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Focus & Coherence
Gateway 1 - Meets Expectations | 77% |
|---|---|
Criterion 1.1: Focus and Coherence | 14 / 18 |
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.
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).
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.
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. Throughout the series, various aspects of the modeling process are present in isolation or combinations, yet opportunities for students to engage in the complete modeling process are absent for the modeling standards throughout the instructional materials of the series.
Examples of where the materials allow students to engage in some aspects of the modeling process with prompts or scaffolding from the materials include, but are not limited to:
In Mathematics I, Unit 2, Lesson 2.6.3, Problem-Based Task, students evaluate two different financial arrangements: one with payment at a constant rate and one with a doubling factor. While students are provided the problem, they are still given the opportunity to formulate, compute and interpret their results to determine the better payment choice. Students do not validate that one financial arrangement is better than the other. (F-LE.3)
In Mathematics I, Unit 5, Lesson 5.5.2, Conceptual Task, Transformation Tests, students use given models to describe how applying a rotation or a reflection to an object changes its location and orientation. The materials direct students to draw lines between corresponding vertices to notice the extent of the rotation and the line of reflection, however the materials inform students of the 180° rotation and reflection over the y-axis in ensuing questions. Students do not create a model and do not validate their work. (G-CO.6)
In Mathematics II, Unit 4, Lesson 4.2.1, Conceptual Task, Allergies and Probabilities, students consider the results of a survey that polled 22 students about their allergies to particular foods. Students respond to a series of exploration questions that focus on interpreting the two-way frequency table and calculating probabilities. As a consequence of these guiding questions, students do not independently formulate and compute probabilities or independently interpret the results of the survey and, therefore, do not fully engage in the modeling process. (S-CP.3,5,6)
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. This task does not present students with an opportunity to design the shape or size of their own alternative area-saving storage container. (G-GMD.3)
In Mathematics III, Unit 1, Lesson 1.5.2, Problem-Based Task, students consider “Unfair Profiling.” Given a specific problem situation, students write a claim and design/implement a simulation to justify their claim. Students do not validate their conclusions or consider improving their model. (S-IC.5)
In Mathematics III, Unit 2 [Unit 2A], Lesson 2A.3.1, Conceptual Task, Engineering Polynomials, students consider an engineer’s proposal to model a roller coaster using multiple “stitched together” polynomials. By responding to exploration questions that lead students through the task, students reinforce their understanding of the problem situation, consider how polynomials could model the height of hills in the coaster, and weigh the advantages and disadvantages of using polynomials to model a roller coaster. Students do not have the opportunity to engage with a model, perform calculations, or interpret their results. (A-APR.3, F-IF.7)
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 partially meet expectations for, when used as designed, allowing students to fully learn each non-plus standard.
Examples of where and how the materials allow students to fully learn a non-plus standard include:
N-CN.7: In Mathematics II, Unit 3, Lesson 3.4.2, students have multiple opportunities to solve quadratic equations with real coefficients that have complex solutions. In the Warm-Up, students apply properties of square roots to simplify radical expressions, including those requiring the imaginary unit. In the Scaffolded Practice, students solve quadratic equations that have complex solutions and sketch graphs of quadratic functions to verify that their solutions are complex. In the Problem-Based Task, students solve quadratics using the quadratic formula and other means. In Practice Sets A and B, students calculate the discriminant to reveal the number and nature of the solutions.
G-CO.5: In Mathematics I, Unit 5, Topic E, Conceptual Task, students specify a sequence of transformations that map one figure onto another. In the Lesson 5.2.1, Scaffolded Practice, students define the translation function in coordinate notation given the diagram. In the Guided Practice, students write the translation of a rotation in terms of a function and write the reflection of the translation of the reflection in terms of a function. In the Unit 5, Lesson 5.2.2, Warm-Up, students describe how the three images have been transformed from a pre-image. In the Lesson 5.2.2, Problem-Based Task, students state the three unique transformations that bring the triangle together. In Scaffolded Practice and Practice Sets A and B, Problems 1-2 and 8-9, students draw transformed figures.
G-CO.11: In Mathematics II, Unit 5, Lesson 5.7.1, students prove many theorems about parallelograms including opposite sides are congruent, diagonals bisect each other, and that diagonals divide parallelograms into two congruent triangles. In the Guided Practice, students prove or disprove that opposite sides are parallel and opposite sides are congruent, verify that consecutive angles are supplementary, prove that diagonals of a parallelogram bisect each other, and prove that a diagonal of a parallelogram divides the parallelogram into two congruent triangles.
Examples of where and how the materials do not allow students to fully learn a non-plus standard include, but are not limited to:
A-SSE.3b: In Mathematics II, Unit 3, Lesson 3.3.3, the materials do not connect the act of completing the square in a quadratic expression with identifying its vertex as the maximum or minimum value of a function. In Scaffolded Practice, Problems 5 and 6 and Guided Practice, Example 3, students convert quadratic functions in standard form to vertex form, although the materials do not prompt students to complete the square.
A-APR.4: In Mathematics III, Unit 2 [Unit 2A], students use polynomial identities, however do not prove them.
A-APR.6: Throughout Mathematics III, Unit 2B, Lesson 2B.1.3, students simplify rational expressions and state restrictions. In the Practice problems, students use synthetic division to rewrite rational expressions. In contrast, in Scaffolded Practice, Problem 10 and Practice Sets A and B, Problem 7, students use long division to rewrite rational expressions in the form q(x) + r(x)/b(x). As such, the materials do not provide students with enough opportunities to fully learn the standard.
A-REI.2: The Mathematics III, Unit 3 [Unit 2B], Lesson 2B.2.1 Presentation suggests that students verify answers. Throughout Lesson 2B.2.1, students solve rational equations. In the Scaffolded Practice and Problem 10, students have the opportunity to identify an extraneous solution. The materials do not provide sufficient opportunities for students to identify extraneous solutions, so students do not fully learn the standard.
F-IF.3: The Mathematics I, Unit 2, Lesson 2.3.1, Teacher Instructional Support indicates that a sequence can be expressed as a function and that the domain is the set of natural numbers. In Guided Practice, Example 1, students determine whether a sequence, given by its formula, is explicit or recursive. The materials do not prompt students to engage with or reason about the domain of sequences.
F-IF.7e: In Mathematics I, Unit 2, Lesson 2.5.2, students sketch graphs of exponential functions. In Mathematics III, Unit 5 [Unit 4A], Lesson 4A.2.4, students sketch graphs of logarithmic functions. In Mathematics III, Unit 5 [Unit 4A], Lessons 4A.3.1 and 4A.3.2, students sketch graphs of sine and cosine functions, respectively. In Practice Sets A and B, Problems 1 and 2, students respond to questions about amplitude and period given graphs of functions. The materials do not give students ample opportunities to engage with the midline of trigonometric functions or graphs of the tangent function to allow them to fully learn the standard.
F-LE.1a: In Mathematics I, Unit 2, Lesson 2.4.2, Problem-Based Task, students calculate the average rate of change for the value of a desk over a four-year period based on two different valuation methods (one linear, one exponential) and explain what they mean in terms of the problem situation. The Problem-Based Task Coaching Sample Responses indicate that the students conclude that the second valuation is exponential because it does not exhibit a constant rate of change. Throughout the Practice problems, students calculate the average rate of change over a given interval for linear and exponential problem situations. Students do not prove that exponential functions grow by equal factors over equal intervals.
G-C.5: In Mathematics II, Unit 6, students do not derive using similarity the fact that the length of an arc intercepted by an angle is proportional to the radius. In Lessons 6.4.1 and 6.4.2, students respond to problems that connect radians, arc length, and arc measure. In Unit 6, Topic D, Conceptual Task, students investigate and explain the relationships between the area of a sector and arc length however do not explicitly derive the formula for the area of a sector.
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 2.6.3, students apply key takeaways from ratios and proportional relationships (7.RP.A) when they interpret key features of linear and exponential functions. Students consider rules, graphs, tables, and descriptions of real-world scenarios to choose between linear and exponential scenarios (F-IF.4, F-LE.5).
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, Practice Set A, students apply understandings of basic statistics and probability (6-8.SP) and operations with rational numbers (7.NS.1, 2) when students use box plots and a table of summary statistics to calculate mean values of sample sets (S-ID.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:
In Mathematics I, Unit 6, students calculate slope to explore properties of geometric shapes in the plane. In Lesson 6.1.1, students calculate slope using integers that are limited to the interval [-10, 15] throughout the print materials. The Geogebra applets allow students to enter numbers (e.g., non-integers) for the coordinates; however the applet performs the calculation and outputs the slope.
In Mathematics II, Unit 3, Topic C, students create quadratic equations and graphs using different forms of the quadratic equation. Throughout Practice Sets A and B of Lessons 3.3.1, 3.3.2, and 3.3.3, the materials rely predominantly on integers. On occasion, an ordered pair includes a rational coordinate, an equation includes a rational term, or a graph includes a rational defining characteristic. When students engage with the applets included in these lessons, students drag a slider to watch the step-by-step procedures for creating an equation or graphing a quadratic equation.
In Mathematics II, Unit 5, students solve problems using right triangles, trigonometry, and proofs. The print problems in this unit have integer and rational answers; they exclude irrational numbers in the problems and answers. Within the applets, students choose which numbers—rational and irrational—to use for calculations or drag a slider to view problem-solving demonstrations.
In Mathematics III, Unit 4 [Unit 3], Lesson 3.1.4, students determine specified trigonometric ratios for angles given in radian measure only. The print materials do not give angles in degrees or angles given in real numbers that would require technology. The applets demonstrate how to find the exact value of a trigonometric ratio given a point that lies on the terminal side of an angle in standard position or given another trigonometric function ratio and the quadrant in which the terminal side of the angle lies.
In Mathematics III, Unit 3, Lesson 2B.2.2 students solve radical equations. Throughout Practice Sets A and B, students’ calculations yield only one irrational solution; an abundance of the other solutions are integers. When students engage with the lesson’s applets, they drag a slider to watch demonstrations of the solution procedure.
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.