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Report Overview
Summary of Alignment & Usability: CCSS Mathematics Integrated Pathway | Math
Math High School
The instructional materials reviewed for the Walch Integrated series partially meet the expectations for Alignment to the CCSSM. The materials partially meet the 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; and making meaningful connections in a single course and throughout the series. The materials partially meet the expectations for Rigor and Mathematical Practices as they partially meet the expectations for Rigor and Balance and partially meet the expectations for Practice-Content Connections. Within Rigor and Balance, the materials did show strengths with providing students opportunities for developing procedural skills and working with applications, and within Practice-Content Connections, the materials showed strengths in developing overarching, mathematical practices (MPs 1 and 6). Since the materials did not meet expectations for Alignment to the CCSSM, evidence concerning Usability was not collected for the series.
High School
View Full ReportEdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
Report for High School
Alignment Summary
The instructional materials reviewed for the Walch Integrated series partially meet the expectations for Alignment to the CCSSM. The materials partially meet the 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; and making meaningful connections in a single course and throughout the series. The materials partially meet the expectations for Rigor and Mathematical Practices as they partially meet the expectations for Rigor and Balance and partially meet the expectations for Practice-Content Connections. Within Rigor and Balance, the materials did show strengths with providing students opportunities for developing procedural skills and working with applications, and within Practice-Content Connections, the materials showed strengths in developing overarching, mathematical practices (MPs 1 and 6).
High School
Alignment (Gateway 1 & 2)
Usability (Gateway 3)
Overview of Gateway 1
Focus & Coherence
Gateway 1
v1.0
Criterion 1.1: Focus & Coherence
The instructional materials reviewed for the Walch Integrated series partially meet the 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; and making meaningful connections in a single course and throughout the series. 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; engaging students in mathematics at a level of sophistication appropriate to high school; and explicitly identifying and building on knowledge from Grades 6-8 to the high school standards.
Indicator 1A
Indicator 1A.i
The instructional materials reviewed for Walch Integrated Math Series meet the 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 1 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
The instructional materials reviewed for the Walch Integrated Math Series partially meet the expectations for attending to the full intent of the modeling process when applied to the modeling standards. Overall, most of the modeling standards are addressed with various aspects of the modeling process present in isolation or combinations. However, opportunities for the full modeling process are absent throughout the instructional materials.
The materials often allow students to incorporate their own solution method to find a particular predetermined quantity or range of quantities. Modeling opportunities in the materials are thus “closed” in beginning and end while “open” in the middle. However, students are rarely given the opportunity to question their reasoning and “cycle” through the modeling process by validating their conclusions and potentially making improvements to their model.
The following examples address much of the modeling process; however, students are not given the opportunity to validate and adjust their model as needed:
- Mathematics I, Unit 4, Lesson 4.1.2 Problem-Based Task (S-ID.2): Students are provided a problem and data from which they need to construct a graph, and they use the graph to interpret differences as they compare two types of cars. Students also compute measures of center and spread in order to further compare cars. The task is completed when students report as to which car would be the better buy.
- Mathematics II Unit 2 Lesson 2.3.1 Problem-Based Task (F-BF.1a): Students must create a model to predict the effect that more wells will have on oil production. Students must then use their model to determine the maximum number of wells needed to maximize oil production.
- Mathematics III Unit 2A Lesson 2A.5.2 Problem-Based Task (A-SSE.2): Students are asked to use a formula to compute a refinanced payment, interpret the payment in terms of the aunt’s current financial situation, validate results by comparing prices over a 15 year time period and over a 30 year time period, and finally make a recommendation to the aunt regarding which refinancing option is the best.
The following examples allow students to engage in only a part of the modeling process:
- Mathematics II Unit 2 Lesson 2.5.1 Problem-Based Task (F-IF.8 ): Students engage in all aspects of the modeling process except formulate. Students are given a problem to consider and formulas representing the scenario (students do not generate the formulas). Students use the formulas to make computations, interpret results, validate their results through comparison, and report on which car to purchase.
- Mathematics II Unit 2 Lesson 2.3.2 Problem-Based Task (F-BF.2): Students develop a function to calculate the amount of paper needed to make each note card and corresponding envelope; however, students do not use this function to actually calculate.
- Mathematics I Unit 1 Lesson 1.2.1 Example 4 (N-Q.2): Students are given several scenarios and need to consider what units would be appropriate to report answers. While this is an important step in the modeling process, this example does not connect to the remaining steps in the modeling process.
- Mathematics III Unit 4B Lesson 4B.5.2 Problem-Based Task (G-MG.2): Students are asked to find a function model for a provided set of data in a graph and a table related to water density as ice melts. While students are asked to find an appropriate model, students do not use the model to complete calculations to finish the modeling cycle.
Indicator 1B
Indicator 1B.i
The instructional materials reviewed for the Walch Integrated Series meet the 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
The instructional materials reviewed for the Walch Integrated Math Series, when used as designed, partially meet the expectation for allowing students to fully learn each non-plus standard. Overall, the series addresses many, yet not all, of the standards in a way that would allow students to fully learn the standards.
The materials combine classroom practice, additional practice, problem-based tasks, supplemental workbooks, and IXL internet links. (It should be noted that the IXL links provide supplemental practice of up to 12 practice problems per IP address per day, as it is only a trial version and does not provide full access.) However, cases exist where the instructional materials devoted to the standard are insufficient.
The following are examples where the materials partially meet the expectation for allowing students to fully learn each standard.
- N-CN.7: In Mathematics II Unit 3 Lesson 3.4.2, there are a limited number of problems that allow students to solve quadratic equations with real coefficients with complex solutions. Seven problems were identified in the practice, additional practice in the student resource book, and the support supplement workbook.
- A-SSE.3a: There are a limited number of problems that allow students to factor quadratic expressions to reveal zeros. Mathematics II Unit 3 Lesson 3.3.1 provides two example problems that show students two methods to solve the quadratic expression provided (factoring and the quadratic formula). The materials advocate for students to use the quadratic formula over factoring as “the quadratic formula always works” (page 122).
- A-SSE.3b: No evidence was found where the materials directly give students the opportunity to complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. However, in Mathematics II students use the vertex form of quadratic equations to determine the maximum or minimum value of the function. In Mathematics II Unit 3 Lesson 3.3.3, students complete the square to convert quadratic functions into vertex form. The materials do not explicitly make the connection from completing the square to revealing the maximum or minimum value of a quadratic expression via usage of the vertex form. As such, students are not provided specific opportunities to practice finding extreme values of quadratics by completing the square.
- A-APR.3: There are a limited number of opportunities for students to use zeros of polynomials to complete a rough sketch. While sufficient practice is provided with quadratics in Mathematics II, students are given few opportunities to work with higher-order polynomials in Mathematics III.
- A-APR.4: Proofs are provided of polynomial identities in Mathematics III Unit 2A Lesson 2A.2. 1 thru Lesson 2A.2.3. While students do use the provided identities sufficiently, students do not prove the identities for themselves as required by the standard.
- A-APR.6: In Mathematics III Unit 2A Lesson 2A.3.2 students use long division to rewrite simple rational expressions in Example 1. The materials do not require students to use long division in the other examples in the lesson, but materials instead require synthetic division. There are a limited number of problems for students to practice long division of polynomials so that students fully learn the material in Lesson 2A.3.2.
- A-REI.2: Solving simple rational equations is first taught in Mathematics II Unit 3 Lesson 3.5.1. Students are introduced to the term extraneous solution; however, none of the examples incorporate a problem with an extraneous solution. Solving simple rational equations is later taught in Mathematics III Unit 2B Lesson 2B.2.1 in which extraneous solutions are once again discussed, and students see examples that result in one or more extraneous solution. Solving radical equations is taught in Mathematics III Unit 2B Lesson 2B.2.2. No examples in this lesson incorporate extraneous solutions with regards to radical equations.
- F-IF.3: In Mathematics I Unit 2 Lesson 2.3.1, students are not asked to identify sequences as functions. The materials only list this fact in the introduction on page 146. Also, the instructional materials do not discuss the domain of a sequence other than in the introduction of the lesson.
- F-IF.7e: Mathematics III Unit 4A Lesson 4A.3.1 provides opportunities for students to graph sine functions, and Lesson 4A.3.2 provides opportunities for students to graph cosine functions. However, graphing of tangent functions is included in one problem in the station activity provided within Unit 4A.
- F-LE.1a: Although a single problem was found in Mathematics I Unit 2 Lesson 2.4.2, the materials do not allow students sufficient opportunity to compare and contrast how linear functions grow by equal differences over equal intervals whereas exponential functions grow by equal factors over equal intervals.
- G-CO.5: In Mathematics I Unit 5 Lessons 5.2.1 and 5.2.2, students are provided limited opportunities to specify a sequence of transformations that will carry a given figure onto another.
- G-CO.10: The standard calls for students to “prove theorems about triangles;” however, many proofs are provided by the materials. For example, the following proofs are provided in the materials: Triangle Sum Theorem in Mathematics II Unit 5 Lesson 5.6.1, base angles of isosceles triangles in Mathematics II Lesson 5.6.2, Midsegment Theorem in Mathematics II Lesson 5.6.3, and medians of a triangle in Mathematics II Lesson 5.6.4.
- G-CO.11: The standard calls for students to “prove theorems about parallelograms;” however, many proofs are provided by the materials. For example, the following proofs are provided in the materials: the opposite sides are congruent, the opposite angles are congruent, and the diagonals are congruent are all proofs found in Mathematics II Lesson 5.7.1 and 5.7.2.
- G-C.5: While the teacher materials incorporate similarity into the definition for arc length, students do not derive by similarity that the length of the arc intercepted by an angle is proportional to the radius in Mathematics II 6.4.1 or 6.4.2.
Indicator 1C
The instructional materials reviewed partially meet the expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, yet they do not vary the types of real numbers being used.
Materials use age appropriate and relevant contexts throughout the series. The following examples illustrate appropriate contexts for high school students.
- Mathematics I Unit 2 Lesson 2.4.1: Students interpret an appropriate domain of a function in the context of buying a vehicle and considering how that vehicle will depreciate over time.
- Mathematics I Unit 1 Lesson 1.1.2: Students decide how best to invest money.
- Mathematics II Unit 5 Lesson 5.42: Students use similarity to determine the diameter of a sinkhole in Louisiana.
- Mathematics II Unit 2 Lesson 2.4: Students find the volume of a swimming pool.
- Mathematics III Unit 2A Lesson 2A.5.2: Students compare home refinancing options and college loan payment options in their work with geometric series.
- Mathematics III Unit 1 Lesson 1.3.1: Students decided how to test if soda is linked to cancer.
- Mathematics III Unit 2B Lesson 2b.1.4: Students discuss fuel economy through rational expressions.
The following problems represent key takeaways from Grades 6-8:
- Proportional relationships are used to show similarity of two triangles in Mathematics II Unit 5 Lessons 4 and 5. Students have to extend the knowledge of ratios and proportions to determine the golden ratio in Mathematics II Unit 3.
- Student knowledge of ratios is built upon as students explore the trigonometric identities of sine, cosine, and tangent in Mathematics II Unit 5 Lessons 5.8.1 and 5.8.2.
- Instructional materials support student development in applying basic function concepts. Students create and graph linear, exponential, quadratic, polynomial, and other types of functions across the series. Particularly, the F-IF standards, which are included in all Mathematics courses, support the takeaways from Grades 6-8.
- Analyzing concepts and skills of geometric measurement is further developed at the high school level within the context of coordinate geometry in Mathematics II Unit 5 and Mathematics III Unit 6. Students also compute perimeter and area using coordinate distances in Mathematics I Unit 6 Lesson 6.1.2.
- Instructional materials support student development in applying concepts and skills of basic statistics and probability first taught in Grades 6-8. Students expand their statistical knowledge as they learn how to represent and interpret data and make inferences from sample surveys, experiments, and observational studies. Students expand their knowledge of probability as they learn about independent and conditional probability and rules to compute probabilities of compound events.
Problems throughout the series provide regular practice with operations on integers and whole numbers. However, problems throughout the series provide limited practice with operations on fractions, decimals, and irrational numbers. The majority of the series uses whole number coefficients and values unless the context involves money, percents, or irrational constants like π or e. Examples include the following:
- In Mathematics I Unit 3, students solve linear equations and inequalities and exponential equations. Problems in this unit typically feature integer answers. There are few problems that have fractional answers, and most decimal answers are present only in problems that reference money. For example, see Practice 3.2.1.
- In Mathematics II Unit 5, students solve problems using right triangles, trigonometry, and proofs. The majority of problems in this unit have integer answers. For example, practice problems using midpoints for Practice 5.1.1 do not feature decimal, fractional, or irrational answers.
- In Mathematics III Unit 2B, students solve problems using rational and radical relationships. Few problems in this unit allow students to work with irrational and decimal answers.
Indicator 1D
The materials meet the expectation 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
The instructional materials reviewed partially meet the expectations that the series explicitly identifies and builds on knowledge from Grades 6-8. Materials include and build on content from grades 6-8, however, the content is not clearly identified or connected to a specific middle school standard. Although the provided content from Grades 6-8 fully supports progressions of the high school standards, the Grade 6-8 standards are not identified in either the teacher or student materials.
The following are examples of where the materials do not explicitly identify and/or build on standards from Grades 6-8.
Mathematics I:
- Stations Activity Set 1 Unit 1 involves ratios and proportions. While the indicated standards are N-Q.1 and A-CED.1, no indication is made to middle school standards or how the material relates to prior grade-levels.
- Each lesson indicates prerequisite skills. For example, in Unit 1 Lesson 1.2.3 students are expected to work with exponents and apply the order of operations within the lesson. However these skills are not identified by standard or connected explicitly to the current material within the lesson.
- Unit 1 Lesson 1.3.1, page 97, under key concepts, lists reviewing linear equations are provided, but the information is presented as new content within the teacher commentary. The information is not identified by a standard.
- Linear study is extensive in Grade 8, and the high school series works throughout to solidify the understanding of linear relationships. Mathematics I Unit 2 includes domain and range, function notation, key features of linear graphs, proving average rate of change, comparing linear functions to one another, and exponential functions.
- A-REI.C: Mathematics I Unit 3 Lesson 3.2 focuses on solving systems of equations which extends from 8.EE.8 “Analyse and solve pairs of simultaneous linear equations.” Students solve systems of equations algebraically and graphically as well as solve systems of equations within a real world context.
- F-IF.A: Mathematics I Unit 2 extends student understanding of the concept of function introduced in 8th grade. Instructional materials have students consistently use function notation and identify domain and range of a function given an equation and within a context.
- S-ID.B: Mathematics I Unit 4 extends 8.SP.A, “Investigate patterns of association in bivariate data.” Instructional materials reinforce students’ knowledge of using a scatterplot to represent data and fitting a line to data At the high school level, students then assess this best fit line using residuals and interpret the model within a given context.
- In Unit 4 Lesson 1 the material discusses finding the median, first and third quartiles, and minimum and maximum values. It does not specifically reference 6.SP.2.
Mathematics II:
- N-RN.A: 8.EE.A, “Work with radicals and integer exponents,” is built upon in Mathematics II Unit 1 Lessons 1.1.1 and 1.1.2 as students extend the properties of exponents to rational exponents.
- G-SRT.A: 8.G.A, “Understand congruence and similarity,” primarily focuses on similarity and congruence within the context of transformations. Mathematics II Unit 5 builds upon this prior knowledge by defining similarity and congruence in terms of transformations. Instructional materials build upon this knowledge to include using the definition of similarity and congruence to prove theorems (particularly related to triangles).
- G-GMD.A: Middle school standards related to calculating volume of three-dimensional figures are built upon in high school as students use Cavalieri’s principle to justify volume formulas in Mathematics II Lesson 6.5.2.
- G-CO.C: Mathematics II Unit 5 extends students’ knowledge about “...facts about supplementary, complementary, vertical, and adjacent angles…” from 7.G.5 as students prove theorems about lines, angles, triangles, and parallelograms at the high school level.
- In Unit 1 Lesson 1 the material discusses evaluating expressions involving integer powers, but it does not reference 8.EE.1 as a prior standard.
- In Unit 5 Lesson 3 the material discusses creating ratios and solving proportions. It does not specifically reference 6.RP.1 or 7.RP.3.
Mathematics III does not contain references to content from Grades 6-8.
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.
Of the 43 plus standards and 5 plus substandards 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.
Overview of Gateway 2
Rigor & Mathematical Practices
Gateway 2
v1.0
Criterion 2.1: Rigor
The instructional materials reviewed partially meet the expectations for Rigor and Balance. The materials display a balance between 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 do not develop conceptual understanding of key mathematical concepts as students do not get opportunities to work with multiple representations of concepts or explain their reasoning about concepts in different formats.
Indicator 2A
The instructional materials reviewed for Walch Integrated Math Series do not meet the expectations for giving attention to conceptual understanding. The materials rarely develop conceptual understanding of key mathematical concepts where called for in specific content standards or cluster headings. The materials rarely offer opportunities for students to engage with concrete and semi-concrete representations, as well as verbalization and writing, when developing conceptual understanding.
The following examples indicate where the materials lack the opportunity to develop conceptual understanding:
- Mathematics III Unit 2A Lesson 2A.3.3 (A-APR.B): The cluster is intended to build an understanding of the relationship between zeros and factors of polynomials. The materials do not connect the concepts of zeros, factors, and the shapes of the graphs. The connection between zeros and factors of polynomials is an area that lacks conceptual problems and conceptual discussion questions. The majority of the problems focus on finding factors of a problem to sketch a graph. The text also provides step-by-step directions for how to graph polynomial functions.
- Mathematics II Unit 5 Lesson 5.8 (G-SRT.6): The instruction portion of the lesson (page 502, TR) leads the student to see the connection between ratios of sides in similar triangles and the definition of trigonometric ratios. Then in Example 3 on page 510 the materials state that “without drawing another triangle, compare the trigonometric ratios of Triangle ABC with those of a triangle that has been dilated by a factor of K=3.” The remainder of the section, including any examples and practice exercises, are just procedural in nature.
- Mathematics I Unit 2 Lesson 2.4 (F-LE.1): Students are not given the opportunity to distinguish between linear and exponential situations; they are usually directed toward a particular model. Mathematics I Lessons 2.5.1 and 2.5.2 also have students work with linear and exponential models. In the Problem-Based Task for Lesson 2.5.2, students needed to determine whether a scenario would best be modeled by a linear or exponential function and then create that model and use that model to answer questions. However, problems in Practice 2.5.2 tell students to “(w)rite an exponential function to model the scenario” rather than providing them the opportunity to determine whether a linear or exponential model is most appropriate based on the scenario. Students are not given independent opportunities to demonstrate their conceptual understanding of this standard.
- Mathematics III Unit 3 Lesson 3.1.3 (F-TF.2): While guided practice connects the unit circle to the coordinate plane, students are not asked to explain or expand on the connection. Whereas problem 10 on page 69 (TR) does ask students to sketch the unit circle and label, this could be memorized and practice problems are procedural in nature.
- Mathematics I Unit 5 Lesson 5.6.2 (G-CO.8): Students are asked to explain why two triangles cannot be deemed similar. The example provided in the guided practice (Example 4) demonstrates what to do if something is not similar. The indicated correct response indicates that the triangles are not similar because “no congruence statement that allows us to state that the two triangles are congruent based on the provided information.” This is a missed opportunity where the text could have developed conceptual understanding geared towards use of transformations.
- Mathematics III Unit 2A Lesson 2A.4.1 (A-REI.11): Problems are procedural in nature and require students to verify if the intersections are solutions. The indicated correct responses in the teacher's’ resource indicate that if the solutions are solutions to the original equations then the answers are solutions. However, students are not asked to explain why, but rather it is provided in the materials.
The following examples indicate where the materials develop conceptual understanding:
- Mathematics I Unit 3 Lesson 3.1 (A-REI.1): Students work with solving linear equations by explaining the connection between each step in solving and a property of equality. In Mathematics I Lesson 3.1.1, the Problem Based Task has students answer a magic number problem: “Think of a number. Then double it. Now add 6. Take half of that number. Finally, subtract the number you started with. Your answer is 3.” As students consider why this works and how this is possible, they are analyzing their reasoning as they progress from one step to the next (A-REI.A).
- Mathematics I Unit 2 Lesson 2.1 (A-REI.10): Through the examples and the practice problems, the students are given several opportunities to discuss how the graph represents all of the solutions.
- Mathematics I Unit 2 Lesson 2.1.3 has students consider whether a relation is a function using multiple means (mapping, analysis of coordinates, and the vertical line test) as they seek to better conceptually understand what a function is, how it is represented, and what it looks like (F-IF.A).
Indicator 2B
The materials meet the expectation 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
The materials meet the expectation of 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 materials partially meet the expectations for not always treating the three aspects of rigor together and not always treating them separately. Each lesson has an application warm-up exercise, procedural concept development section (guided practice), a problem-based task, and procedural individual practice, regardless of the standards addressed in the lesson. Materials rarely incorporate conceptual understanding into a lesson. The three aspects are not balanced with respect to the standards being addressed as minimal evidence of conceptual understanding can be found throughout the content. The majority of lessons are heavily focused on procedural skill and application. Instructional materials balance procedural fluency problems with application problems throughout the entire series. Procedural skills are enhanced when practiced within the context of an application problem. Instructional materials missed opportunities to incorporate conceptual-based problems throughout the series, thus preventing the balance of all three.
Criterion 2.2: Math Practices
The instructional materials partially meet the expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified in the implementation guides for the problem-based tasks. A spiral reference notebook is provided that lists the MPs, but it does not connect the MPs to the materials. The materials give students opportunities to develop overarching, mathematical practices, reasoning, modeling with mathematics, and seeing structure. The materials do not support the intentional development of explaining, using tools, and generalizing.
Indicator 2E
The instructional materials meet the expectation 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
The instructional materials partially meet the expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3).
The materials develop MP2 as students are provided opportunities in which they can develop their mathematical reasoning skills. Examples of the materials providing opportunities for students to develop MP2 include:
- In Mathematics I Unit 1 Station Activities Set 2 Station 4, students are asked to match inequalities to real-world situations. After completing this matching task, students are asked to “explain the strategies you used to match the inequalities to the situations.” In this activity students decontextualize a situation to represent it symbolically, and they contextualize the symbolic representations by considering if the calculated quantities make sense in the given real-world situations.
- In Mathematics II Unit 5 Lesson 6.4, the implementation guide for the problem-based task with this lesson reminds teachers 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 4A Lesson 2.3, students reason abstractly by determining how to organize data presented to them in a paragraph so that they can find a logarithmic function that models the data. Students also reason quantitatively by determining if the corresponding exponential function models the given data.
The materials do not develop MP3 to its full extent. In the materials, students construct viable arguments, but students are not prompted to critique the reasoning of others. As students complete problem-based tasks, they construct arguments to explain their solutions, but there are no questions or directions in the prompts for students to critique the reasoning of other students regarding the task. In addition, teachers are provided with general instructions to have students discuss their own arguments with each other and explain their own reasoning if disagreements arise while students complete problem-based tasks. Examples of how students do not have to critique the reasoning of others include:
- In Mathematics I Unit 2 Lesson 4.1, students construct an argument that supports their conclusions about an exponential graph, and the accompanying function, for the depreciating value of a car based on given information. The implementation guide for the problem-based task has teachers "encourage students to share their thoughts and ideas with others and to defend their ideas should disagreements arise."
- In Mathematics II Unit 3 Lesson 6.1, students construct an argument to determine which of two options is better for investing $20,000 over 5 years. The implementation guide for the problem-based task has teachers "encourage students to discuss their arguments with each other and explain their reasoning if they do not agree with each other."
- In Mathematics III Unit 2A Lesson 1.3, students construct an argument while determining an expression that can be used to represent the area of a walkway. The implementation guide for the problem-based task has teachers "encourage students to describe their own solution methods, listen to and critique their classmates’ methods, and discuss which method, if any, is best." The implementation guide also states that students could possibly answer "we agreed that there was no best method because all of our methods worked."
Indicator 2G
The instructional materials partially meet the expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5).
The materials fully develop MP4 as students build upon prior knowledge to solve problems, and they create and use models in the problem-based tasks provided with most lessons. The materials pose problems connected to previous concepts and a variety of real-world contexts. Students are provided meaningful real-world problems to model using mathematics as they identify important relationships when there are opportunities to compare and contrast or draw conclusions. In the implementation guides for the problem-based tasks, references to MP4 typically describe how students will translate the context of the tasks into either an algebraic or graphical model.
The materials do not fully develop MP5 as students are not given the opportunity to choose their own tools, but rather, tools are provided to them. The materials do not encourage the use of multiple tools to complete investigations even though tools are incorporated throughout the series as students engage in mathematics; for example, students use a compass and patty paper to perform geometric constructions, a graphing calculator and pencil/paper to graph equations, and a graphing calculator to determine statistics of a data set. While tools are appropriately modeled throughout the series (step-by-step instructions are provided), limited opportunities exist for students to discuss their benefits/limitations and when to use one tool over another. In the implementation guides for the problem-based tasks, references to MP5 typically describe how students will use a form of graphing technology to help them complete the given task.
- In Mathematics I Unit 3 Station Activities Set 3 Station 3, students use a graphing calculator to solve a system of equations rather than allowing students to use the method they choose to solve the system of equations.
- Mathematics I Unit 5 has a strong emphasis on performing geometric constructions. While lessons include step-by-step instructions on how to perform constructions with a compass and straightedge in addition to using patty paper, all examples, problem-based tasks, and practice exercises have students use a compass and straightedge to perform constructions. Directions throughout the unit explicitly state, “Use a compass and straightedge to …” Mathematics I Unit 5 Station Activity Set 2 Station 1 has students show that two triangles are congruent using a ruler and then show the same two triangles are congruent using a protractor. In doing so, students can compare the tools, but the materials do not support students to evaluate the benefits or limitations of each tool or which tool is “better” to use in the given context.
- In Mathematics I Unit 6 Station Activities Set 2 Station 2, the materials state, “Do NOT use your protractor” to determine whether or not two lines are perpendicular. Students are instructed to “...use your protractor to determine whether or not the lines are perpendicular” in the second part of the station task.
- In Mathematics II, Unit 5 Station Activities students use the provided tools, rather than choosing their own, to investigate angles formed by parallel lines.
- In Mathematics II Unit 6 Station Activities Set 2, the materials state students "will be given a ruler, a compass, a protractor, and a calculator" as they work in groups to answer questions investigating and drawing conclusions about secant and tangent lines to circle theorems.
- In Mathematics III Unit 1 Lesson 1.1, students engage with S-ID.4 as they determine the mean and standard deviation of data sets. Every example has the students follow a set of steps on a TI calculator to solve for the values. The materials offer little opportunity for students to choose an appropriate tool.
Indicator 2H
The instructional materials partially meet the expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8).
The materials develop MP7 as students are provided opportunities in which they can look for and make use of structure. Examples of the materials providing opportunities for students to develop MP7 include:
- In Mathematics I Unit 1 Lesson 4.1, students look for and make use of the structure of the information provided about two types of skates to create two linear inequalities in two variables. Students use the linear inequalities to determine possible combinations of the two types of skates that could be made.
- In Mathematics II Unit 5 Lesson 3.1, 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 2A Lesson 5.1, students look for and make use of the structure of the information provided to write an explicit formula for a geometric sequence that models the number of people that will hear a positive restaurant experience n weeks after the positive experience was had.
The materials do not develop MP8 to its full extent. In the materials, there are many tasks where students engage in repeated calculations or reasoning, but students do not use the repeated calculations or reasoning to make mathematical generalizations. Examples where students do not use repeated calculations or reasoning to make mathematical generalizations include:
- In Mathematics I Unit 2 Lesson 3.1, the implementation guide for the problem-based task states that students engage in MP 8 by "noticing that the same calculations are performed repeatedly in order to achieve the desired results and recognize that the same domain value is used in order to evaluate the sequences for all three species of trees." The repeated calculations are used to answer questions about the diameters of trees and determine which type of trees should be purchased, but the repeated calculations are not used to make any mathematical generalizations. The calculations were made using general formulas provided for each species of tree.
- In Mathematics II Unit 4 Lesson 1.3, the implementation guide for the problem-based task states that students engage in MP 8 by "using repeated reasoning as they determine a pattern of possible outcomes when two coins are tossed and using the repeated process of calculating probabilities for each event." The pattern of possible outcomes and calculated probabilities are not used to make any mathematical generalizations.
- In Mathematics III Unit 4A Lesson 1.1, the implementation guide for the problem-based task states that students "express regularity in repeated reasoning as they explain and justify their steps involved in determining the inverse of the function representing the motion of the overhang of rocks." The inverse function that is created is specific to the problem-based task and does not represent any mathematical generalization.