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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).
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.