Alignment: Overall Summary

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for Alignment to the CCSSM. The materials meet expectations for Focus and Coherence as they show strengths in: attending to the full intent of the mathematical content for all students; spending the majority of time on content widely applicable as prerequisites; engaging students in mathematics at a level of sophistication appropriate to high school; and making meaningful connections in a single course and throughout the series. The materials meet expectations for Rigor and Mathematical Practices as they meet expectations for Rigor and Balance and meet expectations for Practice-Content Connections. Within Rigor and Balance, the materials show strengths in providing students opportunities to develop conceptual understanding, procedural skills, and application, and the materials balance the three aspects of Rigor. Within Practice-Content Connections, the materials show strengths in developing: overarching, mathematical practices (MPs 1 and 6); reasoning and explaining (MPs 2 and 3); and seeing structure and generalizing (MPs 7 and 8).

See Rating Scale Understanding Gateways

Alignment

|

Meets Expectations

Gateway 1:

Focus & Coherence

0
9
14
18
14
14-18
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Gateway 2:

Rigor & Mathematical Practices

0
9
14
16
15
14-16
Meets Expectations
10-13
Partially Meets Expectations
0-9
Does Not Meet Expectations

Usability

|

Meets Expectations

Not Rated

Gateway 3:

Usability

0
21
30
36
30
30-36
Meets Expectations
22-29
Partially Meets Expectations
0-21
Does Not Meet Expectations

Gateway One

Focus & Coherence

Meets Expectations

Criterion 1a - 1f

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).
14/18
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Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for Focus and Coherence. The materials meet expectations for: attending to the full intent of the mathematical content for all students; spending the majority of time on content widely applicable as prerequisites; engaging students in mathematics at a level of sophistication appropriate to high school; and making meaningful connections in a single course and throughout the series. The materials partially meet expectations for the remaining indicators in Gateway 1, which include: attending to the full intent of the modeling process; allowing students to fully learn each standard; and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards.

Indicator 1a

The materials focus on the high school standards.*
0/0

Indicator 1a.i

The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
4/4
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The instructional materials address most aspects of all non-plus standards across the courses of the series.

Examples of non-plus standards fully addressed by the series include, but are not limited to:

  • N-RN.1,2: In Mathematics I, Lesson 5-1, multiple examples address square root and rational relationships and identify the product of powers property. Students work with the properties throughout the series, as in Mathematics II, Lesson 1-2, and Mathematics III, Lesson 4-1.
  • A-SSE.3c: In Mathematics I, Lesson 5-3, students work with changing the format as indicated by the standard, and in Mathematics II, Lesson 1-3, students work with basic interest problems. In Mathematics III, Lesson 5-2, Example 1, students apply the power rule to show the change of form of the equations, and Example 2 includes rewriting an expression with exponents using the properties to transform the expressions.
  • F-IF.4: In Mathematics I, Lesson 5-2, Example 1 defines asymptotes, the end behavior of a graph as x approaches negative infinity, and the behavior of a graph as y approaches 0. Mathematics I, Lesson 5-5, Example 3 discusses end behavior as the graph approaches -1. In Mathematics II, Lesson 3-3, multiple examples address y-intercepts, vertex, axis of symmetry, and a context problem. Mathematics II, Lesson 6-1, Example 3 addresses increasing absolute value functions and the axis of symmetry, and Mathematics III, Lesson 6-4, Example 1 addresses the graph of a periodic function. 
  • G-CO.7: Mathematics I, Lesson 9-3, Example 1 includes a video that explains how students prove triangles are congruent through rigid motions, and Theorem 9-4 makes this explicit.

 Non-plus standards partially addressed by the series include:

  • F-LE.1a: In Mathematics I, Lessons 3-2 (Linear) and 5-2 (Exponential), students are shown linear and exponential functions growing by equal differences and equal factors, but no proof is included in the materials for either type of function.
  • G-CO.2: In Mathematics II, Lesson 9-1, students work with dilations, but no evidence was found where students are shown or compare the transformations that preserve distance and angle measures to those that do not.
  • G-CO.8: In Mathematics I, Lessons 9-3 and 9-4, rigid motions are referenced, but there was no evidence that the students had to explain the connection between the criteria for triangle congruence and the definition of congruence in terms of rigid motions.
  • G-C.3: In Mathematics III Lesson 9-1, Part A of the lesson, (marked "Extension"  and may not be required) students construct the circles but do not prove properties for quadrilaterals. They are not asked for or shown proof in Part B or Part C of the lesson.
  • G-GPE.7: In Mathematics I, Lesson 9-7, students use coordinates to compute perimeter and area of polygons, however, there was no evidence found of computing perimeter or area using coordinates of a rectangle.
  • G-GMD.1: In Mathematics II, Lessons 13-2 and 13-3, students work with the formulas for the volume of a cylinder, pyramid, and cone and apply Cavaleri's Principle to pyramids and cones. There was no evidence in the series where circumference and area of a circle were addressed.
  • S-ID.4: In Mathematics III, Lesson 8-3, multiple examples introduce students to the appropriateness of procedures for estimating population proportions. The series does not address using technology and calculators to estimate the areas under the Normal Curve.

Standard not addressed by the series include:

  • A-REI.10: No evidence was found in the series with regard to a curve (or a line) representing all of the solutions plotted in the coordinate plane to an equation in two variables.

Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.
1/2
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 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.1 and A.SSE.3c: In Mathematics I, Lesson 1-3, students use a formula to find the dimensions of the flag with the variable given. Students do not make conjectures and are not asked to validate or interpret answers. No evidence was found where students have the opportunity to engage in the full modeling cycle with these standards.
  • N-Q.2: In Mathematics I, Lesson 1-2, students have opportunities to define quantities, formulate equations, and compute and interpret results. Students answer specific questions that guide them but do not conjecture or validate their choices.
  • A-SSE.1b: In Mathematics I, Lesson 5-3, Practice and Problem Solving, Problem 27, students use an exponential growth function to model and explain how they found the solution, but the students do not make a conjecture because the conjecture is already given in the problem. In Mathematics II, Topic 1, students use tools to find an exponential model for the given data but do not interpret results.
  • A-SSE.3b: In Mathematics II, Lesson 5-2, Performance Task, students formulate a function that represents increasing area by adding to given dimensions. Students compute using the function to find the new dimensions. In Mathematics II, Lesson 1-3, Practice & Problem Solving, Problems 14 and 30, are described as "model with mathematics." However, students formulate and compute, but students do not make assumptions, predictions, or interpret results.
  • A-SSE.4: In Mathematics III, Lesson 5, students formulate and compute in many problems aligned to this standard. Students do not encounter extraneous solutions and do not summarize conclusions, make assumptions and conjectures, or validate answers.
  • A-REI.11: In Mathematics I, Lesson 4-1, students are led through the formulation of an equation set and creation of a graph which they interpret. In the Try it! problems, students repeat the steps themselves. In Mathematics III, Lesson 1-5, Practice and Problem Solving, Problem 32, students solve a system of equations related to revenue and expenses. However, students are provided what to do rather than engaging in the complete modeling cycle in either course.
  • F-IF.5: In Mathematics I, Lesson 3-2, Practice and Problem Solving, Problem 29, students are directed what to do in Part A to solve the main question written in Part B. Students do not make conjectures, but are directed in how to proceed.
  • F-IF.7a: In Mathematics II, Lesson 3-3, Performance Task, Problem 42, students use the information given to write a quadratic model. Students are provided with a quadratic function, a quadratic graph, and data points for a third quadratic relationship. Students identify maximum profits for each scenario and explain their results. Students formulate a function for the third model, compute and interpret maximized results for all three situations, and explain/validate their results. However, each situation has one solution, and students do not predict the best model or justify their responses.
  • G-GPE.7: In Mathematics I, Lesson 9-7, Practice and Problem Solving, Problem 26, students find one solution. In Practice and Problem Solving, Problem 28, students formulate, compute, interpret, validate and report, but since there is one specific answer, students do not revise their work.

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

  • In Mathematics I, Topic 2, Mathematical Modeling in 3 Acts, students determine a basketball player's height in unconventional ways, which culminates in determining the height based on the height of foam cups. Students use their knowledge of linear functions and data given about smaller stacks of cups to determine his height. Students make predictions, report their findings, and compare their solutions to the findings presented in the final video. Students represent the relationship between the number of cups and the height of the stacks to generate more data. (A-CED.1,3,4)
  • In Mathematics III, Topic 8, STEM project, students conjecture, make a model, calculate, justify, validate, and report their conclusions. Students use statistical surveys to decide how best to use public spaces. Students plan a new public space by deciding how to gather data, analyze the data, and report the conclusions. (S-IC.B)

Indicator 1b

The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
0/0

Indicator 1b.i

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.
2/2
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for (when used as designed) spending the majority of time on the CCSSM “widely applicable as prerequisites (WAPs),” for a range of college majors, postsecondary programs, and careers.

  • In Mathematics I, the materials address the WAPs in the conceptual categories of Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability. The majority of the lessons in Mathematics I address the WAPs, and there were only a few lessons that did not include a WAP.
  • In Mathematics II, the materials address the WAPs in the conceptual categories of Number and Quantity, Algebra, Functions, and Geometry. The majority of the lessons in Mathematics II address the WAPs, and there are fewer lessons that focus on WAPs compared to Mathematics I.
  • In Mathematics III, the materials do not spend the majority of the lessons on the WAPs. The materials address the WAPs in the conceptual categories of Number and Quantity, Algebra, Functions, and Statistics and Probability. 

Examples of how the materials allow students to spend the majority of their time on the WAPs include:

  • Throughout the series, students get several opportunities to work with standards from F-IF. In Mathematics I, Topic 1, students solve linear equations and inequalities in one variable and work with absolute value equations. In Topic 2, students graph, solve, and manipulate linear equations. In Mathematics II, Topic 4, students solve quadratic equations by factoring, using square roots, graphs, and tables. In Mathematics III, Topic 5, students convert equations between logarithmic and exponential forms, solve logarithmic and exponential equations, graph logarithmic equations, and explore properties of logarithms.
  • In Mathematics I and II, students engage with G-SRT.5 by exploring the properties of similarity. In Mathematics I, Lessons 9-2 through 9-6, students explore the relationship between similarity and triangle congruence. In Mathematics II, Lessons 7-5, 8-1, 8-3 through 8-6, and 9-2 through 9-4, students explore similarity relationships with quadrilaterals.
  • The series addresses A-SSE.2 throughout Mathematics II and III in a variety of places with a variety of practice and expressions (exponential, cubic, quadratic, etc.). In Mathematics II, Lesson 1-3, Example 2 includes using the structure to rewrite an interest formula. In Lesson 2-7, Example 1 explores perfect-square trinomials and writing equivalent expressions for those containing perfect-square trinomials. In Lesson 5-5, Example 2 includes the use of polynomial identities to multiply polynomials after rewriting the expression using its structure. In Mathematics III, Lesson 5-2, students use the structure of exponential functions to write and rewrite models.

Indicator 1b.ii

The materials, when used as designed, allow students to fully learn each standard.
2/4
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for (when used as designed) letting students fully learn each non-plus standard. The instructional materials for the series (when used as designed) do not enable students to fully learn some of the non-plus standards. For the most part, students do not have the opportunity to independently prove theorems throughout the series.

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

  • N-Q.1: Mathematics I, Lessons 1-3 and 9-7, and Mathematics II, Lesson 6-1 are examples of lessons throughout the series in which units and scales are provided for students, but students do not “choose and interpret units consistently in formulas.”
  • N-Q.3: In Mathematics III, Lesson 9-1, students perform basic constructions using a ruler in Example 1, but students do not connect the constructions with taking appropriate measurements. Throughout the series, a level of accuracy appropriate to limitations on measurement is not discussed or modeled, and students are not given sufficient opportunities to practice choosing a level of accuracy when reporting quantities.
  • A-APR.4: In Mathematics II, Lesson 5-5, students prove the sum of cubes and difference of cubes' identities. Two other identities are proven, but students do not use them to describe numerical relationships.
  • A-REI.1: In Mathematics I, 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.
  • A-REI.5: In Mathematics I, Lesson 5-4, students practice solving systems of equations by elimination; however, students do not prove (given a system of two equations in two variables) that 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 Mathematics I, Lesson 3-4, Example 1 shows a sequence being a function. However, there are no other opportunities found for students to recognize that sequences are functions. In Mathematics III, Lesson 1-4, students determine whether each sequence is arithmetic, but students do not develop a connection between sequences and functions.
  • F-IF.7: Throughout the series, students do not use technology to create graphs, as graphs and screenshots from technology are consistently provided for students.
  • F-LE.3: In Mathematics II, Lesson 3-5, students compare linear and exponential graphs to determine which function will exceed the other in one problem, and in Mathematics III, Lesson 5-3, Problem 17, students determine when an exponential function will exceed a linear function and a quadratic function.
  • F-TF.8: In Mathematics II, Lesson 9-7, students use the Pythagorean identity to find trigonometric values, and in Mathematics III, Lesson 6-1, Example 2, students use one trigonometric ratio to find another trigonometric ratio using the Pythagorean Theorem. In Mathematics III, Lesson 6-3, students also use the Pythagrean identity, but students do not prove the Pythagorean identity.
  • G-SRT.2: In Mathematics II, Lessons 9-1and 9-2, students determine whether or not triangles are similar. Students do not use similarity transformations to explain the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Indicator 1c

The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 Mathematics I, Lesson 4-2, students encounter problems about a lawn mowing business and going to an amusement park. 
  • In Mathematics II, Topic 4, STEM Project, students use quadratic functions to model parabolic vertical motion and design an appropriate t-shirt launcher.
  • In Mathematics III, Topic 4, Problem 34, students write a function for the cost of hiring a DJ for an event.

Some examples where students apply key takeaways from Grades 6-8 include:

  • In Mathematics I, Lesson 2-1, Model and Discuss, students use proportional relationships to determine which payment plan would be better.
  • In Mathematics I, Lesson 3-2, Problem 30, students write a linear function to model the data and determine the total cost of a job that took 4 hours and 15 minutes of labor. 
  • In Mathematics II, Lesson 1-1, Problems 8 and 9, students order a set of numbers from least to greatest including integers, rational numbers, and irrational numbers.
  • In Mathematics II, Lesson 8-2, Problem 22, students use a diagram of a map to determine a 2 mile run for the team, applying a conversion of yards to miles and interpreting the map.
  • In Mathematics II, Lesson 9-1, Explore and reason, students use similarity understanding to build dilated figures.
  • In Mathematics II, Lesson 10-2, students calculate probabilities to develop understanding of the probability of mutually exclusive events.
  • In Mathematics III, Lesson 8-7, students determine if games of probability and chance are fair and determine percentages based on probabilities.

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

  • In Mathematics I, Lesson 1-1, students solve linear equations including all types of rational numbers (positive and negative integers, decimals, and fractions) within examples and practice problems.
  • In Mathematics I, Lesson 2-2, the problem set has integers and fractions in the equations for students to graph.
  • In Mathematics I, Lesson 6-3, students calculate midpoints for number sets that include negative numbers, decimals, and fractions.
  • In Mathematics II, Lesson 4-4, students solve quadratic equations including irrational numbers, and Lesson 5-1 includes operations with complex numbers.
  • In Mathematics III, Lesson 2-6, students discuss the rational root theorem for polynomial functions, along with irrational and complex roots.
  • In Mathematics III, Lesson 4-3, students solve equations containing square and cube roots.

Indicator 1d

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.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. The instructional materials make meaningful connections within and across courses, and the materials provide guidance so that multiple lessons are not repeated across the courses of the series.

Examples of the materials making meaningful connections within courses include:

  • In Mathematics I, Lesson 1-1, students use linear equations in one variable to solve problems (A-CED.1). In Lessons 2-1 and 2-2, students use linear equations in two variables to represent relationships between two quantities and graph those relationships on the coordinate plane (A-CED.2). In Lesson 4-4, students represent inequalities in two variables and solve problems in context (A-CED.3).
  • In Mathematics II, Lessons 7-1, 7-2, and 7-3, students prove relationships about lines and angles (G-CO.9), and in Lessons 8-3 and 8-4, students use those angle relationships to prove properties of parallelograms(G-CO.11). In Lesson 9-3, students prove triangle similarity relationships using angle and line relationships (G-CO.10).
  • In Mathematics III, Lesson 1-1, students review the key features of functions (F-IF.7). In Lesson 1-3, students graph absolute value and piecewise functions by identifying the maximum value, minimum value, and other key features of the graph (F-IF.7b). In Lesson 2-5, students find the zeros of polynomials and describe their behavior at each zero (F-IF.7c).

Examples of the materials making meaningful connections across courses include: 

  • In Mathematics I, Lesson 5-2, students graph and label key features of exponential functions. In Mathematics II, Lesson 3-1, students identify key features of quadratic functions, and in Mathematics III, Lesson 2-1, students determine the end behavior of polynomial functions (F-IF.4).
  • In Mathematics I, Lesson 3-3, students describe how adding k affects the graph of a line. In Mathematics II, Lesson 3-1, students describe how the leading coefficient of a quadratic equation affects the graph of that equation. In Mathematics III, Lesson 5-1, students identify the effect on graphs by replacing values for x (F-BF.3). 
  • In Mathematics II, Lesson 6-1, students use their knowledge of linear distance to develop an understanding of absolute value as a function and that each point has a corresponding point equidistant from the vertex of the graph of an absolute value function. In Mathematics III, Lesson 6-1, students use similar triangles, ratio reasoning, and trigonometric identities to develop trigonometric functions (F-IF.7b; F-TF.2). 
  • In Mathematics I, Chapter 9, students prove congruent relationships in triangles and other geometric figures. In Mathematics II, Lesson 9-2, students use similarity transformations to determine if polygons and triangles are similar. In Mathematics III, lesson 6-1, students apply their knowledge of triangle similarity and congruence to side ratios in right triangles (G-SRT.5,6). 
  • In Mathematics I, Lesson 1-4, students write linear equations for different contexts. In Mathematics II, Lesson 3-4, students create a quadratic equation with two or more variables given a context. In Mathematics III, Lesson 1-5, students create linear, absolute value, and quadratic equations from given contexts (A-CED.1,2).

Below is the list of lessons in Mathematics III that are duplicated from previous courses in the series. In the Teacher Resources for each course, there is a document entitled enVision Integrated CC Pathway. The document includes the lessons that should be included in each course and the courses from which the repeated lessons should be omitted. The guidance provided by the document eliminates disruptions to the coherence of the materials that would occur if lessons were repeated from one course to another.

  • Mathematics III Lesson 9-1 is a duplicate of Mathematics I Lesson 6-2 (N-Q).
  • Mathematics III Lesson 5-2 is a duplicate of Mathematics II Lesson 1-3 (A-SSE.3).
  • Mathematics III Lesson 2-3 is a duplicate of Mathematics II Lesson 5-5 (A-APR.4, A-REI.4).
  • Mathematics III Lesson 9-4 is a duplicate of Mathematics II Lesson 11-2 and Lesson 10-1 is a duplicate of Mathematics II, Lesson 12-1. (G-CO.1)
  • Mathematics III Lesson 9-3 is a duplicate of Mathematics II Lesson 11-1. (G-CO.10)
  • Mathematics III Lessons 10-1 through 10-5 are duplicates of Mathematics II Lessons 12-1 through 12-5. (G-C.2,4,5)
  • Mathematics III Lessons 9-5 and 9-6 are duplicates of Mathematics II Lessons 11-3 and 11-4. (G-GPE.1-3)
  • Mathematics III Lessons 11-1 through 11-4 are duplicates of Mathematics II Lessons 13-1 through 13-4. (G-GMD.1-3)
  • Mathematics III Lessons 11-2 and 11-4 are duplicates of Mathematics II Lessons 13-2 and 13-4. (G-MG.1,2)

Statistics and Probability standards (S-ID.C, S-IC, S-CP, and S-MD) are primarily addressed in Mathematics I or Mathematics III and are not often connected to other categories of standards within or across courses.

Indicator 1e

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials do not explicitly identify content from Grades 6-8. There are limited places where “6-8” is written, but specific standards are not mentioned in the teacher or student materials.

Each chapter throughout the series has Math Background Coherence, which includes Looking Back, This Topic, and Looking Ahead. In these sections, concepts and skills from Grades 6-8 are mentioned, but domains, clusters, or standards are not explicitly identified.

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

  • In Mathematics I, Teacher Edition, the Topic 6 Overview states, “In Grade 8, students used lines, segments, rays, and angles.”
  • In Mathematics II, Teacher Edition, the Topic 3 Overview indicates the grade 8 content of analyzing graphs of functions as background knowledge for connecting to quadratic functions.
  • In Mathematics III, Teacher Edition, the Topic 8 Overview states, “In Grade 6 students used dot plots, box plots, and histograms to represent data.”

Some examples where the materials make connections between Grades 6-8 and high school concepts and build on students’ previous knowledge include: 

  • In Mathematics I, Topic 3 Overview, Teacher Edition, students explore linear and nonlinear functions as well as key features of linear functions that include slope and rate of change. Students will extend their understanding to determine the domain and range of linear functions, write linear function rules, and transform linear functions.
  • In Mathematics I, Topic 10 Overview, Teacher Edition, students use dot plots, boxplots, and histograms to represent data. Students will extend their understanding of these data displays to interpret and compare data.
  • In Mathematics II, Topic 2 Overview, Teacher Edition, students factored expressions by identifying the greatest common factor and using the distributive property (in Grade 6). This will be used in Topic 2 when factoring polynomials, including trinomials with binomial factors.
  • In Mathematics II, Topic 7 Overview, Teacher Edition, students studied the radius of the circle (in Grade 7), and in grade 8, they learned to identify congruent figures using transformations. In Topic 7, students study the relationships and side lengths in a single triangle. In Mathematics III, students will extend the relationship of triangles and circles when they make sense of trigonometric ratios.

Indicator 1f

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.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics explicitly identify the plus standards and use the plus standards 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 Edition explicitly identify the plus standards. At the beginning of each lesson is 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 Mathematics II, Lesson 5-1, students complete operations with complex numbers, including using the conjugate to rationalize a complex denominator and finding solutions to quadratic equations with imaginary solutions.
  • N-CN.8: In Mathematics III, Lesson 2-6, students are provided one complex root and determine the equation with that root.
  • N-CN.9: In Mathematics II, Lesson 5-5, students prove the fundamental theorem of algebra for quadratic equations in Example 2 and use this example to show that two equations have two complex roots in Try it Problem 2. In Mathematics III, Lesson 2-6 defines the fundamental theorem of algebra and shows its application to quadratic equations in Example 4. 
  • A-APR.5: In Mathematics II, Lesson 5-5, students explore the binomial theorem and its application to expanding binomials. In Example 4 and Try it!, Problem 4, students expand powers of binomial expressions using Pascal's Triangle. This lesson is repeated in Mathematics III, Lesson 2-3.
  • A-APR.7: In Mathematics III, Lesson 3-3, students multiply and divide rational expressions, and in Lesson 3-4, students add and subtract rational expressions.
  • A-REI.8,9: In Mathematics III, Lesson 1-6 (digital only), students write a system of equations as a matrix equation and solve the matrix equation. This lesson is supposed to be repeated in Mathematics III, Lessons 6-6 and 6-7, but those lessons do not exist. 
  • F-IF.7d: In Mathematics III, Lessons 3-1 and 3-2, students graph a rational function, including identifying asymptotes, intercepts, and end behavior.
  • F-BF.1c: In Mathematics III, Lesson 4-5, students compose functions, write a rule for composite functions, and use a composite function model. 
  • F-BF.4b,c: In Mathematics III, Lesson 4-6, students use compositions to verify inverse functions and tables and graphs to explore and find inverse functions. In Mathematics III, Lesson 5-5, students explore inverses of logarithmic functions. 
  • F-BF.4d and F-TF.6: In Mathematics III, Lesson 7-1, students find the inverse of trigonometric functions by restricting the domain.
  • F-BF.5: In Mathematics III, Lesson 5-5, students explore how exponential and logarithmic functions have an inverse relationship. In Lesson 5-7, students use the inverse relationship between logarithms and exponents to solve problems.
  • F-TF.4: In Mathematics III, Lesson 7-4, Example 1 explains that f(x)=sin x is an odd function since f(-x) = -f(x).
  • F-TF.7,9: In Mathematics III, Lessons 7-1 and 7-4, students use inverse trigonometric functions to solve problems and evaluate them with technology. In Lesson 7-4, students prove the sum and difference formulas for sine and cosine and use them to solve problems.
  • G-SRT.9: In Mathematics III, Lesson 7-3, Example 4, the materials guide students to use sine to find the area of non-right triangles. 
  • G-SRT.10,11: In Mathematics III, Lesson 7-2, Example 1 proves the law of sines, and Example 4 proves the law of cosines. Students use the laws to solve problems in practice problems. 
  • G-C.4: In Mathematics III, Lesson 10-2, Example 5 shows the construction of a tangent line to a circle.
  • G-GMD.2: In Mathematics II, Lesson 13-2 uses Cavalieri's Principle with square and rectangular prisms and cylinders. Lesson 13-3 uses Cavalieri's Principle with cones and pyramids, and in Lesson 13-4, Example 1 connects the volume of a sphere to the volume of a cylinder and Cavalieri's Principle.
  • S-CP.8: In Mathematics II, Lesson 10-3, students apply the conditional Probability formula and use the conditional probability formula to make a decision. 
  • S-CP.9: In Mathematics II, Lesson 10-4 addresses permutation and combinations.
  • S-MD.1: In Mathematics II, Lesson 10-5, Example 1 develops a theoretical probability distribution including assigning numerical values to events in a sample space and graphs the possibilities. Example 2 develops an experimental probability distribution, graphs the distribution, and interprets the results. 
  • S-MD.5: In Mathematics II, Lesson 10-6, students evaluate and apply expected values using percent sold as probability, fine expected payoffs, and evaluate strategies for an auto insurance policy.
  • S-MD.6,7: In Mathematics II, Lesson 10-7, students use probability to make fair decisions. This lesson is duplicated in Mathematics III, Lesson 8-7.

The (+) standards that are partially addressed include:

  • F-TF.3: In Mathematics III, Lesson 6-1, students evaluate trigonometric ratios in special triangles using degrees but not radians. In Lesson 7-4, students again use special right triangles, but the connection between special right triangles and the unit circle is not made.
  • G-GPE.3: In Mathematics II, Lesson 11-3, students work with circles but not with ellipses. In Lesson 11-4, students work with parabolas. 
  • S-MD.2-4: In Mathematics II Lesson 10-5, students calculate expected values but do not connect them to probability distributions.

The (+) standards that are not addressed include:

  • N-CN.4-6
  • N-VM.1-12

Gateway Two

Rigor & Mathematical Practices

Meets Expectations

Criterion 2a - 2d

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.
8/8
+
-
Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for Rigor and Balance. The materials meet expectations for providing students opportunities in developing conceptual understanding, procedural skills, and application, and the materials also meet expectations for balancing the three aspects of Rigor.

Indicator 2a

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.
2/2
+
-
Indicator Rating Details

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

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

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

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

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

Indicator 2b

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.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for providing intentional opportunities for students to develop procedural skills, especially where called for in specific content standards or clusters. The instructional materials develop procedural skills while also providing opportunities for students to independently demonstrate procedural skills throughout the series.

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

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

Indicator 2c

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.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Throughout the series, students engage in a variety of application-based problems through STEM projects for each topic, Apply in Practice & Problem Solving, Performance Tasks in Assessment practice, and Mathematical Modeling in 3 Acts embedded into topics.

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

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

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

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

Indicator 2d

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.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations that the three aspects of rigor are not always treated together and are not always treated separately.

All three aspects of rigor are present independently throughout the program materials. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Each topic in this series includes a Topic Opener, STEM Project, and Mathematical Modeling in 3 Acts (which relates to the Topic Opener). Each lesson includes: Explore and Reason, Understand and Apply (which guides students through examples and problems to try on their own), Concept Summary, Practice and Problem Solving, and a formative quiz to determine understanding and mastery. This structure of the materials lends itself to balancing the three aspects of rigor.

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

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

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

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

Criterion 2e - 2h

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
7/8
+
-
Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for Practice-Content Connections. The materials intentionally develop the following mathematical practices to their full intent: make sense of problems and persevere in solving them (MP1), reason abstractly and quantitatively (MP2), construct viable arguments and critique the reasoning of others (MP3), model with mathematics (MP4), attend to precision (MP6), look for and make use of structure (MP7), and look for and express regularity in repeated reasoning (MP8).

Indicator 2e

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.
2/2
+
-
Indicator Rating Details

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

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

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

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

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

Indicator 2f

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.
2/2
+
-
Indicator Rating Details

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

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

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

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

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

Indicator 2g

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.
1/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for supporting 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.

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 Mathematics I, Lesson 3-4, students explain what information is needed to write a linear equation that represents a pattern.
  • In Mathematics I, Topic 8, Mathematical Modeling in 3 Acts, students use geometric transformations to solve design problems.
  • In Mathematics II, Lesson 1-2, students determine if they can use the same properties of exponents for expressions with rational exponents as they do with integer exponents.
  • In Mathematics II, Lesson 1-3, students use functions to describe how the value of land varies over time and draw conclusions about when a bowl of soup is cool enough to consume. 
  • In Mathematics III, Lesson 4-6, students write a rule for number path f, which is a set of written mathematical commands. Students also write a rule for following the number path backward and compare the rules. 
  • In Mathematics III, Lesson 7-6, Mathematical Modeling in 3 Acts, students write an equation to model a given diagram or graph of trigonometric functions.

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 Mathematics I, Lesson 6-2, 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 Mathematics I, 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 Mathematics I, Lesson 8-1, there are no tools or choice of tools for students to use when working with reflections.
  • In Mathematics III, 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 Mathematics III, Lesson 7-3, students are directed to use diagrams and sketches to plan solutions to trigonometric problems.
  • Students do not use multiple tools to represent information in a situation or demonstrate modeling effectively with tools in the data sections of the Mathematics III 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.

Indicator 2h

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.
2/2
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-
Indicator Rating Details

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

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

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

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

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

Gateway Three

Usability

Meets Expectations

Criterion 3a - 3e

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.
7/8
+
-
Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

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.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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, Model & Discuss, or Critique & Explain 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

Design of assignments is not haphazard: exercises are given in intentional sequences.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for having a design of assignments that is not haphazard 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 full understanding of new mathematics. Within each set of exercises, students are brought back to the essential question for understanding. Students progress in the lesson by starting with problems that focus on understanding, move to practice exercises which are more procedural, and complete application problems. 

Indicator 3c

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.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet 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 Mathematics I, lesson 8-5, 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 Mathematics II, lesson 3-2, 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 Mathematics III, lesson 5-1, Problems 3 and 11, students analyze an error and explain or correct it. In problem 5, students identify the different parts of an exponential function. In problem 30, students create an exponential function to model a radioactive isotope and make a prediction. 

Indicator 3d

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.
1/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet 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 connected 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 Mathematics I, but there was no evidence of their use in Mathematics II or Mathematics III to make connections across the courses. 
  • 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

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.
0/0
+
-
Indicator Rating Details

The instructional materials reviewed for the enVision Integrated Mathematics 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, Examples, Concept Summary, Do You Understand?, Do You Know How?, and Practice and Problem Solving.

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 3f - 3l

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.
6/8
+
-
Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 includes explanations and examples of the course-level mathematics specifically for teachers so that they can improve their own knowledge of the subject.

Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.
2/2
+
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for supporting teachers in planning and providing effective learning experiences by providing quality questions 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 & Reason, Model & Discuss, or Critique & Explain question in the student edition and teachers are provided an assortment of questions to ask their students to encourage discourse, conceptual understanding, support for productive struggle, and differentiation. For example, in Mathematics II, lesson 7-3, teachers are given the following to ask students: “Draw 5 new points on your map. How can you tell which middle school each point is closer to?” In addition, “What do you notice about the points that are the same distance from each middle school?”

Indicator 3g

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.
2/2
+
-
Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics meet expectations for containing 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’s 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 common errors. 

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

  • In Mathematics I, lesson 8-2, the student edition describes how to write a translation rule. The teacher’s edition suggests two questions to ask students to help them think about and make sense of these directions. 
  • In Mathematics III, lesson 3-3, the teacher’s edition provides a question to ask students at the beginning of the Explore & Reason activity and specifies that this question is intended for the whole class. The teacher’s edition provides one more question to ask students as they are completing the activity, specifying that the additional question 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’s 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

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.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for 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’s Edition and in the Answer and Solutions Application.The Teacher’s Edition includes 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. Examples include:

  • In Mathematics I, Teacher Edition, Lesson 4-5, Problem 12, teachers are provided with the following: “Answers 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.
  • In Mathematics I, Lesson 9-5, Try It!, the Answer and Solution Application states, “Yes, there are two cases. If the congruent legs are included between the congruent acute angle and the right angles, the triangles are congruent by ASA. If the congruent legs are not included between the congruent acute angles and the right angles, the triangles are congruent by AAS.”
  • In Mathematics II, Lesson 5-2, Answers and Solution Application, 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 vertices.”

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

  • In Mathematics I, Topic 5, Topic Overview, Math Background, Coherence, Looking Ahead connects the content of this topic to additional topics in Mathematics II and III: “Quadratic Functions Students will extend modeling with exponential functions to model using quadratic functions. They will compare models using linear, exponential, and quadratic functions. Students will apply what they learned about rational exponents to solve quadratic equations containing radicals. Radical Functions Students will apply their knowledge of radical exponents to solve radical equations. Students will also extend their knowledge of the effects of constants on functions as they transform radical functions. Rational Exponents Students will build upon their skills of writing radicals and creating, solving, and graphing exponential functions, when writing, solving, and graphing radical functions.”
  • In Mathematics II, Topic 8, Topic Overview, Math Background, Coherence, Looking Ahead identifies Topic 11 and states: “Students will re-examine some Properties learned in Topic 8 and prove these properties using Coordinate Geometry. Topic 12 Students will use the polygon interior Angle-Sum Theorem when studying inscribed angles and polygons. Trigonometry In Mathematics III, students will use polygon angle sums to generalize a formula for the side length of regular polygons.”
  • In Mathematics III, Topic 7, Topic Overview, Math Background, Coherence, Looking Ahead connects the content to “Precalculus Trigonometric Identities Students verify and apply trigonometric identities and the sum and differences of formulas. In Precalculus, they will verify and evaluate functions involving Multiple-Angle Formulas, Product-to-Sum Formulas, and Sum-to-Product Formulas.”

Indicator 3i

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for containing a teacher’s 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. In the beginning of each topic, the teacher’s edition provides a page that discusses 3 types of connections: how the topic connects with what students have learned earlier in the course or in previous courses, how different concepts are connected throughout the topic, and how that topic is connected to what students will learn later in the course or in future courses. For example, in Mathematics II, Topic 5, Quadratic Equations and Complex Numbers, there is an explanation of how this topic connects to Mathematics I work on functions as well as linear systems in Topic 5. There is also an explanation of how this topic will connect later in Mathematics III to polynomial functions. These descriptions do not reference specific mathematical standards.

Indicator 3j

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).
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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. 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

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.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 smart phones. 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 Mathematics I, Lesson 1-4, How do you solve and graph a two-step inequality?: “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

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 used the Effective Mathematical Teaching Practices as a framework within which probing questions were developed.

Criterion 3m - 3q

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

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 that is often 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

Materials provide support for teachers to identify and address common student errors and misconceptions.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 to teachers for addressing common student errors and/or misconceptions. In the Teacher’s 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 Mathematics I, 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 guide them through a review of this property and how it applies to this context.
  • In Mathematics II, lesson 9-2, example 3, the teacher’s edition notes that students may have difficulty identifying a reflection as a similarity transformation. The teacher’s edition suggests that teachers have students trace the figures and mark congruent angles while considering the rigid motion that maps figures with different orientations. 
  • In Mathematics III, lesson 2-4, example 2, the teacher’s edition notes that students may multiply the zero of the divisor, 1, by -5 and place that answer in the wrong location. The teacher’s edition suggests that teachers have students use long division to do the same problem and consider the relationships between the values in long division and synthetic division. 

Indicator 3o

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 topic review for Mathematics III, Topic 4. One question states, “A relation has one element in its domain and two elements in its range. Is the relation a​ function? Is the inverse a​ function? Explain.” Students select a multiple choice answer. 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

Materials offer ongoing assessments:
0/0

Indicator 3p.i

Assessments clearly denote which standards are being emphasized.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics partially meet expectations for offering 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 some 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 is no follow-up suggestions for any of the other assessments.

Indicator 3q

Materials encourage students to monitor their own progress.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics do not encourage students to monitor their own progress.

Criterion 3r - 3y

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.
9/10
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Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.
1/2
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials provide teachers with strategies for meeting the needs of a range of learners.
2/2
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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’s 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’s edition for advanced students, which contains more complex problems for more advanced students to complete.

For example, in Mathematics III, Topic 7-2, Teacher’s Edition, 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

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 Mathematics I, 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 Mathematics II, Topic 5, STEM project, students design a ballpark and determine what it would take to hit a home run at the park. Students can choose every dimension for their ballpark and determine an appropriate quadratic equation to fit their design.

The teacher’s 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

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).
2/2
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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’s 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 Mathematics III, Lesson 2-7, Teacher’s Edition, page 109A, ELL supports at the bottom of the page provide guidance on vocabulary practice for Beginning, Intermediate, and Advanced speakers. Each practice section begins with “Do you understand”, from which teachers can modify pacing for special populations if needed.

Indicator 3v

Materials provide support for advanced students to investigate mathematics content at greater depth.
2/2
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 students’ specific needs, including advanced students. Each lesson provides teachers with additional problems for advanced students. Some examples of enrichment for advanced learners include:

  • In Mathematics II, Lesson 9-1, page 417 in the Teacher’s Edition, an extension is provided of a problem on dilations.
  • In Mathematics III, Lesson 2-2, page 71 in the Teacher’s Edition, 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

Materials provide a balanced portrayal of various demographic and personal characteristics.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials provide opportunities for teachers to use a variety of grouping strategies.
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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. Some examples include:

  • In Mathematics II, Topic 13, 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 that choice to other students.
  • In Mathematics I, Lesson 8-2, Teacher’s Edition, page 327N, teachers guide students utilizing whole group, small group, and individual work as noted in the margins.

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.
0/0
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Indicator Rating Details

The instructional materials reviewed by enVision Integrated Mathematics 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’s 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 3z - 3ad

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.
0/0
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Criterion Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 understanding 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 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.
0/0
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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:

  • In Mathematics II, Topic 8, Mathematical Modeling in 3 Acts utilizes a dynamic geometry tool to explore a series of objects to determine how many sides an object has based on interior angle measures.
  • In every lesson, students can complete in-class work in Pearson Realize.
  • Students can access instructional tutorials using the Virtual Nerd app.

Indicator 3aa

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.
0/0
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.
0/0
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials can be easily customized for individual learners.
0/0

Indicator 3ac.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.
0/0
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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

Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.
0/0
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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 upon 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 student score to prescribe differentiated assignments based upon their performance. Assignments can be edited as well as assessments. Teachers have access to problem banks to customize assignments and assessments as well.

Indicator 3ad

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.).
0/0
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Indicator Rating Details

The instructional materials reviewed for enVision Integrated Mathematics 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.

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Additional Publication Details

Report Published Date: 08/13/2020

Report Edition: 2020

Title ISBN Edition Publisher Year
enVision Integrated Mathematics I 2019 Student Edition 9781418283766 Savvas Learning Company 2019
enVision Integrated Mathematics II 2019 Student Edition 9781418283773 Savvas Learning Company 2019
enVision Integrated Mathematics III 2019 Student Edition 9781418283780 Savvas Learning Company 2019
enVision Integrated Mathematics I 2019 Teacher Edition Volume 1 9781418283797 Savvas Learning Company 2019
enVision Integrated Mathematics II 2019 Teacher Edition Volume 1 9781418283803 Savvas Learning Company 2019
enVision Integrated Mathematics III 2019 Teacher Edition Volume 1 9781418283810 Savvas Learning Company 2019
enVision Integrated Mathematics I 2019 Teacher Edition Volume 2 9781418283827 Savvas Learning Company 2019
enVision Integrated Mathematics II 2019 Teacher Edition Volume 2 9781418283834 Savvas Learning Company 2019
enVision Integrated Mathematics III 2019 Teacher Edition Volume 2 9781418283841 Savvas Learning Company 2019
enVision Integrated Mathematics Common Core 2019 Teacher Edition Program Overview 9781418283896 Savvas Learning Company 2019

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Once a review is complete, publishers have the opportunity to post a 1,500-word response to the educator report and a 1,500-word document that includes any background information or research on the instructional materials.

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Rubric Design

The EdReports.org’s rubric supports a sequential review process through three gateways. These gateways reflect the importance of standards alignment to the fundamental design elements of the materials and considers other attributes of high-quality curriculum as recommended by educators.

Advancing Through Gateways

  • Materials must meet or partially meet expectations for the first set of indicators to move along the process. Gateways 1 and 2 focus on questions of alignment. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?
  • Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom. In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Key Terms Used throughout Review Rubric and Reports

  • Indicator Specific item that reviewers look for in materials.
  • Criterion Combination of all of the individual indicators for a single focus area.
  • Gateway Organizing feature of the evaluation rubric that combines criteria and prioritizes order for sequential review.
  • Alignment Rating Degree to which materials meet expectations for alignment, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.
  • Usability Degree to which materials are consistent with effective practices for use and design, teacher planning and learning, assessment, and differentiated instruction.

Math HS Rubric and Evidence Guides

The High School review rubric identifies the criteria and indicators for high quality instructional materials. The rubric supports a sequential review process that reflect the importance of alignment to the standards then consider other high-quality attributes of curriculum as recommended by educators.

For math, our rubrics evaluate materials based on:

  • Focus and Coherence

  • Rigor and Mathematical Practices

  • Instructional Supports and Usability

The High School Evidence Guides complement the rubric by elaborating details for each indicator including the purpose of the indicator, information on how to collect evidence, guiding questions and discussion prompts, and scoring criteria.

The EdReports rubric supports a sequential review process through three gateways. These gateways reflect the importance of alignment to college and career ready standards and considers other attributes of high-quality curriculum, such as usability and design, as recommended by educators.

Materials must meet or partially meet expectations for the first set of indicators (gateway 1) to move to the other gateways. 

Gateways 1 and 2 focus on questions of alignment to the standards. Are the instructional materials aligned to the standards? Are all standards present and treated with appropriate depth and quality required to support student learning?

Gateway 3 focuses on the question of usability. Are the instructional materials user-friendly for students and educators? Materials must be well designed to facilitate student learning and enhance a teacher’s ability to differentiate and build knowledge within the classroom. 

In order to be reviewed and attain a rating for usability (Gateway 3), the instructional materials must first meet expectations for alignment (Gateways 1 and 2).

Alignment and usability ratings are assigned based on how materials score on a series of criteria and indicators with reviewers providing supporting evidence to determine and substantiate each point awarded.

For ELA and math, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to college- and career-ready standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For science, alignment ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for alignment to the Next Generation Science Standards, including that all standards are present and treated with the appropriate depth to support students in learning the skills and knowledge that they need to be ready for college and career.

For all content areas, usability ratings represent the degree to which materials meet expectations, partially meet expectations, or do not meet expectations for effective practices (as outlined in the evaluation tool) for use and design, teacher planning and learning, assessment, differentiated instruction, and effective technology use.

Math K-8

Math High School

ELA K-2

ELA 3-5

ELA 6-8


ELA High School

Science Middle School

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