High School - Gateway 1
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Focus & Coherence
Gateway 1 - Partially Meets Expectations | 72% |
|---|---|
Criterion 1.1: Focus & Coherence | 13 / 18 |
Criterion 1.1: Focus & Coherence
Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).
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
The materials focus on the high school standards.*
Indicator 1a.i
The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
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 materials attend to the full intent of the modeling process when applied to the modeling standards.
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
The 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
The materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.
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 materials, when used as designed, allow students to fully learn each standard.
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 materials require students to engage in mathematics at a level of sophistication appropriate to high school.
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 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 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 materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
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.