High School - Gateway 3

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

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

• In Mathematics I, Unit 2, Lesson 3, Supporting Resource includes Recommended Resources to a few external sites. One of the sites is OEIS.org  “The On-Line Encyclopedia of Integer Sequences,” which is described as a “quirky site (that) will complete and define an entered sequence of integers. It will also provide any historical or quasi-relevant links to the defined sequence.” Lesson 2.3.1 addresses Sequences as Functions. The Supporting Resource in the lesson defines and describes a sequence in detail using abstract examples and referring to mathematical concepts such as, the set of natural numbers and a discrete function. Key Concepts include several bullets of sequence-relevant information so that teachers can improve their own knowledge; these bullets detail explicit and recursive formulas, common differences and ratios, and connections to arithmetic and geometric sequences. The materials provide examples of explicit and recursive formulas in addition to graphs of a sequence.

• In Mathematics II, Unit 6, Lesson 1, Supporting Resources includes Recommended Resources to some external sites:

  • From Math Open Reference, the “site describes the Central Angle Theorem and allows users to explore the relationship between inscribed angles and central angles.”

  • A site at Math Warehouse “reviews the properties of tangent lines and allows users to interactively explore the idea that a tangent line is perpendicular to a radius at the point of tangency. This site also provides limited practice problems, as well as solutions.”

• In Mathematics III, Unit 4 [Unit 3], Lesson 1 focuses on Radians and the Unit Circle. Recommended Resources include the following:

  • “Intuitive Guide to Angles, Degrees, and Radians” (Better Explained.com), which provides a creative explanation of the radian system of angle measure.

  • Khan Academy’s “Unit Circle Definition of Trig Functions,” which includes videos and examples of how to use the unit circle to find trigonometric functions.

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

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

• In Mathematics I, the Standards Correlations and Connections to Future Courses document indicates that the course aligns to standards from all strands of the CCSSM. The Suggested Pacing Guide details days of instruction per unit. All days that do not indicate an Assessment are aligned to a standard. Some examples are the following: Unit 1, Lesson 1: Interpreting Structure in Expressions, Day 3, aligns to standard A-SSE.1b, and Lesson 5: Rearranging Formulas, Day 23, aligns to standards A-CED.1 and A-CED.2.

• In Mathematics II, Unit 4, Lesson 4.1.1, the Supporting Resource (Teacher Instructional Support) for Describing Events, Prerequisite Skills, includes the alignment to middle school standard 7.SP.8b in the print version. The comparable document in the print version of Lesson 4.1.2 indicates alignment to elementary (4.NF.1),  middle school (7.SP.1, 7.SP.8c, 7.EE.3), and high school (S-CP.1) math standards. In Lesson 4.1.3, the print version of the Warm-Up Debrief indicates alignment to S-CP.2.

• In the print version of Mathematics III, Unit 2A, the Standard Correlation follows the Table of Contents; it provides a comprehensive list mapping the sub-lessons to the CCSSM. The Lesson 1, Instructional Resource indicates the CCSSM A-SSE.1 and A-APR.1. Lesson 2A.1.1 Warm-Up Debrief indicates alignment to A-SSE.1. The Teacher Instructional Support indicates the Prerequisite Skills align to 5.OA.2 and 6.EE.2a.  

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

• The Mathematics I, Suggested Pacing Guide, Unit 2: Linear and Exponential Relationships, Unit Overview states the following: “This unit builds on the concepts of functions that were first introduced in Grade 8. Students extend their understanding of functions to include exponential relationships. Students learn how to analyze and model relationships from contexts, graphs, tables, and equations using what they know about exponential and linear relationships.”

• The Mathematics II, Suggested Pacing Guide, Unit 3: Expressions and Equations, Unit Overview states the following: “Students reexamine the basic structures of expressions, but this time these structures are applied to quadratic expressions. Then, students learn to solve quadratic equations using various methods. Next, students create quadratic equations in various forms and learn how to rearrange formulas to solve for a variable that is being squared. The unit builds on previous units by introducing the Fundamental Theorem of Algebra and showing students how complex numbers are solutions to some quadratic equations. At this point, students are introduced to rational functions. Again, students learn to write exponentially structured expressions in equivalent forms. The unit ends by returning to a familiar topic—solving systems of equations—but now complex solutions can be determined.”

• The Mathematics III, Suggested Pacing Guide, Unit 4B: Mathematical Modeling and Choosing a Model, Unit Overview states the following: “Students revisit the process of creating equations in one variable and explore creating constraints and rearranging formulas. They then learn about transforming models and combining functions. Students review various kinds of functions, including linear, exponential, quadratic, piecewise, step, absolute value, square root, and cube root functions, all with an eye to choosing a model for a real-world situation. Finally, students consider geometric models, including two-dimensional cross sections of three-dimensional objects.”

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

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

Examples include, but are not limited to:

• In Mathematics I, Unit 7, End-of-Course Assessment, Problem 2, students are given a scenario involving cupcakes, hot dogs, and a fundraising goal and must identify which inequality represents the scenario. The End-of-Course Assessment Answer Key identifies the standard being assessed for Problem 2 as A-CED.1 but does not identify the practice.

• In Mathematics II, Unit 7, End-of-Course Assessment, Problem 9, students are tasked with identifying the inverse of a function given the domain. The End-of-Course Assessment Answer Key identifies the standard being assessed for Problem 9 as F-BF.4a but does not identify the practice.

• In Mathematics III, Unit 7, End-of-Course Assessment, Problem 41, “The ages of dogs regularly visiting a popular dog park are normally distributed. The mean age is 6, and the standard deviation is 2 years. There are 540 dogs that regularly visit the park. a. Sketch a normal curve for this situation. b. What percentage of dogs are between 2 and 12 years old? c. How many dogs is this?” The End-of-Course Assessment Answer Key identifies the standard being assessed for Problem 41 as S-ID.4 but does not identify the practice.

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

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

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

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

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

The following assessments are within each course:

• The Pre-Assessment, “can be used to gauge students’ prior knowledge and to inform instructional planning.”

• The Progress Assessment consists of “10 multiple-choice questions, as well as one extended-response question that incorporates critical thinking and writing components. This can be used to document the extent to which students grasp the concepts and skills addressed during instruction.”

• The Unit Assessment consists  of “12 multiple-choice questions and three extended-response questions that incorporate critical thinking and writing components. This can be used to document the extent to which students grasp the concepts and skills addressed during the unit.” 

• The Comprehensive Assessment (available in some units) covers material from two units.

• The End-of-Course Assessments are 45 questions each and are a combination of multiple-choice questions and extended-response questions. These assessments are used to document the extent to which students grasp the concepts and skills addressed during the course.

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

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

• The materials do not fully assess F-IF.7. The standard specifies that students will graph and analyze linear, quadratic, square root, absolute value, cube root, piecewise, step , polynomial, exponential, logarithmic, and trigonometric functions. The Mathematics I, Unit 7, End-of-Course Assessment, Problem 13 asks students to match an equation to a given graph of a linear function. Mathematics II, Unit 2, Lesson 2.4, Pre- and Progress Assessments ask students to demonstrate their knowledge of square root, absolute value, step, and piecewise functions. Mathematics II, Unit 7, End-of-Course Assessment, Problems 3 and 4 ask students to match a graph of a quadratic function and a square root function, respectively, to their function. Mathematics III, Unit 7, End-of-Course Assessment, Problem 10 asks students to identify the end behavior of a polynomial; Problems 32 and 33 ask students to identify period and the equation for a trigonometric graph, respectively.  Mathematics III, Unit 5 [Unit 4A], Lesson 5.2, Pre- and Progress Assessments ask students to engage with logarithmic functions. The materials do not assess student knowledge of the tangent or trigonometric cofunctions. Assessments do not ask students to graph logarithmic or trigonometric functions.

• In Mathematics II, Unit 7, the End-of-Course Assessment Answer Key states Problems 23-25 assess G-CO.10, proving theorems about triangles. Students do not prove theorems about triangles but use theorems pertaining to triangles to find missing angles in triangles (Problem 23), justify the last two steps of a triangle proof but do not use triangle theorems (Problem 24) and find perimeter using given midsegment information (Problem 25). In the Pre-, Progress, and Unit Assessments, students use theorems about triangles to solve problems but do not prove theorems about triangles.

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

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

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

• Mathematics II, Unit 4, Lesson 2, Progress Assessment, Problem 10 states, “What is the probability that 2 cards selected from a standard deck of 52 cards without replacement are both multiples of 5? a. 0.00075, b. 0.0045, c. 0.011, d. 0.021”

• Mathematics III, Unit 7, End-of-Course Assessment, Problem 14 states, “Which simplified expression is equivalent to \frac{8x^3-6x}{2x}? a. 4x^2-3, x\ne0, b. 4x^2-6x, x\ne 0 c. 61, x\ne 0 d. 4x-3, x\ne 0, \pm\frac{\sqrt{3}}{2}” (A-SSE.2).

Examples of extended-response questions include:

• Mathematics I, Unit 6, Lesson 1, Progress Assessment, Problem 11 states, “Read the scenario, write an equation to model the walkway, and then solve the problem. Mariah’s house is located at the point -1,2. The equation of the road is y+4x=2. Mariah is building a walkway from her house to the road. What is the approximate length of the shortest walkway she can build? Each unit represents 100 yards.”

• Mathematics II, Unit 7, End-of-Course Assessment, Problem 44 states, “A box contains three blue marbles and two white marbles. Rob pulls a marble from the box at random, then replaces it and selects another. Let B_1,B_2,B_3,W_1, and W_2 represent the different marbles in the box. A. List the complete sample space for this scenario. B. What is the probability that Rob selects a blue marble both times?” (S-CP.1).  

• Mathematics III, Unit 3 [Unit 2B], Unit Assessment, Problem 13 states, “It takes Alden 9 hours to bale all the hay in a field, but his brother, Anthony, can do the same job in 7 hours. If they bale all the hay in an entire field together, approximately how long will it take them to complete the job?”

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

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

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

• Mathematics I, Unit 3, Lesson 3.1.1, Problem-Based Task, students “use properties of equality and properties of operations to justify a solution method to an equation. They will form a generalization from simplifying a numerical expression, and they will construct a viable argument as to why the answer to the equation described in the task is always 3.” The Problem-Based Task, Implementation Guide, Extending the Task, provides two options, one of two suggest, “To extend the task, ask students to work with a partner to create a different magic number game with different steps of operations. Ask them to write down the steps, and also to work through their equation, identifying the properties of equality and operations that they use as they go. Once they have completed their new game, ask them to share the game with the class. Encourage and guide a discussion about each pair’s number game, and encourage students to use the proper terminology when explaining why their magic number game works.”

• Mathematics II, Unit 1, Lesson 1.1.1, Problem-Based Task, “Students will use a town’s current census and estimated 10-year growth rate to estimate the population before the next census is taken.” The Problem-Based Task, Implementation Guide, Extending the Task, provides the following two options: “To extend the task, ask students to find the estimated yearly growth rate and create a table of values showing the population for each year from the current year to 8 years from today. Ask students to demonstrate and verify that the yearly growth rate (1.03) holds for each year.” and/or “Another option is to have students graph the function and then analyze the graph.”

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

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

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

• Math II Honors Supplement, Topic 4, Learning/Performance Task, students seek to prove that two unique package designs have the same volume. Problem 6 asks that students “Draw the design of a third package in a 3-D plane that can be proved by Cavalieri’s Principle to have the same volume as the two packages in the diagram. Provide coordinates and show your calculations.”

• Math III Honors Supplement, Topic 1, Learning/Performance Task, students will be introduced to the process of animation by using matrices to move a figure along a graph. Suggestions for Extending the Activity provides two options: one of two suggests, “Students with knowledge of a 3-D animation program can create a simple animation using programming that involves matrices. Animation software is available to download from several sites, many of which are open-source or offer free trials. Some options include Blender, Synfig, and Toonboom.”