2 Lesson 2-5, students solve a quadratic equation and identify the maximum and minimum values of the function. In example 4, students enclose a pasture in the shape of a rectangle with parameters on fencing materials and square footage available. In Practice problem 42, students find the length and width of a skate park given parameters on perimeter and area.
• A-REI.11: Algebra 1 Topic 9, students solve linear and quadratic systems of equations using methods of substitution, elimination, and graphing. Contexts for students to explain why the intersection 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.
##### Indicator {{'2d' | indicatorName}}
Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

The instructional materials for enVisionMath A/G/A 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:

• Algebra 1 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.
• Geometry Topic 7 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.
• Algebra 2 Topic 2 addresses extending understanding of quadratic functions. In the STEM Project, students explore how the design of a ballpark influences the number and frequency of home runs. Throughout the seven lessons, 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 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.

#### Criterion 2.2: Math Practices

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for mathematical practice-content connections. The instructional materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice by intentionally developing overarching, mathematical practices, reasoning and explaining, and seeing structure and generalizing and partially meet the expectations for modeling and using tools.

##### Indicator {{'2e' | indicatorName}}
The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support 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.

Some 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 Algebra 1 Lesson 4-1, problem 35 poses a question about maximizing volume. Students relate the height to the radius and write a formula to meet the needs of the manufacturer.
• In Algebra 1 Lesson 5-3, students use step functions to make sense of why the functions appear to be different given two scenarios. Students determine which graph is correct and why.
• In Algebra 1 Lesson 6-5, students describe two ways to identify how an exponential function is transformed.
• In Algebra 2 Lesson 7-5, students determine a function that models the height of a triangle to construct a hexagonal floor of a treehouse. Students make sense of the shape of the triangle to fit the floor pattern.

In some places, identification of MP1 does not reflect the intent of the mathematical practice as seen in the following examples:

• In Geometry Lesson 4-5, problem 21, the materials indicate students are making sense and persevering in identifying congruent triangles; however, the problem states that students use the Hypotenuse-Leg Theorem.
• In Geometry Lesson 5-1, problem 22, the materials indicate students are making sense and persevering in finding the perimeter of a garden using perpendicular bisectors; however, students are given a diagram including dimensions which reduces students' making sense of how to determine the amount of fencing needed.

Some examples of where and how the materials use MP6 to enrich the mathematical content and demonstrate the full intent of the mathematical practice include:

• Many of the problem sets contain a problem titled “Communicate Precisely,” which asks students to write a clear explanation. For example, in Algebra 1 lesson 5-1, problem 2, students compare the domain and range of g(x) = a|x| and f(x) = |x|. In Algebra 2 lesson 6-1, problem 4, students compare and contrast exponential growth and exponential decay. Both of these examples require students to be precise in their words.
• In Algebra 1 Lesson 3-4, students explore arithmetic sequences. In problem 48, students are given the number of rows at an outdoor concert and and that two more chairs in each additional row. Students precisely write a recursive formula, an explicit formula, and graph the sequence for setting up rows of chairs at the concert.
• In Geometry Lesson 9-2, students justify reasoning within proofs and are precise in the calculation of needed values.
• In Algebra 2 Lesson 11-6, students communicate precisely by using data and statistical measures to support or reject a hypothesis.
##### Indicator {{'2f' | indicatorName}}
The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support 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 are asked to reason abstractly and quantitatively and to 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 are asked to construct an argument to support their answer.

Some 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 1 Topic 5, Mathematical Modeling in 3 Acts, students are presented with a video showing a person running on uneven terrain. Students generate a graph that matches the presented situation. Students rely on their understanding of concepts related to nonlinear functions to develop a representative model. Students engage in abstract and quantitative reasoning to identify constraints which will affect the graph of the scenario. Students compare the speed of a runner going uphill, downhill, and on a flat surface.
• In Geometry Lesson 1-3, students reason abstractly to represent real-world scenarios using points on the coordinate plan and calculate the distance and midpoint between those points.
• In Geometry Lesson 3-4, students use symbolic rules for rigid motion. Students also explain the relationships between the rules and the figures which involve application of the rules.
• In Geometry Lesson 5-5, students reason abstractly by using triangles to represent real-world situations. Students also interpret their solutions in terms of the original context.
• In Algebra 2 Lesson 2-7, problem 26, students are given an equation to model the height of a football and an equation to model the height of a blocker’s hands. Students determine if it is possible for the blocker to knock down the ball and what other information is needed. Students must decontextualize the situation in order to think about the function needed to represent the situation and solve the equations as a system. Students must also contextualize the situation in order to interpret both the equations and the solution in context.

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

• Algebra 1 Lesson 2-2 shows two different people writing a linear equation based on a point and a slope, and each person uses a different method. Students determine if the two equations represent the same line and provide a mathematical argument to support their answer. In this problem, students critique the reasoning of someone else by examining the approach each person used. Students construct a viable argument to explain their conclusion about whether or not the two equations represent the same line. In lesson 3-3, problem 12, students examine work with a mistake in it and describe how to correct the mistake.
• In Geometry Lesson 6-4, problem 11, students find the error in an argument of the proof that a quadrilateral with one pair of congruent sides and one pair of parallel sides is a parallelogram.
• In Geometry Lesson 7-4, problem 14, students construct a viable argument by proving the altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
• In Algebra 2 Lesson 2-4, problem 10, students agree or disagree with a statement about the result of raising the number i to an integer power and explain their thinking. In Lesson 2-7, problem 7 presents two possible methods to solve a linear-quadratic system. Students decide which method is correct. In Lesson 2-7, problem 10, students construct a viable argument for why a linear-quadratic system cannot have more than two solutions.
• In Algebra 2 Lesson 9-2, example 5, students examine a system of equations and answer the following question from the teacher edition: “Which of the three methods for solving a system of equations would you be least likely to use? Explain.” Students construct an argument for which method is not possible.
##### Indicator {{'2g' | indicatorName}}
The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for enVisionMath A/G/A partially meet expectations that the materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards. There is intentional development of MP4 that reaches the full intent of the MP. However, across the series, the materials do not develop MP5 to the full intent of the MP.

Some 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 Algebra 1 Topic 3, 3 Acts, students create a linear model to make a decision as they analyze two lines at a grocery store. Students consider how long it will take them to get through the store line. Students select and use a data display in order to see how many variables might be related.
• In Algebra 1 Topic 8, 3 Acts, students design a wheelchair ramp from a sidewalk to a woman’s front porch. Students make an initial conjecture based on preliminary information and refine the conjecture after obtaining more information about the height of the porch. Finally, students watch a video that shows how long the ramp needs to be to verify their solution and discuss any differences between their solution and the actual solution.
• In Geometry Topic 4, 3 Acts, students explore and analyze triangles drawn to specific parameters to determine what it means for two triangles to be “the same” and whether all triangles that meet this criteria are the same.
• In Algebra 2 Topic 4, students are presented with a video of a student filling a swimming pool using two hoses. Students analyze the situation in which the two hoses fill the pool at different rates and explore and apply concepts related to rational functions and equations. Not only are the rates different, but an extension of the problem prompts students to consider the impact if a third hose is utilized.

Some examples where the materials do not develop MP5 to its full intent include:

The instructional materials often list MP5 in topics when students are directed to use tools that are listed in the lesson. There are some opportunities where students could use tools such as graphing calculators and algebra tiles in exercises beyond the ones that students are directed to use. Some examples include:

• In Algebra 1 Topic 7, students are directed to identify when to use tables to organize factors and their sums. Additionally, students are directed to use algebra tiles to verify the correct pair of factors. The tables are fill-in-the-blank tables, so students have no choice on how to organize these factors.
• In Algebra 2 Topic 1, Lesson 1-2, students are directed to use graphing calculators to graph original and transformed functions in the first problem set. Students are also directed to use the calculator to check that their work is correct.
• In Geometry Topic 1, Lesson 1-2, students are directed to use a compass and straightedge to make basic geometric constructions, and there is no opportunity to use other tools for basic constructions.
• In Geometry Topic 8, Lesson 8-5, students use diagrams and sketches to plan solutions to trigonometric problems. Students are directed to make the sketches and diagrams.
• In Geometry Topic 3, Lesson 3-1, there are no tools or choice of tools for students to use when working with reflections.
• In the Algebra 2 materials, graphing technology is the main tool that is used throughout the materials. Students do not choose when and where to use the graphing calculator. Additionally, there was no evidence that students use graphing technology to explore and deepen their understanding of the concepts.
• In Algebra 2 Topic 7 Lesson 7-6, MP5 is listed along with the question, “Why is it important to identify the key features of a function, even if you intend to graph it on a calculator?” This occurs often in Algebra 2 with MP5; students do not choose which tools to use nor use tools to deepen their understanding.
• Students do not use multiple tools to represent information in a situation or demonstrate modeling effectively with tools in the data sections of the Algebra 2 materials.
##### Indicator {{'2h' | indicatorName}}
The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials for enVisionMath A/G/A meet expectations that the materials support 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.

Some 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 1 Topic 2, students learn that linear equations in different forms are composed of several parts that can be manipulated to reveal the slope and intercepts of the equations. Students apply their previous understandings of parallel and perpendicular lines to identify methods for writing the equation of a line parallel or perpendicular to a given line.
• In Algebra 1 Topic 8, students see the standard form of a quadratic function as composed of several objects, including values of a, b, and c that can be used to graph the intercepts, axis of symmetry, and vertex of the parabola that represents the function. Students look at the structure of the vertical motion model and relate it to the standard form of a quadratic equation.
• In Geometry Topic 3, Lesson 3-5, students use the structure of a symmetric figure to determine that it can be mapped onto itself after a reflection, rotation, or series of rotations.
• In Geometry Topic 7, Lesson 7-1, students use the structure of similar triangles to understand relationships in triangles.
• In Algebra 2 Topic 1, Lesson 1-4 students use patterns in arithmetic sequences and series to find the common difference of a series and write both recursive and explicit definitions for each sequence.
• In Algebra 2 Topic 2, Lesson 2-1, students use the structure of a quadratic equation to rewrite an equation from standard form to vertex form.

Some 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 1 Topic 3, students graph transformations of linear functions by multiplying or adding specific values of k to the input or output of a function. Students make generalizations to recognize that multiplying the output of a linear function by k scales its graph vertically.
• In Algebra 1 Topic 8, Lesson 8-1, students use repeated reasoning to make generalizations about the effects of how changing the values of a function affects the graph of that function.
• In Geometry Topic 1, Lesson 1-4, students look for regularity in repeated reasoning when they identify patterns and rules for sequences and look for general rules to define the sequences.
• In Geometry Topic 5, Lesson 5-3, problem 6, students express regularity in repeated reasoning when investigating where the orthocenter is located for any right triangle.
• In Algebra 2 Topic 3, Lesson 3-1, students generalize how the sign of the leading coefficient and the degree of a polynomial affect the end behavior of a graph of a polynomial function. In lesson 2, students generalize the procedures used to add, subtract, and multiply polynomials and recognize whether or not the operations are closed for polynomials. In lesson 5, students generalize the relationship between the multiplicity of zeros and the appearance of the graph of a polynomial function.
• In Algebra 2 Topic 7, Lesson 7-1, students use regularity in repeated reasoning to determine that in isosceles right triangles, trigonometric co-functions are equivalent.

### Usability

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Use & Design

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for being well designed and taking into account effective lesson structure and pacing. In the instructional materials, the underlying design distinguishes between problems and exercises, the design of assignments is not haphazard, and there is variety in how students are asked to present the mathematics.

##### Indicator {{'3a' | indicatorName}}
The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials reviewed for enVisionMath A/G/A meet expectations that the underlying design of the materials distinguish between problems and exercises. The materials distinguish between problems and exercises within each lesson. Most problems or exercises have a purpose. Each lesson starts with Explore & Reason to introduce the new concept for that lesson. Then, there are several examples that guide students through the learning of the content. The lessons end with exercises where students use what they have learned to develop procedural skills, application, and conceptual understanding as appropriate.

##### Indicator {{'3b' | indicatorName}}
Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials reviewed for enVisionMath A/G/A meet the expectations for having a design of assignments that are not haphazard but with problems and exercises given in intentional sequences. Exercises within student assignments are intentionally sequenced to build understanding and knowledge. There is a natural progression within student assignments leading to a full understanding of new mathematics. Within each set of exercises, students are brought back to the essential question for understanding. Students progress in each lesson by starting with problems that focus on understanding, move to practice exercises which are more procedural, and complete application problems.

##### Indicator {{'3c' | indicatorName}}
There is variety in how students are asked to present the mathematics. For example, students are asked to produce answers and solutions, but also, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials reviewed for enVisionMath A/G/A meet the expectations that there is variety in how students are asked to present the mathematics. Students compute numerical answers while also providing diagrams and graphs. There are problems in each lesson that ask students a question about the mathematics and require them to explain their thinking. Each lesson includes at least one error-analysis problem where students must find, describe, and correct an error in mathematical work. Examples include:

• In Algebra 1 Lesson 8-3, students graph quadratic functions using vertex form. In problem 40, students are presented with two ordered pairs identifying the path a soccer ball travels. Students determine the quadratic function in vertex form, defend possible solutions that can not be determined, and explain why. Students generate a realistic graph using technology to explore undetermined values as well as find values that generate a realistic graph. Students also explain how key features of the graph correspond to the given situation.
• In Geometry Lesson 3-5, students construct an argument on the possibility of a figure having rotational symmetry and no reflectional symmetry. Students explain and give examples to construct arguments.
• In Algebra 2 Lesson 6-2, problems 2 and 12, students analyze an error and explain or correct it. In problem 4, students explain the different parts of an exponential expression. In problems 7, 8 and 16-21, students compute the amount of money in an account at a certain time. In problem 15, students find two different equations to model a list of data and create a graph of the data and the equations.
##### Indicator {{'3d' | indicatorName}}
Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials reviewed for enVisionMath A/G/A partially meet the expectations that manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent, and when appropriate are connected to written methods. Manipulatives are not consistently correlated to written methods when appropriate. Examples include:

• The materials occasionally direct students to use manipulatives within the materials, but the materials do not provide directions for the use of virtual manipulatives such as Desmos. For example, there were no Desmos screenshots or other supports offered in the materials.
• Algebra tiles are used in Algebra 1, but there is no evidence of their use in Algebra 2 to make connections across the courses. Additionally, algebra tiles are used with quadratic expressions and equations, but there are not connections made within Algebra 1 to other concepts.
• The Digital Math Tools include a graphing calculator and geometry tools to explore transformations, evaluate equations, and plot tables of data. The materials state, “Much more is always available to students and teachers at PearsonRealize.com”
##### Indicator {{'3e' | indicatorName}}
The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials reviewed for the enVisionMath A/G/A series have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The materials keep a consistent layout for topics and lessons. In general, the sections appear in the following order: Explore, Understand and Apply, Practice and Problem Solving, and Assess and Differentiate.

Pictures and models used throughout the series support student learning as these representations are connected directly to an investigation or problems being solved. The figures and models used do not distract from the mathematical content.

#### Criterion 3.2: Teacher Planning

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series partially meet expectations for supporting teacher learning and understanding of the Standards. The instructional materials support teachers in planning and providing effective learning experiences by providing quality questions, and the teacher edition contains ample and useful annotations and suggestions on how to present the content in the student edition and in ancillary materials. The instructional materials rarely explain the role of the specific mathematics standards in the context of the overall series, and the teacher edition partially include explanations and examples of the course-level mathematics specifically for teachers so that they can improve their own knowledge of the subject.

##### Indicator {{'3f' | indicatorName}}
Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials reviewed for enVisionMath A/G/A meet the expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Questions are consistently provided to teachers to help guide students’ mathematical development. For each topic throughout the series, teachers are provided a math background section, specific to the focus, coherence, rigor, and mathematical practices addressed, and a topic planner. Within the math background section, teachers are given a clear and concise explanation of what students will be covering, what concepts students should already know, and where the concepts lead. Each lesson begins with an Explore question in the student edition, and teachers are provided an assortment of questions to ask their students to encourage discourse and conceptual understanding, as well as providing support for productive struggle and differentiation. For example, in Algebra 1 Lesson 1-5, teachers are given the following: “How could you use words to describe the relationship between Skylar’s personal best and the district state records?” In addition, “How could you represent the solutions visually so they could be easily understood?”

##### Indicator {{'3g' | indicatorName}}
Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials reviewed for enVisionMath A/G/A meet the expectations that the teacher edition contains ample and useful annotations and suggestions on how to present the content in the student edition and in ancillary materials. The materials provide an overview at the beginning of each topic explaining the overarching ideas. This overview is broken up among conceptual understanding, procedural skills, and applications.

Each lesson also includes useful annotations such as a lesson overview that contains student objectives, connections to previous and future content, common errors, vocabulary, and guiding questions with sample student answers. There is also a section with suggestions for advanced students, struggling students, and English Language Learners.

The teacher edition contains an abundance of teaching supports for both planning and in-class instruction. Within the side margins, teachers find highlights on effective teaching practices, essential questions, probing questions, habits of mind questions, additional examples, differentiated instruction supports for English Language Learners, advanced, and struggling students, and various common errors. However, the images for the student book are reduced to 25 percent, rendering them difficult to read and study.

Examples from the teachers edition that show useful annotations and suggestions include:

• In Algebra 1 Lesson 3-3, the student edition gives the definitions of vertical and horizontal translations. The teacher edition suggests two questions to ask students to help them think about and make sense of these definitions.
• In Geometry Lesson 3-2, the teacher edition provides a question to ask students at the beginning of the opening activity and specifies that this question is intended for the whole class. The teacher edition provides two more questions to ask students as they are completing the activity, specifying that the two additional questions should be asked to small groups. There are also two questions that the teacher could use to extend the thinking of early finishers, followed by one more question to summarize the activity for the whole group. The teacher edition also indicates that this activity could be done using an online tool instead of paper and gives a picture of what this tool looks like.
##### Indicator {{'3h' | indicatorName}}
Materials contain a teacher's edition that contains full, adult--level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials reviewed for enVision Algebra, Geometry, Algebra 2 partially meet expectations for the teacher edition containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge.

The instructional materials provide narrative explanations for answers and solutions in the Teacher Edition and in the Answer and Solutions Application.The Teacher’s Edition include answers for “Do You Understand? And Do You Know How,” and Practice and Problem Solving tasks which include explanations that build teacher understanding of the mathematical content. In the Answer and Solutions Application, teachers are also given narrative explanations for answers and solutions. For example:

• Algebra 1, Teacher Edition, Lesson 4.5, Problem 12, Teachers are provided with the following: “Answer may vary. Sample: A system of two linear inequalities is similar to a system of two linear equations because their solutions are determined by where the graphs of the inequalities or equations intersect or overlap. They are different because a system of linear equations has infinitely many solutions when the two equations in the system are equivalent. A system of linear inequalities can have infinitely many solutions even when the inequalities are not equivalent.
• Geometry Lesson 4.5 Try It! The Answer and Solution Application states “Yes, there are two cases. If the congruent legs are include between the congruent acute angle and the right angles, the triangles are congruent by ASA. If the congruent leds are not included between the congruent acute angles and the right angles, the triangles are congruent by AAS.”
• In Algebra 2, Answers and Solution Application, Lesson 2.5 Practice and Problem Solving Problem 11 “You can compare the zeros of the graph to the solutions you calculated and because completing the square rearranges the equation to vertex form you can also compare the vertex.

The instructional materials make connections in the Topic Overview, Math Bacjground and Coherence between prior knowledge, the lesson, and future content, but do not provide support for building teachers understanding of more advanced mathematical concepts. For example:

• In Geometry Topic 6 Quadrilaterals and Other Polygons, Topic Overview, Math Background Coherence, Looking Ahead identifies Topic 9, Proofs Using Coordinate Geometry and states: “In Lesson 9-2, students will re-examine some Properties learned in Topic 6 and prove these properties using Coordinate Geometry. Inscribed Polygon in Topic 10, students will use the polygon interior Angle-Sum Theorem when studying inscribed angles and polygons. Trigonometry in Algebra 2, students will use polygon angle sums to generalize a formula for the side length of regular polygons.”
• In Algebra 2, Topic 8, Math Background, Coherence, Looking Ahead connects the content of Topic 8, Trigonometric Equations and Identities to Precalculus Trigonometric Identities. “In Lesson 8-3, students verify and apply trigonometric identities and the sum and differences of formulas. In Precalculus, the will verify and evaluate functions involving Multiple-Angle Formulas, Product-to-Sum Formulas, and Sum-to-Product Formulas.”
##### Indicator {{'3i' | indicatorName}}
Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.

The instructional materials reviewed for enVisionMath A/G/A partially meet the expectations that the materials contain a teacher edition that explains the role of the specific mathematics standards in the context of the overall series. The materials rarely explain the role of the specific mathematics standards in the context of the overall series. At the beginning of each topic, the teacher edition provides a page that discusses three types of connections: How the topic connects with what students have learned earlier in the course or previous courses. How different concepts are connected throughout the topic. Finally, it explains how that topic is connected to what students will learn later in the course or future courses. For example, in Algebra 1 Topic 3 Linear Functions, page 86B, there is an explanation of how this topic connects to Grade 8 work on functions as well as solving and graphing linear equations in Topics 1 and 2 of Algebra 1. There is also an explanation of how this topic will connect later in the course to topics such as transformations and quadratic functions and how this topic will connect to topics in Algebra 2. These descriptions do not reference specific mathematical standards.

##### Indicator {{'3j' | indicatorName}}
Materials provide a list of lessons in the teacher's edition, cross-- referencing the standards addressed and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials reviewed for enVisionMath A/G/A provide a list of lessons in the teacher edition, cross-referencing the standards addressed and providing an estimated instructional time for each lesson, topic, and unit. At the beginning of each topic, there is a Topic Planner that lists each lesson, the suggested number of days to spend on each lesson, the objective and essential understanding of each lesson, and the content and practice standards addressed in each lesson. There is no year-at-a-glance pacing guide to give teachers a one-page look at how many days each topic should take, but teachers can find this information by looking at each individual topic.

##### Indicator {{'3k' | indicatorName}}
Materials contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials reviewed for enVisionMath A/G/A provide some information for informing students, parents or caregivers about the mathematics program and suggestions for how they can support student progress and achievement. The materials include Virtual Nerd tutorials for every lesson. These videos can be accessed on any device including smartphones. Students and parents can download the free Virtual Nerd Mobile Math app to access tutorial videos at any time. There are no parent letters in the materials.

Some videos do not support the intentional development of the mathematical reasoning and explanation in connection to the high school content standards, as noted in the video for Algebra 1, Solving and Graphing Linear Inequalities: “Just perform the order of operations in reverse. Don't forget that if you multiply or divide by a negative number, you must flip the sign of the inequality. That's one of the big differences between solving equations and solving inequalities.” This video provides a shortcut for students to use (flip the sign of the inequality) without providing mathematical reasoning or explanation for the shortcut.

##### Indicator {{'3l' | indicatorName}}
Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.

The instructional materials reviewed for enVisionMath A/G/A contain explanations of the instructional approaches of the program and identification of the research-based strategies. The materials state the program is built around three principles: a balanced pedagogy, a focus on visual learning, and a focus on effective teacher and learning. The teaching supports were created using NCTM’s Guiding Principles for School Mathematics, in particular Teaching and Learning. The program uses the Effective Mathematical Teaching Practices as a framework within which probing questions are developed.

#### Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series partially meet expectations for offering teachers resources and tools to collect ongoing data about student progress. The instructional materials provide strategies for gathering information about students' prior knowledge, support for teachers to identify and address common student errors and misconceptions, and clearly denote which standards are being emphasized on assessments. Ongoing review and practice is available in the digital materials but not in the print materials, and the materials do not include guidance for teachers to interpret student performance.

##### Indicator {{'3m' | indicatorName}}
Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for providing strategies for gathering information about students’ prior knowledge within and across courses. There is a pre-assessment that addresses prior knowledge, and this assessment includes content that is not on course level. These online pre-assessments are editable. Answer keys are provided along with a list of the prior standards associated with each item on the assessment. Each topic in the series includes a Topic Readiness assessment, also found online, that provides the same features as the pre-assessment.

##### Indicator {{'3n' | indicatorName}}
Materials provide support for teachers to identify and address common student errors and misconceptions.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for providing support for teachers to identify and address common student errors and misconceptions. The materials highlight common student errors and/or misconceptions for teachers. The materials also provide strategies for teachers in addressing common student errors and/or misconceptions. In the teacher editions there are red boxes titled “Common Error” that describe common errors and misconceptions. There are at least two different descriptions of common errors in each lesson. Examples include:

• In Algebra 1 Lesson 4-2, students solve systems of equations using the substitution method. In example 2, students compare the graphical representation to the substitution method. The materials note that students may incorrectly simplify when removing parentheses and solving for the variable using the substitution method. Teachers are prompted to remind students to use the distributive property and to guide them through a review of this property and how it applies to this context.
• In Geometry Lesson 4-1, example 3, the teacher edition notes that students may have difficulty identifying why one polygon is not congruent to the other polygon when they are very close to being congruent. The teacher edition suggests that teachers focus students’ attention on two specific sides of the polygons and compare the rates of change between the endpoints of each segment.
• In Algebra 2 Lesson 9-1, the teacher edition notes that students may substitute values into the wrong variable in the equation. The materials give information about creating a visual to help students not make this common error.
##### Indicator {{'3o' | indicatorName}}
Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for providing support for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials do not provide support for ongoing review and practice for students in learning concepts. The materials also do not provide support for teachers to provide feedback.

The problems in each lesson address the content taught in that lesson. There is no ongoing review and practice built into the materials. However, the online materials allow teachers to assign a 10-question mixed review for each section. This is not included in the print materials, only in the online materials. Students are given immediate feedback on whether each answer is correct or not. If the answer is incorrect, there is an explanation of the content, and students can try again to get the correct answer. After two or three attempts, the correct solution is shown, and students can choose to try a similar question or move on to the next question. These problems mainly address procedural skills. An example of a conceptual question from this digital source can be found in the mixed review for Algebra 2 Lesson 5-1. The first question states, “If a rational exponent represents the cube root of $$x^m$$, where m is a positive integer, how does the rational exponent change as m increases?” Students choose from a drop-down menu with choices that either the rational exponent increases or decreases as m increases. The remaining nine questions in the mixed review address procedural skills. There is no support for teachers in grading these assessments, as the feedback is provided by the digital device when answers are incorrect. There is no support for teachers in using this information as students progress through these digital online reviews.

##### Indicator {{'3p' | indicatorName}}
Materials offer ongoing assessments:
##### Indicator {{'3p.i' | indicatorName}}
Assessments clearly denote which standards are being emphasized.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for assessments clearly denoting which standards are being emphasized. There are lesson quizzes at the end of each lesson, topic assessments and performance assessments at the end of each topic, four benchmark tests throughout the year, a mid-year assessment, and an end-of-course assessment for each course. Each of these assessments include an answer key and the standards being assessed for each item of the assessment. The benchmark tests, mid-year assessments, and end-of-course assessments are found online, not in the print materials.

##### Indicator {{'3p.ii' | indicatorName}}
Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations that the materials offer ongoing assessments that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The materials do not include guidance for teachers to interpret student performance. The materials provide some suggestions for follow-up after students complete various assessments. Each lesson quiz provides suggestions of differentiated assignments based on scores from the quiz. These assignments include a print or digital assignment called Reteach to Build Understanding, an Additional Practice worksheet, an Enrichment worksheet, and a Vocabulary worksheet. There is no guidance for interpreting student performance, and there are no follow-up suggestions for any of the other assessments.

##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

The instructional materials reviewed for enVisionMath A/G/A do not encourage students to monitor their own progress.

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide teachers with strategies for meeting the needs of a range of learners, tasks with multiple entry-points that can be solved using a variety of solution strategies or representations, support, accommodations, and modifications for English Language Learners and other special populations, and support for advanced students to investigate mathematics content at greater depth.

##### Indicator {{'3r' | indicatorName}}
Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials reviewed for enVisionMath A/G/A partially meet expectations for providing teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners. The materials provide some strategies to scaffold lessons, but there are no general statements about sequencing provided. The lessons do include common misconceptions for the teacher to point out, as well as a section that provides instructions on how to assist struggling students and advanced students; however, these sections only contain additional questions and explanations. They do not contain any information on how to sequence the lesson for any learner.

##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for providing teachers with strategies for meeting the needs of a range of learners. Some general statements for the teacher about meeting the needs of all learners are included. There is also a section in the teacher edition with ideas and guiding questions to support struggling students with each lesson. The questions in this section are typically questions that help reinforce concepts students need to be successful in the lesson or questions that help build a student’s conceptual understanding of the lesson. There is also a section in the teacher edition for advanced students, which contains more complex problems for them to complete.

For example, in Geometry Topic 8-4, Teacher Edition page 36B, advanced students deepen their understanding as they explore the right triangle case of the Law of Cosines. Struggling students review how to use the Law of Cosines based on the abbreviations for triangle congruence.

##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for embedding tasks with multiple entry points that can be solved using a variety of solution strategies or representations. The materials provide teachers with guidance on helping students solve problems with multiple entry points. Some of the Mathematical Modeling in 3 Acts and STEM projects give students multiple entry points to a problem as well as allow students to try a variety of methods to solve the problem. Examples include:

• In Algebra 1 Topic 3, Mathematical Modeling in 3 Acts, students determine whether it would be faster to check out in a regular line or an express line at a grocery store. Students are given information about a hypothetical situation involving a certain number of customers in each line as well as how many items each customer has. There are various ways students could approach this problem.
• In Geometry Topic 3, STEM project, students draw a figure of any shape in a coordinate grid and write a script about this figure, including several transformations. Students can choose how simple or complex their figure will be and how simple or complex their script will be.

The teacher edition embeds Mathematical Modeling in 3 Acts into the pacing guide, but it does not include the STEM projects into this guide. There are no suggestions of where to use the STEM projects or how much time to allow for them.

##### Indicator {{'3u' | indicatorName}}
Materials provide support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials reviewed for enVisionMath A/G/A meet expectations for providing support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in the learning of mathematics. Within each topic in the teacher edition, the series provides guidance for beginning, intermediate, and advanced ELL students. Students are also given access to read and listen for English and Spanish definitions. Each lesson provides a Concept Summary which includes definitions of the introduced vocabulary. Students are provided multiple representations of the concepts addressed within the lesson.

There are strategies for special populations to practice vocabulary as seen in Algebra 2 Lesson 4-2, Teacher Edition page 209B, Mathematical Literacy and Vocabulary, which help ELL and other special populations to develop and reinforce understanding of key terms and concepts. Each practice section begins with “Do you understand,” from which teachers can modify pacing for special populations as needed.

##### Indicator {{'3v' | indicatorName}}
Materials provide support for advanced students to investigate mathematics content at greater depth.

The instructional materials reviewed for enVisionMath A/G/A meet expectations for providing support for advanced students to investigate mathematics content at a greater depth. The materials provide multiple opportunities for advanced learners to investigate the course-level mathematics at a greater depth. There are no instances of advanced students doing more problems than their classmates. Each topic begins with a Topic Readiness Assessment which, based on students’ performance, assigns a study plan tailored to student’s specific needs, including advanced students. Each lesson provides teachers with additional problems for advanced students. Examples of enrichment for advanced learners include:

• In Geometry Lesson 7-1, Teacher Edition page 305, an extension is provided of a problem on dilations.
• In Algebra 2 Lesson 3-2, Teacher Edition page 141, an extension is provided for advanced students to extend their work with polynomial functions.

Teachers are also provided an assignment guide for advanced students for each lesson.

##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials reviewed for enVisionMath A/G/A provide a balanced portrayal of various demographic and personal characteristics. The photos and illustrations of people show a variety of demographics. The names and situations portrayed in the series are diverse.

##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials reviewed for enVisionMath A/G/A provide opportunities and directions for teachers to use a variety of grouping strategies. The materials focus on mathematical discourse, collaboration, teamwork, individualized work, and whole group. Examples include:

• In Geometry Topic 11, STEM project, students work with a small group to design a package for a product of their choice. The students are given various factors to design their package and defend their choice to other students.
• In Algebra 2 Lesson 3-2, Teacher Edition page 130, teachers guide students utilizing whole group, small group, and individual work as noted in the margins.
##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

The instructional materials reviewed by enVisionMath A/G/A do not encourage teachers to draw upon home language and culture. At the beginning of every lesson, there is a Vocabulary Building activity in the teacher edition that focuses on both mathematical vocabulary and academic vocabulary. The activities that launch each lesson promote and reinforce key language skills of speaking and listening as students defend their solutions strategies. Learner strategies sometimes provide guidance for teachers on how to engage students with different levels of language acquisition; however, the materials do not provide guidance on how to integrate home language into daily classroom activities.

#### Criterion 3.5: Technology Use

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series integrate technology in ways that engage students in the Mathematical Practices and are web-based and compatible for multiple internet browsers. The instructional materials also include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology, to personalize learning for all students, and to easily customize for local use. The instructional materials do not include or reference technology that provides opportunities for teachers and/or students to collaborate with each other.

##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Mac and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The instructional materials reviewed for enVisionMath A/G/A are web-based and compatible for multiple internet browsers. In addition, materials are “platform neutral” and allow the use of tablets and mobile devices.

• The Virtual Nerd Mobile Math app is accessible on iOS and Android devices.
• Materials are compatible with various devices including iPads, laptops, Chromebooks, and other devices that connect to the internet with a browser.
##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials reviewed for enVisionMath A/G/A include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. In Pearson Realize, teachers can assign the same examples from the materials. From these examples, students receive immediate feedback on whether their answers are correct or incorrect. Teachers can assign additional problems that address procedural skills and conceptual understanding for the students to complete for each individual lesson. In addition to assigning problems for each individual lesson, teachers can also assign quizzes and assessments within each topic and across topics.

##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners.
##### Indicator {{'3ac.i' | indicatorName}}
Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.

The instructional materials reviewed for enVisionMath A/G/A include opportunities for teachers to personalize learning for all students using adaptive or other technological innovations. Adaptive practice and homework powered by Knewton provide personalized practice based on each student’s strengths and areas for improvement.

##### Indicator {{'3ac.ii' | indicatorName}}
Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The instructional materials reviewed for enVisionMath A/G/A can be easily customized for local use. For each topic throughout the entire series, students are provided an online Topic Readiness Assessment. This assesses students' understanding of prerequisite concepts and skills. These are auto-scored, online assessments, and based on their performance, students could be assigned a study plan tailored to their specific learning needs. Additionally, after each lesson of each topic throughout the series, students can complete a quiz to assess their understanding of the mathematics in the lesson. Teachers may use the students' scores to prescribe differentiated assignments. Assignments and assessments can be edited. Teachers have access to problem banks to customize assignments and assessments as well.

##### Indicator {{'3ad' | indicatorName}}
Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The instructional materials reviewed for enVisionMath A/G/A do not include or reference technology that provides opportunities for teachers and/or students to collaborate with each other. There is no reference to any type of technology that allows for collaboration.

##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The instructional materials reviewed for enVisionMath A/G/A integrate technology such as interactive tools, virtual manipulatives/objects, and dynamic mathematics software in ways that engage students in the Mathematical Practices. Each lesson has embedded digital math tools that are accessible during instruction. Online practice powered by MathXL offers differentiated assignments and daily homework and assignments. For example:

• Geometry Lesson 4-1 utilizes Desmos to explore a sequence dividing a circular area with chords to find a pattern.
• In every lesson, students can complete in-class work in Pearson Realize.
• Students can access instructional tutorials using the Virtual Nerd app.

## Report Overview

### Summary of Alignment & Usability for enVision A/G/A | Math

#### Math High School

The instructional materials reviewed for the enVision A/G/A: Algebra 1, Geometry, & Algebra 2 series meet expectations for alignment to the CCSSM for high school, Gateways 1 and 2. In Gateway 1, the instructional materials meet the expectations for focus and coherence by being coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM). In Gateway 2, the instructional materials meet the expectations for rigor and balance by reflecting the balances in the Standards and helping students meet the Standards' rigorous expectations, and the materials meet the expectations for mathematical practice-content connections by meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice.

##### High School
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

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###### Usability
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