Walch CCSS Integrated Math I, II, III (2022)

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Report for High School

Alignment Summary

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for alignment to the CCSSM for high school. 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; making meaningful connections in a single course and throughout the series; and explicitly identifying and building on knowledge from Grades 6-8. In Gateway 2, for rigor, the series showed strengths in the following areas: opportunities for students to develop procedural skills, working with applications, and displaying a balance among the three aspects of rigor. The materials intentionally develop all of the mathematical practices to their full intent.

High School

Gateway 1

Focus & Coherence

Gateway 2

Rigor & Mathematical Practices

Alignment (Gateway 1 & 2)

Meets Expectations

Gateway 3

Usability

Usability (Gateway 3)

Partially Meets Expectations

Overview of Gateway 1

Focus & Coherence

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for Focus and Coherence. The materials meet the 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, making meaningful connections in a single course and throughout the series, and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The materials partially meet the 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 engaging students in mathematics at a level of sophistication appropriate to high school.

Gateway 1

v1.5

Gateway 1

Focus & Coherence

Criterion 1.1: Focus and Coherence

Meets Expectations

Criterion 1.1: Focus and Coherence

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 CCSSM).

Indicator 1A

Materials focus on the high school standards.

Indicator 1A.i

Materials attend to the full intent of the mathematical content contained in the high school standards for all students.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. There are a few instances where all of the aspects of the standards are not addressed. Overall, nearly every non-plus standard is addressed to the full intent of the mathematical content by the instructional materials.

The following are examples of standards that are fully addressed:

A-SSE.1a: In each of the three courses, parts of expressions are reinforced when dealing with different types of expressions as they are introduced (i.e. linear expressions in Mathematics 1 Unit 1 Lesson 1.1.1). Materials also move beyond simple identification of terms into an explanation of what terms, factors, and coefficients represent.
F-IF.5: The domain of a function is emphasized throughout the entire series. Students determine the domain for functions from all function families and are asked to describe what the domain represents in a given context. For example, in Mathematics II Lesson 2.2.2, students are asked to “describe the domain of the function” and determine a reasonable domain within the context of a diver jumping from a platform into the pool.
S-IC.3: In Mathematics III Unit 1 Lessons 1.3.1 and 1.3.2, students recognize the purposes of and the differences between sample surveys, experiments, and observational studies by analyzing a variety of methods of study.

The following standards are partially addressed:

N-RN.1: Mathematics II Unit 1 Lesson 1.1.1 contains material related to rational exponents; however, no opportunity is provided for either the student or teacher to give an explanation of how rational exponents follow from integer exponents.
N-RN.3: Mathematics II Unit 1 Lesson 1.1.2 contains problems that ask if a sum or product is rational or irrational; however, neither student nor teacher materials provide an explanation of how a sum or product is rational or irrational. An overview in the teacher’s resource manual simply states “rational + rational = rational” as well as other sums and products.
A-REI.5: While students do solve equations using elimination by way of replacing one equation by the sum of that equation and a multiple of the other in Mathematics 1 Unit 3 Lesson 3.2.1, proof by a comparison of methods or how this method works is not provided nor alluded to in materials.
F-IF.8a: Mathematics II Unit 2 Lesson 2.1.2 and Lesson 2.3.1 have students identify zeros, extreme values, and the axis of symmetry within terms of a context. However, completing the square is not used in order to reveal these properties of quadratic functions.
F-BF.2: Students write arithmetic and geometric sequences recursively and explicitly in Mathematics I Unit 2 Lessons 2.9.1 and 2.9.2 and use them to model situations. While students do convert from a recursive formula to an explicit formula, students are not given the opportunity to convert from an explicit formula to a recursive formula.
G-CO.8: Students solve problems about triangle congruence using ASA, SAS, and SSS in Mathematics I Unit 5 Lesson 5.6.2. An introduction paragraph is provided on page 337 of the teacher’s resource manual, but it does not explain how these criteria for triangle congruence follow from the definition of congruence in terms of rigid motions.

Indicator 1A.ii

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

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The materials include various aspects of the modeling process in isolations or combinations, but opportunities to engage in the full modeling process are absent from the materials. Examples in the materials with various aspects of the modeling process in isolation or combinations include, but are not limited to:

A-REI.6: In Mathematics I, Unit 3, Lesson 3.2.1, Problem-Based Task, students use ticket sales data to formulate models of two problem situations, solve systems of equations, and interpret the results in determining how many two-day adult and child tickets were sold as well as the prices of 3-day adult and child tickets. Students do not revise models as needed or validate their conclusions within the given context.
F-IF.7a: In Mathematics II, Unit 2, Lesson 2.1.1, Problem-Based Task, students answer the question, “Is it possible for the frog to catch the fly, given the equations of the frog’s jump?” The materials identify variables and provide students with a quadratic equation that models the height of the frog throughout the course of the jump. Students create a graphical model and analyze it to determine if the frog catches the fly. Students do not have opportunities to choose variables, create an algebraic model, or revise their work.
F-TF.5: In Mathematics III, Unit 5 [Unit 4A], Lesson 4A.3.2, Conceptual Task, students match defining characteristics (e.g., period and amplitude) to sine and cosine functions. The materials provide students with several representations (graphs, tables, and equations) of several sine and cosine functions, hints that relate to the functions, and guiding Exploration Questions that suggest how to apply the hints. Students analyze, interpret, and revise choices as they proceed through the task; however they do not formulate their own models or reflect upon their process.
G-GMD.3: In Mathematics II, Unit 6, Lesson 6.5.2, Problem-Based Task, students determine how much area will be saved by building a new cylindrical container to store piles of sand. Students substitute given dimensions into formulas to find the area and volume of three cones and a cylinder. Students interpret the results of their calculations when they find the area saved. The materials do not provide students with an opportunity to design the shape or size of their own alternative area-saving storage container.
S-IC.2: In Mathematics III, Unit 1, Lesson 1.2.3, Performance-Based Task, students determine if receiving a free nutritious breakfast can help a student learn. To help students answer this question, the materials provide three tables of data concerning average academic grades, gender, and eligibility for free breakfast in addition to the mathematical model. Students do not have the opportunity to reflect or validate their response nor analyze the given results or suggest an interpretation.

Indicator 1B

Materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.

Indicator 1B.i

Materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.

The materials reviewed for Walch CCSS Integrated Series meet expectations for allowing 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 (WAPs). (Those standards that were not fully attended to by the materials, as noted in indicator 1ai, are not mentioned here.)

In Mathematics I, students spend most of their time working with WAPs from the Algebra, Functions, Statistics and Probability, and Geometry categories. The Mathematics II course focuses on the WAPs in the Functions, Algebra, and Geometry categories. During Mathematics III, students spend most of their time working with WAPs from Statistics and Probability, Algebra, and Functions. Throughout all three courses, students also spend time on the Number and Quantity WAPs.

Examples of students engaging with the WAPs include:

Mathematics I Unit 2 Lesson 2.4 provides multiple opportunities to explore and interpret key features of linear and exponential relationships with scenarios such as interest on investments and depreciation of a vehicle to make wise decisions with money based on the relationships. (F-IF.4 and F-IF.5) Unit 2 Lesson 4 extends the study of functions with analyzation of the key features of a linear and exponential graph with exercises using contexts such as school fundraisers, investment growth, and appreciation of assets. Both Lessons 4 and 5 provide science applications with bacteria, population growth, decay, and half-life.
In Mathematics II Unit 3, the majority of the time is spent in the Algebra category with a focus on A-SSE. The students begin by developing a sense of the structure of quadratic functions and equations. The focus shifts to using the structure to devise multiple methods of solving quadratics. The unit ends with students examining the structure of rational equations and exponential equations with a goal of finding ways to solve them.
In Mathematics II Unit 5, students extend prior knowledge of transformations from Mathematics I to work with dilations and scale factor (G-SRT.1). Focus shifts to triangle similarity (G-SRT.2-5) in Lessons 5.2 and 5.3 as materials make connections to dilations. Lessons 5.8 and 5.9 address problem solving with trigonometric ratios (G-SRT.6,7,9) as an extension of similarity.
Mathematics III Unit 1 Lesson 2 allows students to expand upon 7.SP.A “Use random sampling to draw inferences about a population.” Students use their prior knowledge of sampling in order to draw inferences about population parameters for the widely applicable prerequisite S-IC.1. Instruction in the materials provides students the opportunity to address any sampling errors that may occur that could result in a biased sample.

Indicator 1B.ii

Materials, when used as designed, allow students to fully learn each standard.

The materials reviewed for Walch CCSS Integrated Math Series, when used as directed, partially meet expectations for allowing students to fully learn each standard. Examples of the non-plus standards that would not be fully learned by students include, but are not limited to:

N-RN.3: In Mathematics II, Unit 1, Lesson 1.1.2, students have opportunities to find sums and products of rational numbers, irrational numbers, and a rational number and an irrational number. The Supporting Resource and Slide Presentation explain the outcome of adding or multiplying combinations of rational and irrational numbers. In Guided Practice, Example 3, students evaluate an expression and determine whether the answer is rational or irrational, noting that the expression cannot be written as a ratio of integers. In Problems 4-6 of Problem Sets A and B, students simplify the sum and product of combinations of rational and irrational numbers and determine whether the answer is rational or irrational. The materials do not provide students independent opportunities to explain why the sums and products are rational or irrational.
A-SSE.1b: In Mathematics I, Unit 1, Lesson 1.1.2, Guided Practice, Examples 1-3, students determine the effect, if any, of changing the value of a single entity of a linear or exponential expression. In Practice Sets A and B, students re-engage with similar problems. In Mathematics II, Unit 3, Lesson 3.1.2, Guided Practice, students expand expressions and identify the a, b, and c terms of a quadratic expression in standard form. In the Problem-Based Task and Practice Sets A and B, students do not have the opportunity to interpret complicated expressions, rather they substitute values for variables and solve in order to determine the effect of an indicated change. In Mathematics III, Unit 2A, Lessons 2A.2.1-2.3 and in Unit 2B, Lesson 2B.1.1, students have multiple opportunities to expand and factor expressions. The materials do not provide students opportunities to interpret complicated expressions by viewing one or more of their parts as a single entity.
A-APR.3: In Mathematics III, Unit 2A, Lesson 2A.3.3, Scaffolded Practice, Examples 6-10, students find zeros of functions and then graph the functions to verify their answers. Guided Practice, Example 2, Question 4 suggests that students use a table of values to sketch the graph or use a graphing calculator to create a graph. This expectation to use a graphing calculator to create a graph is reinforced in Problem-Based Task Coaching, Question e. Practice Sets A and B provide students with additional opportunities to find zeros of functions and then graph the functions to verify their answers. The materials do not provide students opportunities to find zeros and use them to sketch a rough graph of a function defined by a polynomial.
F-IF.8a: In Mathematics II, Unit 2, Lesson 2.1.2, students identify properties of quadratic functions and interpret them in terms of a context. The Supporting Resource focuses on how to use the various forms of quadratic functions to show the key features of the graph of a function. In the Guided Practice, students identify the key features for one equation in each form. Student engagement is consistent throughout the Scaffolded Practice, Problem-Based Task, and Practice Sets. The materials do not provide students opportunities to use the process of factoring and completing the square in a quadratic function to reveal different properties of the function.
F-TF.5: In Mathematics III. Unit 4 [Unit 3], Lesson 3.3.1, Scaffolded and Guided Practice, students determine the period, frequency, midline, and amplitude for given trigonometric graphs. In Practice Sets A and B, students do the same given graphs and equations of trigonometric functions, including in the context of a spring. In Lesson 3.3.2, Scaffolded Practice, students write trigonometric functions given descriptions or images of graphs of trigonometric functions. In the Problem-Based Task and Practice Sets A and B, Problems 8-10, students write trigonometric functions to model periodic behavior in terms of a context. The materials do not provide students opportunities to engage with the tangent function or the three reciprocal functions.
G-SRT.4: In Mathematics II, Unit 5, Lesson 5.4.1, Scaffolded Practice, students explain the SAS and SSS Similarity Statements and answer the question, “What is a proof in Geometry?”. In the Guided Practice and the Problem Sets, students apply SAS and SSS similarity to prove triangles are similar, to determine whether triangles are similar, and to find unknown lengths. In Lesson 5.4.2, students engage with the Triangle Proportionality Theorem, Properties of Congruent Segments, and the Triangle Angle Bisector Theorem: in the Scaffolded Practice, students explain them; and in the Guided Practice and Problem Sets, students apply them within proofs, to find unknown lengths, and to determine parallelism. In Lesson 5.4.3, the Supporting Resource details how using the proportions of corresponding sides of similar triangles leads to the Pythagorean Theorem. Students prove the converse of the Pythagorean Theorem in Practice Sets A and B, Problem 10. The materials do not provide students opportunities to prove other theorems that relate to triangle similarity.
G-CO.8: In Mathematics I, Unit 5, Lesson 5.6.2, Scaffolded Practice, Problems 5-10, students identify congruent corresponding parts for two triangles and determine which congruence statement can be used to show that the triangles are congruent. Presentation Slide 8 indicates, “When a series of rigid motions is performed on a triangle, the result is a congruent triangle.” In Problem Sets A and B, students continue to determine triangle congruence, sometimes within a context, and which congruence statement can be used to prove triangle congruence. The materials do not provide students opportunities to explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motion.
S-ID.6a:Throughout Mathematics I, Unit 4, Lesson 4.2.2, students create scatter plots for data sets, determine/explain whether a linear or exponential model better estimates the data, and solve problems in context. The materials do not provide students opportunities to fit a quadratic function to data or to use quadratic functions fitted to data to solve problems in the context of the data.

Indicator 1C

Materials require students to engage in mathematics at a level of sophistication appropriate to high school.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for requiring students to engage in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts and regularly provide opportunities for students to apply key takeaways from Grades 6-8, yet do not regularly use various types of real numbers.

Examples of where the materials use age-appropriate contexts include:

In Mathematics I, Unit 2, Lesson 2.4.1, Guided Practice, Example 1, students determine the key features of the graph of a linear function that represents the cost of a taxi ride as a function of miles traveled.
In Mathematics II, Unit 4, Station Activities, Station 2, students discover concepts and skills related to the counting principle and simple and compound probabilities for independent and dependent events within the context of creating a student’s class schedule.
In Mathematics III, Unit 2, Lesson 2A.5.2, Scaffolded Practice, Problem 6, students identify the geometric series that represents the number of people in a school who would be infected after six iterations of the flu spread pattern.

Examples of where the materials use key takeaways from Grades 6-8 include:

In Mathematics I, Unit 2, Lesson 2.4.2, students apply the key takeaways of reading and interpreting data from charts and tables (6.EE.9) and understanding slope (8.EE.5) when they calculate the rate of change for exponential functions and functions within an interval (F-IF.6, F-LE.1).
In Mathematics II, Unit 5, Lesson 5.3.1, students apply knowledge of ratios and proportional quantities (7.RP.2a) to find scale factors, calculate side lengths of similar triangles, and prove similarity in triangles (G-SRT.2).
In Mathematics III, Unit 1, Lesson 1.2.2, students apply knowledge of calculating means, standard deviations and proportions in data sets (6.SP.5c) and distinguishing between a sample and a population and when to use a sample or a population (7.SP.1) to determine how to make a sample unbiased and have more reliable data (S-IC.2).

Throughout the series, the print materials rely heavily on integers, with other sets of numbers included when they are necessary due to the nature of the lesson. It is through the inclusion of GeoGebra applets that the materials allow students exposure to various types of numbers. Thus, while students may, at times, engage with various types of numbers through the applets, the opportunities for independent practice and reasoning with various types of numbers are insufficient. Examples of where and how the materials do not use various types of numbers include, but are not limited to:

In Mathematics I, Unit 3, Lessons 3.1.2 and 3.1.3, students solve linear equations and linear inequalities, respectively. In Lesson 3.1.2, Problem-Based Task, students encounter fractions (i.e., one-third and one-fourth) as they translate a verbal expression to an algebraic equation and proceed to solve the equation. In Lesson 3.1.3, students solve linear inequalities that continue to consist mostly of integers. The limited exceptions include five problems from Problem Sets A and B.
In Mathematics II, Unit 1, Lesson 1.2.1, students add and subtract polynomials with integer coefficients. In Lesson 1.2.2, students multiply polynomials with integer coefficients. In Station Activities: Operations with Complex Numbers, Station 2, Problem 4, students engage with a rational real part when multiplying two complex numbers; all other terms are integers or exponential terms that evaluate to an integer. Students do not have opportunities to compute with complex non-integer values.
In Mathematics II, Unit 6, Lesson 6.5.2, students calculate volumes of cylinders, pyramids, cones, and spheres. As these volume formulas often include $\pi$ , students encounter irrational base areas and volumes. In all but a few instances throughout the lesson---including the Problem-Based Task, which details the circumference and diameter of three piles of sand---the parameters for volume calculations are integer values; the exceptions include dimensions specific to the tenth-place digit in Problem Sets A and B.
In Mathematics III, Unit 6 [Unit 4B], Lesson 4B.2.1, students transform parent graphs of different functions. In Guided Practice and Scaffolded Practice, students encounter a fraction within the argument and in the domain of logarithmic functions. In the Problem-Based Task, students use integer values to calculate slope to the hundredth place. In Problem Set B, Problem 8, the flow rate is changed by a fractional value. Students do not have sufficient opportunities to engage with function transformations that involve non-integer values.
In Mathematics III, Unit 3, Lesson 2B.2.2, students solve rational and radical equations. In Scaffolded Practice, Problem 9, students encounter a fractional exponent within the radicand; and in Problem 10, they solve a cube root equation. In Guided Practice, Example 4, student calculations include an irrational value although the result is an integer solution. Throughout Practice Sets A and B, students solve two cube root equations and perform calculations that yield only one irrational solution. Students do not have ample opportunity to calculate with irrational values and nth roots where n > 2.

Indicator 1D

Materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for fostering coherence through meaningful connections in a single course and throughout the series. Overall, connections between and across multiple standards are made in meaningful ways. Each course in the series includes a “Topics for Future Courses” in the program overview. This section describes when a topic is introduced, where the topic can be addressed in future courses, and how the topic can be addressed. Each lesson includes a list of prerequisite skills and a warm-up exercise intended to connect previously learned concepts. Materials often refer to previously taught concepts in the “Connection to the Lesson” section and in the “Concept Development” section of the lesson.

Examples of connections made within courses are:

In Mathematics I Unit 2 Lesson 2.1, students connect graphs as solution sets (A-REI.10,11) and as functions. (F-IF.1,2). Unit 1 Lesson 2.1 (A-CED.1, N-Q.2, and N-Q.3) has students create linear equations in one variable. Unit 1 Lessons 1.3.1 and 1.3.2 (A-CED.2 and N-Q.1) has students create and graph linear and exponential equations. In Unit 2 Lesson 2.4.2 (F-IF.6 and F-LE.1a) students prove average rate of change, and Lesson 2.4.3 makes connections among F-IF.6, F-LE.1b, and F-LE.1c.
In Mathematics II Unit 3 Lesson 3.2, students create and solve quadratics (A-CED.1 & A-REI.4) while using the structure of the equations (A-SSE.2). Unit 3 Lesson 3.3 (A-SSE.3a and A-CED.2) has students create and graph equations.
In Mathematics III Unit 4B Lesson 4B.4.1 thru Lesson 4B.4.3 students work on choosing models. They are asked to create graphs (A-CED.2), identify key features of a graph (F-IF.4), and work with the effects of graph transformations (F-BF.3). Mathematics III Unit 2A Lesson 2a.2.1, 2a.2.2, and 2a.2.3 ( A-SSE.1b, A-APR.4) has students identify and use polynomial identities. Unit 2A Lesson 2a.3.4 has students find zeros using A-APR.3 and F-IF.7c. Unit 2B Lessons 2b.1.2 thru 2b.1.4 (A-SSE.2 and A-APR.7) has students work operations with rational expressions.

Examples of connections made between the courses include the following:

Mathematics I Unit 1 Relationship between Quantities: Vocabulary and expressions connect Math II Unit 3 and Math III Units 1 and 2 as the topics are extended to include more complex expressions and higher polynomials.
Treatment of Geometric topics builds across the courses as students work with segments, angles, and triangles in Mathematics I, more advanced triangle relationships such as trigonometry in Mathematics II, and the unit circle and law of sines and cosines in Mathematics III.
The treatment of F-IF standards builds throughout the coursework. Students work with linear equations, inequalities, and exponential equations in Mathematics I. In Mathematics II students continue to work with functions using quadratics, and finally in Mathematics III students work with radical, rational, and polynomial functions.
Mathematics I Unit 2 Linear and Exponential Relationships: Linear graphs and exponential graphs are extended to the study of other types of equations that are more complex, such as logarithmic, radical, and rational, in Math II Units 2 and 3 and in Math III Units 2 and 4.

Indicator 1E

Materials explicitly identify and build on knowledge from Grades 6-8 to the high school standards.

The instructional materials reviewed for Walch CCSS Integrated Math Series meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. The instructional materials explicitly identify the standards from Grades 6-8 in the print Teacher resources as Prerequisite Skills. These resources are not present in the online platform, the Curriculum Engine, and are not included in the student materials.

Examples where the print teacher materials explicitly identify content from Grades 6-8 and build on them include:

In Math I, Unit 3, Lesson 3.1.3, the Teacher Resource indicates that the lesson requires the use of 8.EE.7b, 7EE.4b, and 6.EE.3. Examples include: In the Warm-up and Problem-Based Task, when students write and solve linear inequalities to represent real-world problem situations and to answer real-world questions (A-REI.3), they build on 7.EE.4b, where students solved word problems that involved linear inequalities. In the Practice activities, when students solve linear inequalities of different forms, they revisit their earlier experience with 8.EE.7b.
Math I, Unit 4, Lesson 4.1.3 indicates a connection to 6.SP.4 and 6.SP.5c,d as students focus on identifying outliers and understanding their impact, or not, on measures of center and spread. Students create box plots and interpret outliers in terms of the context (S-ID.3).
In Math II, Unit 3, Lesson 3.5.2, students build on two standards from Grades 6-8: 7.EE.3 (students write equivalent fractions, decimals, and percentages) and 8.F.1 (students plot points of a function given a function rule). During this lesson, students graph rational functions, manually and using technology; describe its end behavior and behavior near the asymptotes; and write/analyze rational functions to model real-world contexts (A-CED.2, F-IF.7d).
In Math II, Unit 5, Lesson 5.4.4, students build on 8.G.7 and 8.G.8, where students use the Pythagorean Theorem to determine unknown side lengths and to find the distance between two points in a coordinate system. Within the lesson, students use congruence and similarity criteria for triangles to solve problems and to prove similarity in various contexts (G-SRT.5).

In Math III, Unit 2A, Lesson 2A.1.1, students build on previous knowledge of 6.EE.2a, which involved writing unknown quantities with variables. In the Scaffolded Practice, students focus on the structures of expressions; in the Problem-Based Task, students write a polynomial expression in standard form and review associated vocabulary (A.SSE.1a).

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.

The materials reviewed for Walch CCSS Integrated Math Series explicitly identify the plus (+) standards and do use the plus (+) standards to coherently support the mathematics which all students should study in order to be college and career ready.

Of the 43 plus standards and 5 plus sub-standards included in the CCSSM, the materials work with 18 of them: N-CN.3, N-CN.8, N-CN.9, A-APR.5, A-APR.7, F-IF.7d, F-BF.4d, F-BF.5, F-TF.3, G-SRT.9, G-SRT.10, G-SRT.11, G-C.4, S-CP.8, S-CP.9, S-MD.2, S-MD.6, and S-MD.7. The materials attend to the depth required by these standards with the exception of A-APR.7, G-SRT.9, and G-SRT.10. In general, the materials treat these 18 standards as additional content that extends or enriches topics within the unit and do not interrupt the flow of the course. No plus standards were located within the first course of the series, Mathematics I.

The following are examples of components of the materials address the full intent of the plus standards:

Mathematics II Unit 1 Lesson 1.3.3: Students find the conjugates of complex numbers. (N-CN.3)
Mathematics II Unit 6 Lesson 6.3.1: Students complete constructions that include the construction of a tangent line to a point outside the circle and a construction of a tangent line to a point on the circle.(G-C.4)
Mathematics III Unit 1 Lesson 1.6.1: Students calculate the expected value of a random variable. (S-MD.2)

The following components of the materials do not address the full intent of the plus standards:

A-APR.7: In Mathematics III Unit 2B Lesson 2B.1.2 (add/subtract rational expressions), 2B.1.3 (multiply rational expressions), and 2B.1.4 (divide rational expressions), practice is provided performing all of these operations; however, materials do not provide evidence that rational expressions are closed under these operations.
G-SRT.9: In Mathematics III Unit 3 Lesson 3.2.1, students do not derive the formula for the area of a triangle using the sine function but are coached through it in Example 4. However, students do use to formula to solve problems.
G-SRT.10: In Mathematics III Unit 3 Lesson 3.2.1 and Lesson 3.2.2, students do not prove the law of sines and cosines themselves; however, practice is provided for students to solve problems using the law of sines and cosines.

Criterion 1.1: Focus and Coherence

Overview of Gateway 2

Rigor & Mathematical Practices

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for rigor and balance. The materials meet expectations for providing students with opportunities in developing procedural skills, and application, and the materials also meet expectations for balancing the three aspects of rigor. The materials meet expectations for Practice-Content Connections as the materials intentionally develop all of the mathematical practices to their full intent.

Gateway 2

v1.5

Gateway 2

Rigor & Mathematical Practices

Criterion 2.1: Rigor and Balance

Criterion 2.2: Practice-Content Connections

Meets Expectations

Criterion 2.1: Rigor and Balance

Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for Rigor and Balance. The materials display a balance between conceptual understanding, procedural skills, and applications. The materials give students sufficient opportunities to utilize mathematical concepts and skills in engaging applications as students complete problems in real-world contexts and engage with non-routine, contextual problems. The materials also provide intentional opportunities for students to develop procedural skills as there are sets of practice problems for each lesson. The materials partially develop conceptual understanding of key mathematical concepts as they do not provide opportunities for students to independently demonstrate conceptual understanding throughout the series.

Indicator 2A

Materials support the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Across the series, conceptual understanding is developed in the instructional portion of the lessons as students are guided through Scaffolded Practice and Guided Practice. Occasional online Interactive Practices and Conceptual Activities give students opportunities to compare their understanding by sharing their answers to prompts with other students in the class. The materials do not provide opportunities for students to independently demonstrate conceptual understanding throughout the series.

Examples of where and how the materials do not provide opportunities for students to independently demonstrate conceptual understanding throughout the series include, but are not limited to:

A-SSE.3b: In Mathematics II, Unit 3, Lesson 3.3.3, students create and graph quadratic equations using vertex form. Beginning with the Scaffolded Practice and through to the Problem Sets, students identify the vertex of functions already in vertex form and state whether the vertex is a minimum or maximum. Students write quadratic equations, convert functions from standard to vertex form, and sketch graphs. In Guided Practice, Example 3, materials show students how to complete the square to convert a quadratic function to vertex form. The materials do not provide opportunities for students to independently demonstrate completing the square of a quadratic will reveal the minimum or maximum of the function.
A-APR.3: In Mathematics III, Unit 2 [Unit 2A], Lesson 2A.3.3, students find zeros of polynomials and sketch graphs. In the Scaffolded Practice, Problems 6-10 and Practice Sets, Problems 4-6, students find all the zeros in polynomial functions where linear factorizations are possible and use a graphing calculator to graph each function. The materials do not provide students to independently use the zeros to construct a rough graph of the function defined by each polynomial.
F-IF.A: In Mathematics I, Unit 2, Lesson 2.1.3, students learn to differentiate between various representations of relations and functions and to state the domain and range of a function. In the Scaffolded Practice, students determine whether a given relation (e.g., table of values, graph, or set of ordered pairs) represents a function and explains why or why not. In the Guided Practice, Example 3 (and in the Presentation), the materials tell students that a graph of a relation is a function when the “vertical line only intersects the curve at one point at a time” and is not a function when the “line intersects the figure at two different points.” The materials do not provide students with opportunities to independently demonstrate their understanding of how the vertical line test connects to the definition of a function.
G-SRT.4: In Mathematics II, Unit 5, Lesson 5.4.1, students engage with triangle similarity. In the Scaffolded Practice, students explain the Side-Angle-Side (SAS) and Side-Side-Side (SSS) triangle similarity statements and respond to prompts including “What does every good proof include?” In the Guided Practice, students use SAS and SSS to determine if three pairs of triangles are similar and to find a missing length. In Practice Sets A and B, students continue to determine similarity, write similarity statements, and find missing lengths and angle measures. The materials do not provide students with the opportunity to independently prove theorems about triangles.

Indicator 2B

Materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for providing intentional opportunities for students to develop procedural skills and fluency. Within the lessons, students are provided with opportunities to develop procedures for solving problems that begin to develop fluency. A practice set that includes 10-15 problems is present for each lesson. These practice sets are often “naked number” problems with no context and provide students the opportunity to practice procedural skills.

Some highlights of strong development of procedural skills and fluency include the following:

A-APR.1: Mathematics II Unit 1 Lesson 1.2.1 and Mathematics III Lesson 2A.1.2 provide opportunities for students to add, subtract, and multiply polynomials. Mathematics III questions extend students’ procedural fluency from those problems students were exposed to in Mathematics II by using larger exponents and more terms in a polynomial expression.
A-SSE.2: Mathematics II Unit 3 Lesson 3.1 on pages 63-104 reinforces vocabulary and concepts of the parts of expressions and develops skill with writing expressions in different ways in the practice tasks. Mathematics III Unit 2A (pages 46-91) and Unit 2B (pages 5-79), in the Station Activities Set 1, have students build on previous concepts of simplification to rewrite complicated expressions. These stations develop procedural skills as students are required to work from both representations of expressions.
F-BF.3: There are opportunities provided throughout the series for students to identify the effect of replacing f(x) by f(x)+k, k f(x), f(kx), or f(x + k). Mathematics I Unit 2 Lesson 2.8.2 provides practice with linear and exponential functions; Mathematics II Unit 4 Lessons 2.4.1, 2.4.2, 2.6.1, and 2.6.2 provide practice with quadratic, square root, cube root, and absolute value graphs; and Mathematics III Lessons 3.3.1 and 4B.2.1 provide practice with trigonometric, quadratic, exponential, logarithmic, and linear functions.
G-GPE.4: Mathematics II Unit 5 Lessons 5.7.1 and 5.7.2 provide opportunities (examples, problem-based task, and practice exercises) for students to use the slope formula, distance formula, and midpoint formula to classify quadrilaterals. Several cases are considered (not only proving a quadrilateral is a parallelogram), and students also work with multiple problems during the practice and guided practices to develop the procedural steps required to prove geometric theorems using coordinates in Mathematics I Unit 6 Lesson 6.1.2.

Indicator 2C

Materials support the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for 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. Students work with mathematical concepts within a real-world context. Each lesson contains a problem-based task at the end of the lesson. The problem-based task includes material found in each lesson in a contextualized situation. Single-step and multi-step contextual problems are used throughout all series’ materials. Non-routine contextual problems are also present within the materials. The problem-based tasks also require greater levels of problem solving sophistication as the series progresses.

Examples of mathematical concepts found in application are as follows:

G-SRT.8: In Mathematics I Unit 5 Lesson 5.9.3, students use trigonometric functions to solve angle of elevation and depression problems. Practice problems include word problems where students must sketch a diagram of the situation and then solve the problem. The problem-based task for the lesson requires students to complete two trigonometric functions and then subtract to find the answer, therefore creating a multi-step contextual problem. In Mathematics II Unit 5 Lesson 5.8, students are asked to determine the dimensions of a ramp using right triangle trigonometry.
G-MG.2: In Mathematics III Unit 4B Lesson 4b.5.2, a non-routine contextual problem is located on page 363 as students relate the density of ice to a graph and table and determine an equation to represent the data.
A-SSE.3: In Mathematics I Unit 1 Lesson 1.2.1 Problem Based Tasks, scaffolding practice, and student practice stress the application of mainly linear relationships. For example, in Guided Practice 3, students must create linear equations to determine when two cars will meet. In Mathematics II Unit 3, the unit has various application problems where station activities and large group discussions provide for application scenarios.
F-IF.4: In Mathematics I Unit 2 Lesson 2.4, students are asked to use information about the purchase of a car to construct a graph of the value of the car over time and identify key features of the graph.
A-CED.4: In Mathematics III Unit 4B Lesson 4b.1, students are given formulas which relate the frequency and length of the strings on stringed instruments and asked to create a combined formula to determine the tension on the string.

Indicator 2D

The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for the three aspects of rigor being present independently throughout the program materials, and multiple aspects of rigor being engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Each lesson includes a common set of components: Warm-Up, Scaffolded Practice, Guided Practice, a Problem-Based Task, Interactive Applets, and Practice Sets. Conceptual development is predominantly addressed independently in tasks specifically called out as Conceptual Tasks. Procedural skills are developed throughout the materials. Engaging applications, although included in the Warm-Up and regularly in a few problems in each Practice Set, are principally addressed in Problem-Based Tasks.

Examples of where the materials independently engage aspects of rigor include:

In Mathematics I, Unit 2, Lesson 2.4.3, the Interactive Practice Problem GeoGebra “Average Rate of Change” supports the conceptual development of average rate of change. Students interact with sliders and entry fields to alter the parameters of a function (linear or quadratic) and the interval of change. Students see changes in the graph, slope triangle, and average rate of change calculations.
The Mathematics II Program Overview indicates that “activities incorporate concept and skill development and guided practice, then move on to the application of new skills.” Those applications can often be found at the end of the Problem Sets. In Mathematics II, Unit 6, Lesson 6.6.1, Problem Set A, Problem 10, students apply their knowledge of completing the square to find “the geometric description of the region” of the park for which Marco, a park ranger, is responsible.
In Mathematics III, Unit 4 [Unit 3], Lesson 1, the Essential Questions focus on concept knowledge and procedures. For example, the list includes “What is a reference angle and how is it found?” and “What are the special angles and how do you find their trigonometric ratios?” In keeping with these Essential Questions, the Problem Sets in Lessons 3.1.2–3.1.4 predominantly consist of procedural skill practice: students convert between degrees and radians, find reference angles, and find the coordinates for terminal sides of angles.

Examples of where the materials engage multiple aspects of rigor simultaneously include:

In Mathematics I, Unit 4, Lesson 4.1.1, students engage with two-way frequency tables in a way that supports their ability to respond to real-world applications. Every table and every question throughout the lesson, including the applets, pertains to a real-world context. Students practice finding marginal and conditional frequencies throughout the lesson. In one of the final Problem Sets (Problem Set B, Problems 1-6), students create a two-way frequency table that shows buildings preferred by residents of each location, find marginal and conditional frequencies, describe trends, and explain how the information could be used to decide where to build each of the three buildings.
In Mathematics II, Unit 3, Topic B, Conceptual Task, students engage with two student work samples: Asked to solve a quadratic equation, Avi completed the square while Ben used the quadratic formula. Students begin by investigating the two solutions for errors. They answer questions by explaining the student work, supporting the choice of a solution method, explaining what solutions have in common, identifying commonalities between solutions, and describing how the methods might be related.
In Mathematics III, Unit 6 [Unit 4B], Lesson 4B.3.1, Problem-Based Task, students engage with multiple representations (i.e., data table, equations, and graph) of a function that models levels of carbon dioxide in the atmosphere over a six-decade period. Specifically, students use graphing technology to plot data from a table and compare the key features of the computer-generated equation with the given sine function. In addition, they combine the functions and use the newly created function to make a prediction. Monitoring and Coaching questions included in the Implementation Guide encourage students to interpret key features of the function in terms of the context and explain the disparities in the two graphs. Throughout this task, students apply what they know about sine functions to this real-world scenario.

Criterion 2.1: Rigor and Balance

Criterion 2.2: Practice-Content Connections

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for Practice-Content Connections as the materials intentionally develop all of the mathematical practices to their full intent.

Indicator 2E

Materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6). Overall, the instructional materials develop both MP1 and MP6 to the full extent of the mathematical practice standards. Accurate and precise mathematical language and conventions are encouraged by both students and teachers as they work with course materials. In each of the units there is also a set of station activities that includes a discussion guide. These discussion guides prompt the instructor to ask discussion questions to help students to make sense of the task and to provide responses including precise vocabulary. Emphasis is placed on using units of measure and labeling axes throughout the series (explicit instruction in Mathematics I Lesson 1.2.1 and expectations of using correct units carried throughout the rest of the series). Making sense of answers within the context of a problem is also emphasized. Students also persevere in problem solving in each problem-based task at the end of each lesson.

Mathematics II Unit 3 Lesson 2.3. (A-SSE.2): Students solve a quadratic equation and determine whether both solutions make sense in the context of a throwing a basketball.
Mathematics I Unit 2 Lesson 10.1 (F-LE.5): Students need to interpret what the parameters represent in the context of a problem in order to determine whether a solution makes sense. In the implementation guide for the problem-based task with this lesson, teachers are reminded to "check to make sure that students understand how the pricing on the cell phone plans works. Ask them to determine how much someone would pay under each plan for a given number of minutes."
Mathematics I Unit 5 Lesson 6.1 (G-CO.7): Students identify corresponding parts of congruent triangles as they are introduced to symbolic notation and markings used to represent congruent side and angles in geometric figures. In the implementation guide for the problem-based task with this lesson, teachers are reminded to "encourage students to discuss their prior knowledge of angle pairs formed by a transversal that intersects parallel lines."
Mathematics III Unit 2 Lesson 2.2: Students use A-REI.2 and the Pythagorean Theorem (G-SRT.8) to solve problems involving radicals. Students are asked to solve applications in both the warmup and problem-based task that require sense-making and perseverance to initiate and precision of units and language to solve.
Mathematics III Unit 4A Lesson 4a.3: Students work on F-IF.7e. The problem-based task involves creating functions that model the voltage in a three phases of AC coming from a generator. Students will need to persevere to begin the problem and will need to be precise in mathematical language to finish the problem.

Indicator 2F

Materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards. The Program Overview contains a Correspondence to Standards for Mathematical Practice that focuses on the relevant attributes of Problem-Based Tasks (PBTs). Each PBT “uses a meaningful real-world context that requires students to reason both abstractly about the situation/relationships and quantitatively about the values representing the elements and relationships” (MP2). In addition, because each PBT “provides opportunities for multiple problem-solving approaches and varied solutions, students are required to construct viable arguments to support their approach and answer,” which, in turn, “provides other students the opportunity to analyze and critique their classmates’ reasoning” (MP3). Although the Program Overview focuses on the PBTs, instances of where and how the materials attend to the intentional development of MPs 2 and 3 is not limited to the PBTs.

The materials develop MP2 as students are provided opportunities to develop their mathematical reasoning skills in connection to course-level content across the series. Examples where students reason abstractly and quantitatively include, but are not limited to:

In Mathematics I, Unit 1, Station Activities, Set 2, Station 4, students match inequalities to real-world situations. After completing the matching task, students explain the strategies used to match the inequalities to the situations. In this activity, students decontextualize a situation to represent it symbolically and contextualize the symbolic representations by considering if the calculated quantities make sense in the given real-world scenarios.
In Mathematics II, Unit 5, Lesson 5.6.4, the Problem-Based Task Implementation Guide indicates that “students will reason abstractly as they make sense of the information represented in the scenario … and will reason quantitatively as they calculate the midpoints and slopes of each side length of the triangle.”
In Mathematics III, Unit 5 [Unit 4A], Lesson 4A.2.3, Problem-Based Task, students reason abstractly as they recognize the need to create a natural logarithmic function to describe the exponential growth of global consumer Internet traffic from 2006 to 2014. Students also reason quantitatively as they “substitute the given values from the calculator into the natural logarithmic equation and use the equation to evaluate the number of petabytes per month in a given year.”

The materials attend to the intentional development of MP3 in connection to course-level content across the series. Examples where students construct viable arguments and critique the reasoning of others include, but are not limited to:

In Mathematics I, Unit 3, Lesson 3.1.1, the Problem-Based Task Implementation Guide indicates: “The focus of the task is to construct a viable argument as to why the answer to the magic number game is always 3. Students will construct their arguments based on their creation of an equation, which is based on following steps and performing calculations on a numerical expression. They will use properties of equality and properties of operations to justify their steps and explanations. Ask students to put their ideas into writing, and encourage students who disagree with each other on any of the steps in the process to discuss and explain their thinking.”
In Mathematics II, Unit 4, Topic D (online) Learning/Performance Task: Mathematics Assessment Resource Service “Representing Conditional Probabilities 1,” students work together in small groups of two to three to share ideas about the task and plan a joint solution. Materials indicate that the teacher prompts students to listen carefully to explanations and “ask questions if you don’t understand or agree with the method.” Later in the task, students compare different solution methods. The materials indicate that students “compare two arguments and determine correct or flawed logic and prompts students to evaluate peer arguments.”
In Mathematics III, Unit 6 [Unit 4B], Station Activities: Choosing a Model, students engage with four different activities that support their ability to distinguish between linear, quadratic, and exponential functions. As part of the debrief, students construct viable arguments when they explain how to “distinguish a linear function, an exponential function, and a quadratic function from one another using a table of data.” Implementing the Think-Pair-Share routine provides students with the opportunity to critique the reasoning of others.

Indicator 2G

Materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

Throughout the materials, the most notable aspect of MP4 is the intentional use of representations to model and to interpret the results of mathematical situations. Examples of where and how students model with mathematics include:

In Mathematics I, Unit 2, Lesson 2.1.1, Problem-Based Task: Saving for College, students use representations (i.e., equations and graphs) to compare two different methods of compensation: commission-based wages or hourly wages. Given the equation that represents commission-based wages, students formulate an equation to represent hourly wages, graph the two methods of compensation, and describe the earning potential based on the two types of wages.
In Mathematics II, Unit 2, Lesson 2.1.1, Problem-Based Task: How High Can a Frog Jump?, students represent a mathematical situation with a graph and interpret the results. Specifically, given a quadratic function that models a frog’s height above the water as it jumps across a creek, students determine if it is possible for the frog—with and without jumping—to catch a fly that is “cruising at a height of 5 feet above the water.” The Problem-Based Task Coaching indicates (in order) that students consider and justify if the frog can catch the fly without jumping, indicate and justify whether the extremum is a minimum or maximum, state the vertex, consider if the frog can catch the fly by jumping, and finally sketch the graph of the paths of the frog and the fly. The Implementation Guide indicates that students might opt to sketch the graph of both functions first, then answer the rest of the questions.
In Mathematics III, Unit 6 [Unit 4B], Lesson 4B.1.1, Guided Practice, students represent mathematics with an equation (in one variable) and interpret the results. Students begin by writing an equation in words to model the total cost of producing personalized cases before creating a variable equation to model the cost of n cases. Using this equation, students then determine how much money will be left to spend on cases after paying a fee and how many cases can be purchased within the context of the scenario. Students use the equation to check the result.

Examples of where and how students use appropriate tools strategically (MP5) include:

In Mathematics I, Unit 2, Lesson 7, Conceptual Task, students consider questions related to meeting the 1979 World Record for consecutive jumping jacks. Given a pattern of completed jumping jacks in a bar graph, students determine how many jumping jacks will be completed if the pattern of jumps is extended to two weeks. Exploration Questions prompt students to make a table, create a graph, and create an equation to extend the pattern before choosing a method to calculate the number of days it would take to exceed the 1979 World Record of 27,000 consecutive jumping jacks, explaining their method, and extrapolating if the same method would be reasonable beyond a year’s time.
In Mathematics II, Unit 1, Lesson 1.1.1, Problem-Based Task, students use a given algebraic model to approximate a town’s population eight years from today. Given the complexity of the model, students use a calculator to perform the calculation. Students will be required to use the calculator features strategically to enter the model correctly with rational exponents.
In Mathematics III, Unit 1, Lesson 1.2.2, Problem-Based Task, students determine if it is reasonable to assume that making eight consecutive foul shots for a player who typically makes 80% of her free throws can be attributed to chance variation alone. Students choose their means (e.g., graphing calculator or deck of cards) of running 20 simulations of a player shooting eight foul shots with an 80% chance of success before calculating the percent of simulations in which all eight shots are made and interpreting the results.

Indicator 2H

Materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for supporting 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.

Math Practice Standards MP7 and MP8 are used to enrich the mathematical content as these practices are not treated as isolated experiences for the students. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

Examples of where and how students look for and make use of structure (MP7) include:

In Mathematics I, Unit 1, Lesson 1.4.1, Problem-Based Task, students look for and make use of the structure in the information provided about two types of skates to create two linear inequalities in two variables. Students use the system of linear inequalities and constraints of the situation to determine some possible combinations of the number of figure skates and hockey skates that can be made.
In Mathematics II, Unit 5, Lesson 5.3.1, Problem-Based Task, students use the structure of similar figures to determine the two possible locations for a vertex of a triangle on the coordinate plane.
In Mathematics III, Unit 3 [Unit 2B], Lesson 2B.1.1, Problem-Based Task, students use structure in the expressions found in the numerator (difference of two squares) and denominator (quadratic trinomial that has linear factors) of a rational expression to write a simplified equivalent expression.

Examples of where and how students look for and express regularity in repeated reasoning (MP8) include:

In Mathematics I, Unit 5, Lesson 6, Conceptual Task, students determine if all triangles that share a right angle and a common side length (3 in.) are congruent. Students use geometric models and the Pythagorean theorem repeatedly to justify their conclusion. In addition, students recall congruence rules learned earlier in the unit to state conditions under which all triangles are congruent.

In Mathematics II, Unit 1, Lesson 1.2.1, Problem-Based Task, students write an expression to represent the perimeter of a small cabin that has both known and unknown dimensions. Students express regularity in repeated reasoning as they simplify the expression for perimeter in terms of x.
In Mathematics III, Unit 3 [Unit 2B], Lesson 2B.1.3, Problem-Based Task, students calculate the area of a right triangle given rational expressions $\frac{3x}{x+1}$ and $\frac{x-1}{x^2}$ for the lengths of the legs. Students build on their previous knowledge of calculating the area of a triangle with only numerical values, by repeating the same reasoning and process to calculate the area with algebraic expressions.

Criterion 2.2: Practice-Content Connections

Overview of Gateway 3

Usability

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for Usability. The materials partially meet expectations for Teacher Supports (Criterion 1), Assessment (Criterion 2), and Student Supports (Criterion 3).

Gateway 3

v1.5

Gateway 3

Usability

Criterion 3.1: Teacher Supports

Criterion 3.2: Assessment

Criterion 3.3: Student Supports

Criterion 3.4: Intentional Design

Partially Meets Expectations

Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the materials, and provide explanations of the instructional approaches of the program and identification of the research-based strategies. The materials partially contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject, and partially includes standards correlation information that explains the role of the standards in the context of the overall series. The materials do not provide a comprehensive list of supplies needed to support instructional activities.

Indicator 3A

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Consistent structure and features help teachers enact the student materials. Examples include:

The first section of all three courses is the Program Overview, which contains the following resources split into individual sections: Table of Contents for Instructional Units, Introduction to the Program, Correspondence to Standards for Mathematical Practice, Correspondence to CCSS Publishers’ Criteria, Correspondence to NCTM Principles to Actions Teaching Practices, Unit Structure, Standards Correlations and Connections to Future Courses, Conceptual Activities, Station Activities Guide, Digital Enhancements Guide, Standards for Mathematical Practice Implementation Guide, Instructional Strategies, Suggested Pacing Guide, Formulas, and Bilingual Glossary.
The Program Overview-Unit Structure describes the units, lessons, and sub-lesson structure throughout the program. “All of the instructional units have common features. Each lesson begins with an overview, listing relevant Essential Questions, vocabulary (titled “Words to Know”), and recommended websites to be used as additional resources, along with a pre-assessment. Each sub-lesson begins with a list of identified prerequisite skills that students need to have mastered in order to be successful with the new material in the upcoming lesson. This is followed by an introduction, key concepts, common errors/misconceptions, guided practice examples, a problem-based task with coaching questions and sample responses, a closure activity, scaffolded practice problems, and practice worksheets with answer keys. Each lesson ends with a progress assessment to evaluate students’ learning.”
The Support Supplements for Mathematics I and Mathematics II and the Honors Supplements for Mathematics II and Mathematics III include instructional strategies in the areas of literacy, annotation, graphic organizers, mathematical discourse, and mathematical modeling.

The materials provide teacher guidance with useful annotations and suggestions throughout the courses. Examples include:

Most Problem-Based Tasks are accompanied by an Implementation Guide, Coaching Questions, and Sample Responses. In Mathematics I, Unit 6, Lesson 6.2.1, the Problem-Based Task Implementation Guide details such useful information as the overall focus of the task, suggested secure vocabulary, SMPs that should be reinforced as students work, tips on addressing common errors/misconceptions, suggested monitoring and coaching moves (e.g., “Ask students if they have questions about areas of the problem that are not clearly understood, and allow students to clarify these points for each other.”), how to debrief the task (e.g., “Compare students’ strategies and ways of justifying responses. …”), how to connect to key concepts, and extend the task, and notes about alternate strategies or solutions.
In Mathematics II, Unit 2, Lesson 2.3.1, instructional components intended for teacher guidance include the Warm-Up Debrief, Teacher Instructional Support, Scaffolded Skills Practice Answer Key, a PowerPoint Presentation (which provides optional versions of the Warm-Ups, Warm-Up Debriefs, Introductions, Key Concepts, and Guided Practices as well as descriptions of what concepts students will glean, skills they will need in the upcoming lesson, and “Connections” to help transition from the Warm-Up to instruction), the aforementioned Problem-Based Task resources, and Problem Set Answer Keys.

Station Activities are accompanied by an Instruction and Answer Key. The Instruction and Answer Key includes the following: Student Activities Overview and Answer Key, Materials List/Setup for each station, and a Discussion Guide. In Mathematics III, Unit 1, Instruction and Answer Key: Distributions and Estimating with Confidence Instruction, the Discussion Guide includes:
- Prompts/Questions, such as, “How is the confidence interval of a population mean affected by an increase in sample size?”
- A Think, Pair, Share suggestion, “Have students jot down their own responses to questions, then discuss with a partner (who was not in the same station group), and then discuss as a whole class.”
- Suggested Appropriate Responses, such as, “The larger sample size greatly reduced the margin of error. The confidence interval was therefore much smaller, and narrowed down the possibilities for the population mean.”
- Possible Misunderstandings/Mistakes, such as, “Omitting qualifying language, such as ‘within my sample’ or ‘in our group’.”

Indicator 3B

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject. The materials do not contain adult-level explanations and examples of concepts beyond the current course so that teachers can improve their own knowledge of the subject.

In each lesson throughout the series, the materials provide a Supporting Resource that includes Recommended Resources, which, if accessed, can help teachers improve their own knowledge of the more complex grade/course-level concepts. Instances of these adult-level explanations and examples include:

In Mathematics I, Unit 2, Lesson 3, Supporting Resource includes Recommended Resources to a few external sites. One of the sites is OEIS.org “The On-Line Encyclopedia of Integer Sequences,” which is described as a “quirky site (that) will complete and define an entered sequence of integers. It will also provide any historical or quasi-relevant links to the defined sequence.” Lesson 2.3.1 addresses Sequences as Functions. The Supporting Resource in the lesson defines and describes a sequence in detail using abstract examples and referring to mathematical concepts such as, the set of natural numbers and a discrete function. Key Concepts include several bullets of sequence-relevant information so that teachers can improve their own knowledge; these bullets detail explicit and recursive formulas, common differences and ratios, and connections to arithmetic and geometric sequences. The materials provide examples of explicit and recursive formulas in addition to graphs of a sequence.
In Mathematics II, Unit 6, Lesson 1, Supporting Resources includes Recommended Resources to some external sites:
- From Math Open Reference, the “site describes the Central Angle Theorem and allows users to explore the relationship between inscribed angles and central angles.”
- A site at Math Warehouse “reviews the properties of tangent lines and allows users to interactively explore the idea that a tangent line is perpendicular to a radius at the point of tangency. This site also provides limited practice problems, as well as solutions.”
In Mathematics III, Unit 4 [Unit 3], Lesson 1 focuses on Radians and the Unit Circle. Recommended Resources include the following:
- “Intuitive Guide to Angles, Degrees, and Radians” (Better Explained.com), which provides a creative explanation of the radian system of angle measure.
- Khan Academy’s “Unit Circle Definition of Trig Functions,” which includes videos and examples of how to use the unit circle to find trigonometric functions.

Indicator 3C

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. The materials do not include explanations of the role of the specific course-level mathematics in the context of the series.

Within each course in the series, the Program Overview includes a Standards Correlations and Connections to Future Courses document and a Suggested Pacing Guide. The Standards Correlation and Connections to Future Courses document correlates each sub-lesson to the CCSSM. The Suggested Pacing Guide aligns a day’s instruction to the area of content/study and the relevant CCSSM standard. The print version of the series notes the alignment of the content to the CCSSM in more detail and places than the digital version. Examples include:

In Mathematics I, the Standards Correlations and Connections to Future Courses document indicates that the course aligns to standards from all strands of the CCSSM. The Suggested Pacing Guide details days of instruction per unit. All days that do not indicate an Assessment are aligned to a standard. Some examples are the following: Unit 1, Lesson 1: Interpreting Structure in Expressions, Day 3, aligns to standard A-SSE.1b, and Lesson 5: Rearranging Formulas, Day 23, aligns to standards A-CED.1 and A-CED.2.
In Mathematics II, Unit 4, Lesson 4.1.1, the Supporting Resource (Teacher Instructional Support) for Describing Events, Prerequisite Skills, includes the alignment to middle school standard 7.SP.8b in the print version. The comparable document in the print version of Lesson 4.1.2 indicates alignment to elementary (4.NF.1), middle school (7.SP.1, 7.SP.8c, 7.EE.3), and high school (S-CP.1) math standards. In Lesson 4.1.3, the print version of the Warm-Up Debrief indicates alignment to S-CP.2.
In the print version of Mathematics III, Unit 2A, the Standard Correlation follows the Table of Contents; it provides a comprehensive list mapping the sub-lessons to the CCSSM. The Lesson 1, Instructional Resource indicates the CCSSM A-SSE.1 and A-APR.1. Lesson 2A.1.1 Warm-Up Debrief indicates alignment to A-SSE.1. The Teacher Instructional Support indicates the Prerequisite Skills align to 5.OA.2 and 6.EE.2a.

Although, the Standards Correlations and Connections to Future Courses document includes a Connections to Future Courses section that “provides a map between topics introduced in each unit of this course and subsequent courses where each topic is revisited and built upon.” The map provided lacks explanations of the role the topics have in the context of the overall series. Some explanations of the role of the specific course-level mathematics are present in the context of the series, in the Suggested Pacing Guide, Unit Overview. Examples include:

The Mathematics I, Suggested Pacing Guide, Unit 2: Linear and Exponential Relationships, Unit Overview states the following: “This unit builds on the concepts of functions that were first introduced in Grade 8. Students extend their understanding of functions to include exponential relationships. Students learn how to analyze and model relationships from contexts, graphs, tables, and equations using what they know about exponential and linear relationships.”
The Mathematics II, Suggested Pacing Guide, Unit 3: Expressions and Equations, Unit Overview states the following: “Students reexamine the basic structures of expressions, but this time these structures are applied to quadratic expressions. Then, students learn to solve quadratic equations using various methods. Next, students create quadratic equations in various forms and learn how to rearrange formulas to solve for a variable that is being squared. The unit builds on previous units by introducing the Fundamental Theorem of Algebra and showing students how complex numbers are solutions to some quadratic equations. At this point, students are introduced to rational functions. Again, students learn to write exponentially structured expressions in equivalent forms. The unit ends by returning to a familiar topic—solving systems of equations—but now complex solutions can be determined.”
The Mathematics III, Suggested Pacing Guide, Unit 4B: Mathematical Modeling and Choosing a Model, Unit Overview states the following: “Students revisit the process of creating equations in one variable and explore creating constraints and rearranging formulas. They then learn about transforming models and combining functions. Students review various kinds of functions, including linear, exponential, quadratic, piecewise, step, absolute value, square root, and cube root functions, all with an eye to choosing a model for a real-world situation. Finally, students consider geometric models, including two-dimensional cross sections of three-dimensional objects.”

Indicator 3D

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Walch CCSS Integrated Math Series do not provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials do not contain strategies for informing students, parents, or caregivers about the mathematics their student is learning. Additionally, no forms of communication with parents and caregivers and no suggestions for how stakeholders can help support student progress and achievement were found in the materials

Indicator 3E

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

In each course of the series, the Program Overview includes correspondences to standards, criteria and practices. These correspondences offer a look into how the program was designed and how it supports and reinforces practices and standards. For example, the Correspondence to NCTM Principles to Actions Teaching Practices states, “the courses were designed by experienced educators and curriculum developers, informed by best-practice research … These materials support and sustain good teaching practices.”

Each course also includes a Supporting Resource: Instructional Strategies document that notes that the instructional strategies within each lesson are “research-based and best practices that work for all students” and “fall into four main categories: Literacy, Mathematical Discourse, Annotation, and Graphic Organizers.” Descriptions of each strategy address the what, how, and when as well as the cross-correlation to other standards (e.g., SMP, WIDA, ELD, and ELA) and source information. Examples include:

Novel Ideas, a literacy strategy, is described as “a classroom activity that explores students’ understanding of important Tier 2 vocabulary words they will use in math class.” The materials provide an 8-step implementation process, suggest when and how to use it with students who need additional support, align it to nine other standards, and attach two electronic sources.
Reverse Annotation Protocol, an annotation strategy is described as “a strategy that asks students to identify and write down key information from math problems.” The materials then address how and when to use it with students. Additionally, the materials include a Reverse Annotation Template that guides a student through analyzing the problem, recording an initial estimate of the solution, and “Checking It Over” (a sequence of questions that help a student reflect upon their initial estimate) section.
Sentence Starters are suggested to encourage Mathematical Discourse. “A Sentence Starter is a common phrase or mathematical sentence frame that can help students begin and sustain academic conversations around mathematical content. It helps guide students through the discussion and bring out pertinent ideas that can lead to greater understanding.” In addition to addressing when and how to use it with students, the materials suggest Discourse Sentence Starters and Math Sentence Starters.

In addition, these research-based strategies are noted occasionally throughout the lessons. Examples include:

The Station Activities, Instruction & Answer Key Discussion Guide states, “To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to “debrief” the station activities.” Think, Pair, Share is recommended: “Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class.”
In Mathematics II, Unit 2, Lesson 2.2.1, Supporting Resource, Key Concepts includes a graphic organizer image, suggesting its use in organizing the key features of a quadratic function (e.g., intercepts and increasing/decreasing behavior). This graphic organizer image corresponds to the image and explanation provided in the Program Overview, Supporting Resource: Instructional Strategies.
In Mathematics II, Unit 3, Lesson 3.4.1, Problem-Based Task, the inclusion of the Modeling image suggests that teachers demonstrate “how to make or generate mathematical representations or models, how to validate them, and how to use them to solve real-world problems.” The Program Overview, Instructional Strategies: Mathematical Modeling section suggests a list of questions and answers in order to create a mathematical modeling classroom environment.
In Mathematics III, Unit 4 [Unit 4B], Lesson 4B.4.1, Problem-Based Task, Close Reading and Mathematical Discourse (“the process of applying mathematics to a real-world problem with a view of understanding the connection”) images suggest their implementation with this task.

Indicator 3F

Materials provide a comprehensive list of supplies needed to support instructional activities.

The materials reviewed for Walch CCSS Integrated Math Series do not meet expectations for providing a comprehensive list of supplies needed to support instructional activities. Although each Station Activity is accompanied by an Instruction & Answer Key that includes a Materials List/Setup, there are many instances throughout the course where materials are noted—for example, within Supporting Resources and Instructional Guides—but without a comprehensive list. Examples of a comprehensive list provided by the Station Activities include:

In Mathematics I, Unit 1, the Station Activities, Instruction & Answer Key, lists the materials needed per station. The list includes: 12 index cards with specific measurement labels, a calculator, green and yellow marbles, and large and small algebra tiles.
In Mathematics III, Unit 3 [Unit 2B], the Station Activities Instruction & Answer Key lists the materials needed per station. The list includes: three sheets of card stock in different colors, six-sided number cube, white paper, and ten index cards.

Examples where the supplies are referenced but without a comprehensive list, include, but are not limited to:

In Mathematics II, Unit 6, Lesson 6.3.1, Supporting Resource, indicates that students will use a compass and a straightedge. Students also have the option to substitute a reflective device or patty paper with the compass. These supplies are indicated within the Supporting Resource but do not appear in a comprehensive list for the course.
In Mathematics III, Unit 1, Lesson 1.2.1, Supporting Resource, the materials indicate that a random number generator will be helpful; the Guided Practice requires the use of a graphing calculator “to estimate the mean and standard deviation.” These supplies are indicated within the Supporting Resource and in the student materials but do not appear in a comprehensive list for the course. Lesson 1.2.2, Problem-Based Task requires the use of a deck of cards (to simulate a set of eight foul shots); a standard deck of cards is not included in a comprehensive list for the course.

Indicator 3G

This is not an assessed indicator in Mathematics.

Indicator 3H

This is not an assessed indicator in Mathematics.

Criterion 3.1: Teacher Supports

Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for Assessment. The materials partially include assessment information that indicate which standards and practices are assessed and partially provide assessments that include opportunities for students to demonstrate the full intent of course-level standards. The materials partially provide multiple opportunities throughout the courses to determine students’ learning or sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Indicator 3I

Assessment information is included in the materials to indicate which standards are assessed.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials identify the standards assessed for some of the formal assessments.

The Program Overview of each course indicates that “each lesson includes a Pre-Assessment and a Progress Assessment, and each unit concludes with a Unit Assessment.” In the digital and print versions of the materials, the Program Overview- Standards Correlations (and Connections to Future Courses) document indicates the standards per lesson. These standards correlate to the entirety of the activities in each lesson, including the Pre-Assessment and Progress Assessment. In the digital version, the standards listed for all the lessons within a unit correlate to the Unit Assessments. The digital End-of-Course Assessment Answer Key includes a chart that enumerates each item, its key, and the standard assessed per item. In the print version, the Warm-up Instruction Resource lists the standards addressed within the lesson, including the Pre-Assessment and Progress Assessment. The digital and print materials differ in how standards are indicated on assessments. The materials do not identify practice standards for any assessments.

Examples include, but are not limited to:

In Mathematics I, Unit 7, End-of-Course Assessment, Problem 2, students are given a scenario involving cupcakes, hot dogs, and a fundraising goal and must identify which inequality represents the scenario. The End-of-Course Assessment Answer Key identifies the standard being assessed for Problem 2 as A-CED.1 but does not identify the practice.
In Mathematics II, Unit 7, End-of-Course Assessment, Problem 9, students are tasked with identifying the inverse of a function given the domain. The End-of-Course Assessment Answer Key identifies the standard being assessed for Problem 9 as F-BF.4a but does not identify the practice.
In Mathematics III, Unit 7, End-of-Course Assessment, Problem 41, “The ages of dogs regularly visiting a popular dog park are normally distributed. The mean age is 6, and the standard deviation is 2 years. There are 540 dogs that regularly visit the park. a. Sketch a normal curve for this situation. b. What percentage of dogs are between 2 and 12 years old? c. How many dogs is this?” The End-of-Course Assessment Answer Key identifies the standard being assessed for Problem 41 as S-ID.4 but does not identify the practice.

Indicator 3J

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for including an assessment system that provides multiple opportunities throughout the series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials provide Pre-Assessments and Progress Assessments per lesson, a Unit Assessment per unit, and an End-of-Course Assessment. In addition, the materials include access to the Curriculum Engine Item Bank “Exam View,” which allows teachers to “build formative and summative assessments.” Through the use of Exam View, teachers have the option to create multiple assessments. Answer keys for each assessment provide the correct answer for multiple choice and extended response questions. The Item Analysis and Course Performance reports provide sufficient guidance to teachers for interpreting student performance, but do not provide suggestions for following-up with students.

The Item Analysis and Course Performance Reports can be accessed by clicking the “Reports” or “Users” buttons at the top of the main page of the Curriculum Engine. The Item Analysis Report provides the number of students that have attempted the assessment, the overall highest and lowest score, the average score, and the difficulty level of the assessment. The item breakdown section provides the teacher with the scores of the individual students, whether they received a full score, partial score, no score, not marked, or null score per item in addition to the time spent on each item. Additionally, the response analysis section provides teachers with the total points available for the assessment, a chart showing how every student in the class responded to each item, item type (e.g. multiple choice, match list, long text…), and the ability to view the item and the specific answers provided by the individual students. The Course Performance Report provides an individual report of student performance in three categories: User, Course Content and Standards. The User category can be displayed in a graph or table view, by clicking on the individual unit bars, teachers can drill down further to see which category students place in (Proficient, Developing or Emerging) on that unit assessments. The Course Content category shows the overall performance for each module of a select course, and by clicking the side arrow to expand into the unit, teachers are able to see how much time was spent on assessments and which category students place in (Proficient, Developing, or Emerging) on the course content. Course Content also gives access to recommendations for learning objectives that can be added to the course, but these recommendations are only given for the lessons and not assessments. The Standards category shows the overall performance for each standard as well as learning objectives by standards of the selected course, and by clicking the side arrow to expand into the standard, teachers are able to see which category students place in (Proficient, Developing or Emerging) for each standard and the type of assessment problems the students were given (e.g. 5 multiple-choice problems, 10 multiple-choice problems; 1 extended response, 15 interactive online assessment items…). Student performance within these categories can be interpreted in one of six ways: Not Scorable, Overdue, In Progress, Proficient: Above 85% (green color), Developing: 65% - 85% (yellow color), and Emerging: Below 65% (red color).

Assessments within the materials do not provide suggestions for following-up with students.

Indicator 3K

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

The materials reviewed for the Walch CCSS Integrated Math Series partially meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of course-level standards and practices across the series. Assessments do not include opportunities for students to demonstrate the full intent of course-level standards across the series.

The following assessments are within each course:

The Pre-Assessment, “can be used to gauge students’ prior knowledge and to inform instructional planning.”
The Progress Assessment consists of “10 multiple-choice questions, as well as one extended-response question that incorporates critical thinking and writing components. This can be used to document the extent to which students grasp the concepts and skills addressed during instruction.”
The Unit Assessment consists of “12 multiple-choice questions and three extended-response questions that incorporate critical thinking and writing components. This can be used to document the extent to which students grasp the concepts and skills addressed during the unit.”
The Comprehensive Assessment (available in some units) covers material from two units.
The End-of-Course Assessments are 45 questions each and are a combination of multiple-choice questions and extended-response questions. These assessments are used to document the extent to which students grasp the concepts and skills addressed during the course.

All of the aforementioned assessments are accompanied by answer keys and are provided in both PDF format and as a Learnosity Assessment. Standards information is only included for the End-of-Course Assessments and can be found in the End-of-Course Assessment Answer Key.

Examples of how standards are not assessed or only partially assessed throughout the series include, but are not limited to:

The materials do not fully assess F-IF.7. The standard specifies that students will graph and analyze linear, quadratic, square root, absolute value, cube root, piecewise, step , polynomial, exponential, logarithmic, and trigonometric functions. The Mathematics I, Unit 7, End-of-Course Assessment, Problem 13 asks students to match an equation to a given graph of a linear function. Mathematics II, Unit 2, Lesson 2.4, Pre- and Progress Assessments ask students to demonstrate their knowledge of square root, absolute value, step, and piecewise functions. Mathematics II, Unit 7, End-of-Course Assessment, Problems 3 and 4 ask students to match a graph of a quadratic function and a square root function, respectively, to their function. Mathematics III, Unit 7, End-of-Course Assessment, Problem 10 asks students to identify the end behavior of a polynomial; Problems 32 and 33 ask students to identify period and the equation for a trigonometric graph, respectively. Mathematics III, Unit 5 [Unit 4A], Lesson 5.2, Pre- and Progress Assessments ask students to engage with logarithmic functions. The materials do not assess student knowledge of the tangent or trigonometric cofunctions. Assessments do not ask students to graph logarithmic or trigonometric functions.
In Mathematics II, Unit 7, the End-of-Course Assessment Answer Key states Problems 23-25 assess G-CO.10, proving theorems about triangles. Students do not prove theorems about triangles but use theorems pertaining to triangles to find missing angles in triangles (Problem 23), justify the last two steps of a triangle proof but do not use triangle theorems (Problem 24) and find perimeter using given midsegment information (Problem 25). In the Pre-, Progress, and Unit Assessments, students use theorems about triangles to solve problems but do not prove theorems about triangles.

In Mathematics II, students engage with quadratic functions: they factor and complete the square to show zeros, extreme values, and symmetry of a graph, and interpret these terms of a context F-IF.8a. Unit 2, Lesson 2.1.2, Pre- and Progress Assessments include contextual questions; however they do not ask students to complete the square to reveal the zeros, extreme values, or symmetry of a graph. Mathematics II, Unit 7, End-of-Course Assessment, Problem 42 asks students to arrange the steps of completing the square into the appropriate sequence. Students do not show extreme values and symmetry of the graph.

In End-of-Course Assessment Answer Keys, CCSSM are identified for each assessment item; however Mathematical Practices are not identified for any of the assessment items. Examples of multiple-choice questions include:

Mathematics I, Unit 2, Lesson 3, Pre-Assessment, Problem 1 states, “Identify the pattern in the sequence of numbers and choose the next number in the sequence. 1, 1, 2, 3, 5, _ a.7, b. 8, c. 9, d. 10.”

Mathematics II, Unit 4, Lesson 2, Progress Assessment, Problem 10 states, “What is the probability that 2 cards selected from a standard deck of 52 cards without replacement are both multiples of 5? a. 0.00075, b. 0.0045, c. 0.011, d. 0.021”
Mathematics III, Unit 7, End-of-Course Assessment, Problem 14 states, “Which simplified expression is equivalent to $\frac{8x^3-6x}{2x}$ ? a. $4x^2-3$ , $x\ne0$ , b. $4x^2-6x$ , $x\ne 0$ c. $61, x\ne 0$ d. $4x-3, x\ne 0$ , $\pm\frac{\sqrt{3}}{2}$ ” (A-SSE.2).

Examples of extended-response questions include:

Mathematics I, Unit 6, Lesson 1, Progress Assessment, Problem 11 states, “Read the scenario, write an equation to model the walkway, and then solve the problem. Mariah’s house is located at the point -1,2. The equation of the road is y+4x=2. Mariah is building a walkway from her house to the road. What is the approximate length of the shortest walkway she can build? Each unit represents 100 yards.”
Mathematics II, Unit 7, End-of-Course Assessment, Problem 44 states, “A box contains three blue marbles and two white marbles. Rob pulls a marble from the box at random, then replaces it and selects another. Let $B_1,B_2,B_3,W_1$ , and $W_2$ represent the different marbles in the box. A. List the complete sample space for this scenario. B. What is the probability that Rob selects a blue marble both times?” (S-CP.1).
Mathematics III, Unit 3 [Unit 2B], Unit Assessment, Problem 13 states, “It takes Alden 9 hours to bale all the hay in a field, but his brother, Anthony, can do the same job in 7 hours. If they bale all the hay in an entire field together, approximately how long will it take them to complete the job?”

Indicator 3L

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for the Walch CCSS Integrated Math Series do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The digital version of the Pre-Assessments, Progress Assessments, some Unit Assessments, and End-of-Course Assessment allows users to change background and font color, font size and grant access to zoom in/out features. A popular accessibility tool, Text to Speech Technology, may be accessed through an application or program installed on a local device. The print version of the assessments does not provide user accommodations. Examples of the accommodations offered in the digital version include:

In Mathematics I, Unit 3, Topic A, Progress Assessment, students who select the Accessibility button may adjust the color scheme (e.g, purple on light green, white on black…etc), the font size (to as small as 75% or as large as 175%), and/or zoom in/out.
In Mathematics II, Unit 4, Pre-Assessment, students who select the Accessibility button may adjust the color scheme (e.g, purple on light green, white on black…etc), the font size (to as small as 75% or as large as 175%), and/or zoom in/out.
In Mathematics III, Unit 7, End-of-Course Assessment, students who select the Accessibility button may adjust the color scheme (e.g, purple on light green, white on black…etc), the font size (to as small as 75% or as large as 175%), and/or zoom in/out.

However, accommodations are not provided on all Unit Assessments. Examples include, but are not limited to:

In Mathematics I, four of the seven Unit Assessments do not offer accessibility options for students.
In Mathematics II, four of the seven Unit Assessments do not offer accessibility options for students.
In Mathematics III, four of the seven Unit Assessments do not offer accessibility options for students.

The materials include access to the Curriculum Engine Item Bank, “ExamView,” which allows teachers to create exams according to any grade/course standards. Teachers are able to create assessments by choosing from standard sets or adding their own questions. As a result, these items have the potential to alter course-level expectations due to the fact that these are teacher-created items.

Criterion 3.2: Assessment

Criterion 3.3: Student Supports

The program includes materials designed for each student’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations to support their regular and active participation in learning course-level mathematics, strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning course-level mathematics and provide manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. The materials partially provide extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity.

Indicator 3M

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The materials reviewed for the Walch CCSS Integrated Math Series meet expectations for providing strategies and support for students in special populations to support their regular and active participation in learning series mathematics.

Many strategies and supports for students in special populations can be found in various sections of the Program Overview. The Program Overview, Instructional Strategies, Ensuring Access for All Students section, states the following: “...The strategies presented here can help all students succeed in math, literacy, school, and ultimately, in life. These instructional strategies provide teachers with a wide range of instructional support to aid English as a Second Language (ESL) students, students with disabilities (SWD), and struggling readers…The instructional strategies detailed here fall into four main categories: Literacy, Mathematical Discourse, Annotation, and Graphic Organizers. These strategies provide teachers with research based strategies to address the needs of all students.” Each category has an icon which is located throughout the materials to indicate the appropriate strategies at the point of use. Examples of strategies and supports found in the Program Overview, Instructional Strategies section include:

Understanding the Language of Mathematics: Literacy, “Mathematics has its own language consisting of words…For students with disabilities, the stress on repetition and different representations in this approach is essential to their ability to grasp the math concepts. For ESL students, repetition and different representations can strip out some of the English language barriers to understanding the language of mathematics, as well as provide multiple means of accessing the content. Literacy strategies include: Close Reading, Text-to-Speech, Concept-Picture-Word Walls, and Novel Ideas.”
Communicating Mathematical Content: Mathematical Discourse, “Reading, writing, speaking, and listening are all important ways to learn and express information…Routines and structures are essential to support students with disabilities, as they often benefit from following a routine.This can lead to developing capability in their mathematical skills...Mathematical Discourse strategies include Sentence Starters and Small Group Discussion.”
Understanding Mathematical Content: Annotation, “Understanding mathematical content is an extremely important skill,both in the Math classroom and in life…For students with disabilities, many of whom struggle with the distractions inherent in many high-school level texts, making notes and drawing pictures to explain a problem can help them focus. ESL students will be pointed to certain Tier 3 vocabulary words and determine which Tier 2 vocabulary words they must learn to be proficient in math class and in the English Language. Annotation Strategies include Reverse Annotation and CUBES protocol.”
Organizing Mathematical Content: Graphic Organizers, “Organizing mathematical content is a crucial skill for problem solving…For students with disabilities, visual representations and graphic organizers can help them clarify their thoughts and focus on the math. ESL students also benefit from visual representations and graphic organizers. Organizing mathematical knowledge with visuals can help ESL students navigate math content while learning the language. Graphic organizers include Frayer Models and Tables of Values”

Examples of strategies and supports for students in special populations that can be found in other sections of the Program Overview include, but are not limited to:

The Unit Structure section includes a description of Common Errors/Misconceptions, which “is a list of the common errors students make when applying Key Concepts. This list suggests what to watch for when students arrive at an incorrect answer or are struggling with solving the problems.” The Optional Problem-Based Task Coaching Questions with Sample Responses includes questions that, “scaffold the task and guide students to solving the problem(s) presented in the task. They should be used at the discretion of the teacher for students requiring additional support.”
The Bilingual Glossary section which contains, “vocabulary terms and formulas from throughout the program, organized alphabetically. Each listing provides the term and the definition in both English and Spanish.”

Additional examples of where and how the materials provide strategies and supports for students in special populations include:

The Problem-Based Task Implementation Guides have a section that helps teachers be aware of common student errors and misconceptions associated with the task and offers suggestions for facilitating and monitoring while providing alternative solutions. For example, in Mathematics I, Unit 5, Lesson 5.5.2, Problem-Based Task Implementation Guide, Addressing Common Errors/Misconceptions section indicates that a student may not recognize that rigid motion preserves shape and size and suggests that “once students perform a rigid motion, remind them to check the lengths of the sides of the preimage and the image to make sure the corresponding sides and angles are congruent.” The Monitoring and Coaching section, similarly supports the student by providing suggestions of what to ask students before and during the task.
Additional support series are available for Mathematics I and II only that can “help teachers provide necessary differentiation to support all students in meeting the standards of CCSS Integrated Pathway: Mathematics (I and II). These resources may be used in Support, ELL, or Special Education classes, in RTI programs or Math Labs, or with individuals or groups requiring additional support.”

Indicator 3N

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

The materials reviewed for Walch CCSS Integrated Math Series partially meet expectations for providing extensions and/or opportunities for students to engage with course-level mathematics at higher levels of complexity. The materials provide some opportunities to investigate mathematics at a higher level of complexity, however the opportunities for extension do create more work for students.

Some Problem-Based Tasks have implementation guides which include an “Extending the Task” section; these sections do allow students to engage with course-level mathematics at higher levels of complexity. However, these students would be completing more assignments than their classmates.

Examples of the materials providing opportunities for students to investigate the content at a higher level of complexity include:

Mathematics I, Unit 3, Lesson 3.1.1, Problem-Based Task, students “use properties of equality and properties of operations to justify a solution method to an equation. They will form a generalization from simplifying a numerical expression, and they will construct a viable argument as to why the answer to the equation described in the task is always 3.” The Problem-Based Task, Implementation Guide, Extending the Task, provides two options, one of two suggest, “To extend the task, ask students to work with a partner to create a different magic number game with different steps of operations. Ask them to write down the steps, and also to work through their equation, identifying the properties of equality and operations that they use as they go. Once they have completed their new game, ask them to share the game with the class. Encourage and guide a discussion about each pair’s number game, and encourage students to use the proper terminology when explaining why their magic number game works.”
Mathematics II, Unit 1, Lesson 1.1.1, Problem-Based Task, “Students will use a town’s current census and estimated 10-year growth rate to estimate the population before the next census is taken.” The Problem-Based Task, Implementation Guide, Extending the Task, provides the following two options: “To extend the task, ask students to find the estimated yearly growth rate and create a table of values showing the population for each year from the current year to 8 years from today. Ask students to demonstrate and verify that the yearly growth rate (1.03) holds for each year.” and/or “Another option is to have students graph the function and then analyze the graph.”

Honors Supplements are available for Math II and Math III only. The Program Overview for both supplements indicates that “these materials may be used in honors or accelerated courses, or with individuals or groups of students requiring additional challenge.” The Unit Structure states that “All of the instructional topics … are comprised of one or more lessons and an extension activity. Each lesson begins with pre-assessment problems, followed by lists of Essential Questions and

vocabulary (titled “Words to Know”) for the lesson...Next comes a warm-up problem, debrief, and connections to the lesson. Then, the lesson features a list of identified prerequisite skills that students need to have mastered in order to be successful with the new material in the Honors lesson. This is followed by an introduction, key concepts, common errors/misconceptions, guided practice examples, a problem-based task with coaching questions and sample responses, a closure activity, and practice. Each lesson ends with progress assessment problems to evaluate students’ learning. Each topic culminates with an extension activity providing students the opportunity to apply the topic in a more sophisticated way. The extension activity includes a problem-based task along with suggestions for introducing, facilitating, and debriefing the activity.”

Examples of advanced students doing more assignments than their classmates, include, but are not limited to:

Math II Honors Supplement, Topic 4, Learning/Performance Task, students seek to prove that two unique package designs have the same volume. Problem 6 asks that students “Draw the design of a third package in a 3-D plane that can be proved by Cavalieri’s Principle to have the same volume as the two packages in the diagram. Provide coordinates and show your calculations.”
Math III Honors Supplement, Topic 1, Learning/Performance Task, students will be introduced to the process of animation by using matrices to move a figure along a graph. Suggestions for Extending the Activity provides two options: one of two suggests, “Students with knowledge of a 3-D animation program can create a simple animation using programming that involves matrices. Animation software is available to download from several sites, many of which are open-source or offer free trials. Some options include Blender, Synfig, and Toonboom.”

Indicator 3O

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Walch CCSS Integrated Math Series provide some varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning. The materials provide a variety of approaches for students to learn the content over time but provide limited opportunities for students to monitor their learning. There is no opportunity for self-reflection.

The materials provide a digital presentation for each lesson that teachers may use to deliver instruction. Each lesson consists of a consistent sequence of activities that supports students’ learning the content over time: Scaffolded Practice, Guided Practice, Online Interactive Problems, a Problem-Based Task with optional Coaching Questions, and two sets of Practice Problems. In addition, units include Online Interactive Conceptual Activities and Conceptual Learning/Performance Tasks. Opportunities for students to monitor their learning are limited to small group work and online resources. Examples include:

In Mathematics I Support Supplement, Unit 7, E-Skills and Readiness Assessments, Topic A, Lesson 1, Supporting Resource, Recommended Resource section, “IXL Learning. Simplify expressions using order of operations and parentheses. … This site allows users to practice evaluating expressions using the order of operations. Immediate feedback is provided, as well as instruction on how to arrive at the correct answer.” Support Supplements materials are only provided for Mathematics I and II.
In Mathematics II, Unit 2, Lesson 2.1.1, Problem-Based Task Implementation Guide, Debriefing the Task states, “Compare the strategies of different groups. Ask students to share their strategies in determining the solutions and ask why they used these methods.”

Indicator 3P

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

The materials reviewed for Walch CCSS Integrated Math Series provide some opportunities for teachers to use a variety of grouping strategies. Grouping strategies are suggested within the materials, however these suggestions are not consistently present or specific to the needs of particular students.

The Program Overview, Introduction to the Program indicates that “The Station Activities correspond to the content in the units and provide students with the opportunity to apply concepts and skills, while you have a chance to circulate, observe, speak to individuals and small groups, and informally assess and plan.” The Program Overview, Station Activities Guide provides the following guidelines for Forming Groups of Students: “All activity sets consist of four or five stations. You might divide the class into four or five groups by having students count off from 1 to 4 or 5.If you have a large class and want to have students working in small groups, you might set up two identical sets of stations, labeled A and B. In this way, the class can be divided into eight groups, with each group of students rotating through the ‘A’ stations or ‘B’ stations.” In addition, the Station Activities Guide includes roles that may be assigned to students within a group. The types of interactions, specified within each individual task or activity, are the same throughout the three courses.

Examples of the materials providing opportunities for teachers to use a variety of grouping strategies include:

In Mathematics I, Unit 4, Lesson 4.3.1, Problem-Based Task Implementation Guide, Debriefing the Task suggests to “Compare methods used by the groups and discuss advantages and disadvantages of the varying strategies.” However, the task does not indicate that students should be placed in groups, and teachers are not provided guidance on the grouping format.
In Mathematics II, Unit 6, Lesson 6.4.1, Problem-Based Task Implementation Guide, Extending the Task, suggests “To extend the task, ask students to work with a partner to create another task, using the same scenario as the one given, but with different values for the circumference and the revolution speed per minute” or “Ask the students to work together to determine where they would place 6 carousel horses around the outer edge of the merry-go-round such that they are equally spaced apart.” While grouping students is suggested, no guidance is given to teachers on how to group students based on their needs.
In Mathematics III, Unit 3 [Unit 2B], Station Activities, Instruction & Answer Key suggests that teachers use Think, Pair, Share, by having “students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class.” No guidance is given to teachers on how to group students based on their needs.

Indicator 3Q

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. The Program Overview, Instructional Strategies, provides, “...support for the Mathematics Standards and the Standards of Mathematical Practice (SMP), English Language Development (ELD) Standards, English Language Arts Standards, and WIDA English Language Development Standards. Within each lesson throughout this course, you will find suggested instructional strategies.” Examples of these instructional strategies include, but are not limited to:

Close Reading with Guiding Questions, which “can be used for any activity in which literacy could be a barrier to learning or demonstrating mastery of mathematical concepts… for struggling readers, including ESLs, Close Reading with Guiding Questions can help make an intimidating lesson, word problem, or task much more accessible..Discussions around the questions will help students graph the math concepts.” Addressing implementation it states, “…the teacher should determine the most important information students need to obtain from a text, whether it is a math problem to solve, a task to complete, or an informational lesson or article to read. Then, the teacher should come up with some questions to guide students before they read. … To ensure the questions are accessible for students and to encourage reflection and debate after reading, many of these questions should be designed as either ‘True/False’ or ‘Always True/ Sometimes True/Never True.’ Students can represent their reasoning for their answer in writing, numbers, or graphic/pictorial representations.”
“Concept-Picture-Word Walls can also be written in multiple languages. Especially for students who are on-grade-level in their native language, a multilingual Concept-Picture-Word Wall can help students connect the content they already know in another language to the English vocabulary necessary for success on English-language math activities and tests.” Addressing implementation it states, “Just seeing the vocabulary on a Concept-Picture-Word Wall by itself will help students; more importantly, referring to the words as the teacher uses them in class help students connect the visual to the application. A simple gesture to the wall makes a very explicit reference to the word as it is used and allows students to connect the unfamiliar word to its meaning in context.”
Reverse Annotation is a strategy that, “can help students identify key information, even when certain vocabulary words are not known...This strategy is especially beneficial for ESL students. Using reverse annotation with graphic organizers benefits ESL students by removing a lot of the confusing wording and allowing them to focus on the important pieces of a problem.” Addressing implementation it states, “When students first begin to use Reverse Annotation, the teacher should walk them through the steps individually to ensure they can accurately identify the question, key words, and important information.”
CUBES Protocol, “can help students identify key words and features, even when certain vocabulary words are not known…Using circles, underlines, and boxes can help students organize the key facts and vocabulary, and the elimination of unnecessary information can simplify the problem. This strategy is especially beneficial for ESL students. Combining CUBES with graphic organizers also benefits ESL students by removing a lot of the confusing wording and allowing them to focus on the important facts of a problem. When using CUBES with a graphic organizer, all students, including ESL students, will begin to think about problem solving in a way that helps encourage them to use the appropriate information to find a solution.” Addressing implementation it states, “The steps for CUBES are: 1. C: Circle all the key numbers. 2. U: Underline the question. 3. B: Box in the key words that will determine the operation(s) necessary and write the mathematical symbol for the operation(s). 4. E: Evaluate the information given to determine the strategy needed. Eliminate any unnecessary information. 5. S: Solve the problem, show your work, and check your answer.”

General strategies/supports the materials provide for students who read, write, and/or speak in a language other than English include:

Enhanced Instructional PowerPoint (Presentation), “Each lesson includes an instructional PowerPoint presentation with the following components: Warm-Up, Key Concepts, Guided Practice…These instructional PowerPoints are downloadable and editable.”
The Bilingual Glossary section which contains “vocabulary terms and formulas from throughout the program, organized alphabetically. Each listing provides the term and the definition in both English and Spanish.”

Indicator 3R

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Walch CCSS Integrated Math Series provide a balance of images or information about people, representing various demographic and physical characteristics. While images are not used within materials, the lessons do include a variety of names in problem contexts that are representative of various demographic and physical characteristics. Examples include:

In Mathematics I, Unit 2, Lesson 2.1.1, Practice Set A, Problem 8, students practice drawing a graph that represents the solution set of an equation. “Katya is a caterer. She has a cookie recipe that calls for 2 eggs per batch. Katya wants to know the number of eggs she needs according to how many batches she cooks. What equation can be used to represent the number of eggs Katya needs for any number of batches? Draw a graph to represent this situation. How many eggs would Katya need for 4 batches of cookies?”
In Mathematics II, Unit 2, Lesson 2.1.2, Problem-Based Task, students will use a quadratic function to analyze the path of a hang glider and determine whether the ride is safe. “Shin is a beginner hang glider. He’s practicing jumping from a certain height, dipping initially, and then rising…Will Shin stay above the safe height? How long will it take for Shin to reach the initial height of the jump? Explain your answer using mathematical reasoning.”
In Mathematics III, Unit 1, Lesson 1.1.3, Warm-up, students analyze data from a histogram, to find the mean, median and standard deviation. “Domenic decided to research the word lengths in his favorite novel for his statistics class. He used a sampling technique to select 40 words from the book…Use Domenic’s table and histogram to solve the problems that follow.”
Other names that could represent a variety of cultures are represented in the materials, i.e., Jacob, Antonio, Marisol, Amy, Amit Faith, Jia, Kaya, and Chen.

Indicator 3S

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Walch CCSS Integrated Math Series do not provide guidance to encourage teachers to draw upon student home language to facilitate learning. There is no evidence of promoting home language as an asset to engage students in the content material or purposefully utilizing student home language in context with the materials.

Indicator 3T

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Walch CCSS Integrated Math Series do not provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning. The Program Overview of each course indicates that “The program is useful for striving students and diverse classrooms. This program realizes the benefits of exploratory and investigative learning and employs a variety of instructional models to meet the learning needs of students with a range of abilities.” Although this statement provides a general overview of the cultural relevance within program components, materials do not embed guidance for teachers to amplify students’ cultural and/or social backgrounds to help facilitate learning.

Indicator 3U

Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Walch CCSS Integrated Math Series provide some support for different reading levels to ensure accessibility for students. The Program Overview, Instructional Strategies makes the following recommendations to support struggling readers, Close Reading with Guiding Questions, Text-to-Speech Technology, and Mathematical Modeling. For each of these strategies, the materials discuss the following: What the strategy is, How the strategy is implemented in the classroom, When the strategy is used in the classroom, How the strategy is used with students needing additional support/struggling students, and Standards the strategy address.

Examples where the materials provide supports for struggling readers:

“How can I use Close Reading with Guiding Questions with students needing additional support? For struggling readers, including ESLs, Close Reading with Guiding Questions can help make an intimidating lesson, word problem, or task much more accessible. Questions focusing more on Tier 2 and Tier 3 vocabulary, text features, and real-world concepts can help struggling readers relate to the text and learn how to decipher the text in context. Discussion around the questions will help students grasp the math concepts.”
“How can I use Mathematical Modeling with struggling students? When struggling readers, which includes ELLs and students with learning disabilities, are exposed to rigorous math learning tasks, there must be a level of scaffolding that includes coaching and guided questions that help to make a word problem or learning task much more accessible. Teachers should come up with questions to guide the students before and during the engagement of the task…Allowing struggling readers to explain their answers using words, numbers, or graphics/pictures ensures that they can express their opinion and rationale despite a potential lack of vocabulary. Through these representations and the ensuing discussion, students will begin to learn the necessary math concepts to be successful.”

Although the materials mentions the benefits of Text-to-Speech Technology, this technology is not accessible within the program. An example of how the material suggest Text-to-Speech Technology to support struggling readers:

“How can I use Text-to-Speech Technology with students needing additional support? Text-to-Speech Technology is an important adaptation and accommodation for struggling readers. Students who have read-aloud accommodations sometimes don’t receive them because they are either embarrassed to accept them or because of staffing restrictions. These students can use Text-to-Speech Technology to supplement their math instruction by having text automatically read to them in a manner in which they can process it.”

Indicator 3V

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials reviewed for Walch CCSS Integrated Math Series meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods. The lessons use a blend of external linked websites to complement their digital and print materials. There is extensive use of virtual manipulatives—such as Desmos, GeoGebra, and IXL—to help students develop an understanding of concepts. Examples of the usage of virtual manipulatives include:

Mathematics I, Unit 1, Lesson 2, Learning/Performance Task Conceptual Activity is a Polygraph concerning Linear Inequalities. In this Desmos activity, students partner with another student as they engage with graphs of linear inequalities. In the ensuing GeoGebra activity, students create one-variable inequalities, plot the solution set of a written problem on a virtual number line, and match word phrases to symbols.
Mathematics II, Unit 1, Lesson 3, Interactive Practice Problem “Algebra of Complex Numbers” is a GeoGebra applet that displays the addition, multiplication and roots of complex numbers graphed on the complex plane.
Mathematics III, Unit 4 [Unit 3], Lesson 3.1.2, Guided Practice, students sketch angles on a unit circle. The ensuing GeoGebra applet allows students to see how the angles are drawn on a digital unit circle; students use the slider to show/draw the angle where the given radians would be located. On the digital Practice Set students choose which letter from the picture of the unit circle would best match the given radians.

Examples of how the materials incorporate physical manipulatives that are accurate representations of the mathematical object they represent includes:

In Mathematics I, Unit 5, Station Activities: Corresponding Parts, Transformations, and Proof, Station 1, students use a ruler and a protractor to prove that two triangles and corresponding parts are congruent. For Station 2, students use graph paper, a ruler, push pins, cardboard, and rubber bands to construct triangles to justify congruent triangles and explain their reasoning.
In Mathematics II, Unit 6, Station Activities: Circumference, Angles, Arcs, Chords, and Inscribed Angles, Station 1, students use a tape measure to measure the radius, diameter, and circumference of a plastic coffee can lid. Students then derive the relationships among these three measurements and solve two real-world problems using circumference. For Station 2, students use a compass, a protractor, and a ruler to construct a circle, radii, a central angle, and an arc; derive the relationship between the arc length and the circumference; and find the length of the arc. For Stations 3 and 4, students continue to use the aforementioned manipulatives to derive the relationship between chords, arcs, and the triangles created by chords and radii and derive the relationship between the inscribed angle and its intercepted arc.
In Mathematics III, Unit 3 [Unit 2B], Station Activities: Rational Expressions and Equations, Station 1, students use three sets of index cards to complete a card sort of rational expressions, matching five sets of “original expression,” “simplified expression,” and “restrictions on the variable” cards. For Station 4, students “work in teams to choose a card with a rational equation on it, and determine the number of solutions to the equation. Teams score points equal in number to the number of solutions for the equation solved.”

Criterion 3.3: Student Supports

Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Walch CCSS Integrated Math Series integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in series standards, have a visual design that supports students in engaging thoughtfully with the subject and is neither distracting nor chaotic and provide teacher guidance for the use of embedded technology to support and enhance student learning. The materials do not include or reference digital technology that provides opportunities for collaboration among teachers and/or students.

Indicator 3W

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

Interactive tools are available throughout the series. Using the Learning Object Manager located under the Content heading on the main page, one can search for and filter out all the Interactive Practice Problems within the series. At the lesson level the following interactives are provided: GeoGebra applets of selected Guided Practice examples, and/or applets of Practice problems (these applets are editable). Practice Set A, an “interactive Learnosity activity with Technology-Enhanced Items.” Conceptual Activities, which “links to interactive open education and Desmos resources.” IXL interactive practices, applets and virtual manipulatives can be accessed through the Learning Object Manager. Additionally, at the end of each unit, there is an interactive unit review.

Examples of how the materials integrate technology such as interactive tools, virtual manipulative/objects, and/or dynamic mathematics software in ways that engage students in the series standard include:

Mathematics I, Unit 4, Lesson 4.1.1 provides an interactive Guided Practice problem about representing data sets through the GeoGebra applet. “1. Type in values for the number of customers for each hour span, using whole numbers 0 to 99, inclusive. 2. Show example work steps to create a histogram. 3. Type new values for step 1 and observe the changes. To reset the steps (above) click: Reset Steps (button).” Students are able to edit the number for each hour, and by pulling down a slider below step 2, display the steps to creating a histogram as the histogram is displayed.”
Mathematics II, Unit 2, Lesson 2.1.2 links to a Desmos interactive conceptual activity about finding the shape of a parabola by using its form to reveal its characteristics. “The activity begins with a review of both the characteristics and forms of a parabola. Later, students will determine characteristics of the graph of a parabola given in standard form, vertex form, or intercept form.” The first screen requires students to complete a card sort of the different forms of a parabola. “Match each equation with its type and a property we can determine from its form.” Intercept, standard, and vertex forms are the choices into which one can match/sort the six cards.
Mathematics III, Unit 3 [Unit 2B], provides an interactive practice problem about simplifying rational expressions and stating restrictions through the GeoGebra applet. “Question: Can you simplify the rational expression? Instructions: Complete the puzzle by clicking on a piece that contains the restrictions and a piece that contains the simplified expression.” The puzzle piece that is displayed is $\frac{xt^3+3t^2}{6t^2-3t-3}$ . The puzzle pieces that the student can choose from are the following: $\frac{t^2}{t-1}$ , $\frac{t^2}{t+1}$ , $t\ne1$ , $-\frac{1}{2}$ . Students can click on the puzzle pieces to have them join the original puzzle piece. Once students have selected the two pieces, they can then click on “How Did I Do?”, and the applet will display if they are correct or incorrect as well as a worked solution. Students have the option to “Try Another” problem as well.

Indicator 3X

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Walch CCSS Integrated Math Series do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

While there are opportunities within activities in this series for students to collaborate with each other, the materials do not specifically include or reference student-to-student or student-to-teacher collaboration with digital technology.

Indicator 3Y

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Walch CCSS Integrated Math Series have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

There is a consistent design within the units and lessons that supports learning on the digital platform. The general structure of the series is organized by Units, Topics, Unit Review, Unit Assessment (with Answer Key), and Station Activities (with Instruction and Answer Key). Each Topic contains a Supporting Resource for Teachers, Pre-Assessment (with Answer Key), Lessons, Progress Assessment (with Answer Key) with most Topics containing Conceptual Activities and/or a Conceptual Task. The Lessons within the Topics, includes a Warm-up, Warm-up Debrief, Supporting Resource for Teachers, Scaffolded/Skills Practice (with Answer Key), Guided Practice, Presentation, Problem-Based Task, Problem-Based Task Coaching Questions, Problem-Based Task Sample Responses, Problem-Based Task Implementation Guide and Practice Sets A and B (with Answer Keys). This format is consistent in each unit throughout the series.

When images, graphics, or models are included, they clearly communicate information supporting student understanding of topics, texts, or concepts. The Interactives have clear directions that make them easy to use.

Indicator 3Z

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Walch CCSS Integrated Math Series provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

Teachers are provided support with the materials digitally through the Help, Knowledge Base, which can be accessed on the main page. Examples of quick guides and/or video trainings found with the Knowledge Base that support the embedded technology within the materials include, but are not limited to: Getting Started with the Curriculum Engine, Key Features of Your Walch Courses, Best Practices for Using the Curriculum Engine with Remote Learning, and Four easy ways to use the Curriculum Engine to enhance your Walch print materials.

The Program Overview- Digital Enhancements Guide describes guidelines and suggestions for implementation for the digital components that can be accessed within the materials. For example, under the GeoGebra Applets (Interactive Practice Problems) section, the materials state the following: “One or two interactive GeoGebra applets are provided for most lessons. The applets model the mathematics in the Guided Practice examples for these lessons. Links to these applets are also embedded within the Instructional PowerPoints … Once you’ve accessed the GeoGebra applet, please adjust your view to maximize the image. Each applet illustrates the specific problem addressed in the Guided Practice example. The applets allow you to walk through the solution by visually demonstrating the steps, such as defining points and drawing lines...”

Additionally, the Desmos resources provide guidance for the teacher on how to support and enhance student learning while they engage with the activity.

Criterion 3.4: Intentional Design

Downloadable Resources

2022